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-- Operators used in the wrong way.
module NoParseForApplication where
postulate
X : Set
_! : X -> X
right : X -> X
right x = x !
wrong : X -> X
wrong x = ! x
|
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module Text.Greek.SBLGNT.Gal where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΠΡΟΣ-ΓΑΛΑΤΑΣ : List (Word)
ΠΡΟΣ-ΓΑΛΑΤΑΣ =
word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.1.1"
∷ word (ἀ ∷ π ∷ ό ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.1.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.1.1"
∷ word (ἀ ∷ π ∷ []) "Gal.1.1"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "Gal.1.1"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.1.1"
∷ word (δ ∷ ι ∷ []) "Gal.1.1"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Gal.1.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.1.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.1.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.1.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.1.1"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.1.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.1"
∷ word (ἐ ∷ γ ∷ ε ∷ ί ∷ ρ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Gal.1.1"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.1.1"
∷ word (ἐ ∷ κ ∷ []) "Gal.1.1"
∷ word (ν ∷ ε ∷ κ ∷ ρ ∷ ῶ ∷ ν ∷ []) "Gal.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.2"
∷ word (ο ∷ ἱ ∷ []) "Gal.1.2"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Gal.1.2"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.1.2"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.1.2"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.1.2"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Gal.1.2"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Gal.1.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.1.2"
∷ word (Γ ∷ α ∷ ∙λ ∷ α ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Gal.1.2"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Gal.1.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.3"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Gal.1.3"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.1.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.1.3"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.3"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (δ ∷ ό ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Gal.1.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.1.4"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Gal.1.4"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.1.4"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "Gal.1.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.1.4"
∷ word (ὅ ∷ π ∷ ω ∷ ς ∷ []) "Gal.1.4"
∷ word (ἐ ∷ ξ ∷ έ ∷ ∙λ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Gal.1.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.1.4"
∷ word (ἐ ∷ κ ∷ []) "Gal.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ ο ∷ ς ∷ []) "Gal.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (ἐ ∷ ν ∷ ε ∷ σ ∷ τ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "Gal.1.4"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.1.4"
∷ word (τ ∷ ὸ ∷ []) "Gal.1.4"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "Gal.1.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.1.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.4"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.1.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.1.4"
∷ word (ᾧ ∷ []) "Gal.1.5"
∷ word (ἡ ∷ []) "Gal.1.5"
∷ word (δ ∷ ό ∷ ξ ∷ α ∷ []) "Gal.1.5"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.5"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.1.5"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ ς ∷ []) "Gal.1.5"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.1.5"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ω ∷ ν ∷ []) "Gal.1.5"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Gal.1.5"
∷ word (Θ ∷ α ∷ υ ∷ μ ∷ ά ∷ ζ ∷ ω ∷ []) "Gal.1.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.1.6"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Gal.1.6"
∷ word (τ ∷ α ∷ χ ∷ έ ∷ ω ∷ ς ∷ []) "Gal.1.6"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ τ ∷ ί ∷ θ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Gal.1.6"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.1.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.6"
∷ word (κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Gal.1.6"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.1.6"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.6"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ι ∷ []) "Gal.1.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.6"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Gal.1.6"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.1.6"
∷ word (ὃ ∷ []) "Gal.1.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.1.7"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.1.7"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ []) "Gal.1.7"
∷ word (ε ∷ ἰ ∷ []) "Gal.1.7"
∷ word (μ ∷ ή ∷ []) "Gal.1.7"
∷ word (τ ∷ ι ∷ ν ∷ έ ∷ ς ∷ []) "Gal.1.7"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.1.7"
∷ word (ο ∷ ἱ ∷ []) "Gal.1.7"
∷ word (τ ∷ α ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.1.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.1.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.7"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.1.7"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ σ ∷ τ ∷ ρ ∷ έ ∷ ψ ∷ α ∷ ι ∷ []) "Gal.1.7"
∷ word (τ ∷ ὸ ∷ []) "Gal.1.7"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.7"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.1.8"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.1.8"
∷ word (ἢ ∷ []) "Gal.1.8"
∷ word (ἄ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.1.8"
∷ word (ἐ ∷ ξ ∷ []) "Gal.1.8"
∷ word (ο ∷ ὐ ∷ ρ ∷ α ∷ ν ∷ ο ∷ ῦ ∷ []) "Gal.1.8"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Gal.1.8"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.1.8"
∷ word (π ∷ α ∷ ρ ∷ []) "Gal.1.8"
∷ word (ὃ ∷ []) "Gal.1.8"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.1.8"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.1.8"
∷ word (ἀ ∷ ν ∷ ά ∷ θ ∷ ε ∷ μ ∷ α ∷ []) "Gal.1.8"
∷ word (ἔ ∷ σ ∷ τ ∷ ω ∷ []) "Gal.1.8"
∷ word (ὡ ∷ ς ∷ []) "Gal.1.9"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ι ∷ ρ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Gal.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.9"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "Gal.1.9"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.1.9"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.1.9"
∷ word (ε ∷ ἴ ∷ []) "Gal.1.9"
∷ word (τ ∷ ι ∷ ς ∷ []) "Gal.1.9"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.1.9"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.1.9"
∷ word (π ∷ α ∷ ρ ∷ []) "Gal.1.9"
∷ word (ὃ ∷ []) "Gal.1.9"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Gal.1.9"
∷ word (ἀ ∷ ν ∷ ά ∷ θ ∷ ε ∷ μ ∷ α ∷ []) "Gal.1.9"
∷ word (ἔ ∷ σ ∷ τ ∷ ω ∷ []) "Gal.1.9"
∷ word (Ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "Gal.1.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.1.10"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Gal.1.10"
∷ word (π ∷ ε ∷ ί ∷ θ ∷ ω ∷ []) "Gal.1.10"
∷ word (ἢ ∷ []) "Gal.1.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.1.10"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Gal.1.10"
∷ word (ἢ ∷ []) "Gal.1.10"
∷ word (ζ ∷ η ∷ τ ∷ ῶ ∷ []) "Gal.1.10"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Gal.1.10"
∷ word (ἀ ∷ ρ ∷ έ ∷ σ ∷ κ ∷ ε ∷ ι ∷ ν ∷ []) "Gal.1.10"
∷ word (ε ∷ ἰ ∷ []) "Gal.1.10"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Gal.1.10"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ ι ∷ ς ∷ []) "Gal.1.10"
∷ word (ἤ ∷ ρ ∷ ε ∷ σ ∷ κ ∷ ο ∷ ν ∷ []) "Gal.1.10"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.10"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.1.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.1.10"
∷ word (ἂ ∷ ν ∷ []) "Gal.1.10"
∷ word (ἤ ∷ μ ∷ η ∷ ν ∷ []) "Gal.1.10"
∷ word (Γ ∷ ν ∷ ω ∷ ρ ∷ ί ∷ ζ ∷ ω ∷ []) "Gal.1.11"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.1.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.1.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.1.11"
∷ word (τ ∷ ὸ ∷ []) "Gal.1.11"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.1.11"
∷ word (τ ∷ ὸ ∷ []) "Gal.1.11"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ θ ∷ ὲ ∷ ν ∷ []) "Gal.1.11"
∷ word (ὑ ∷ π ∷ []) "Gal.1.11"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Gal.1.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.1.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.1.11"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.1.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.1.11"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Gal.1.11"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.1.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.1.12"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.1.12"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Gal.1.12"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Gal.1.12"
∷ word (π ∷ α ∷ ρ ∷ έ ∷ ∙λ ∷ α ∷ β ∷ ο ∷ ν ∷ []) "Gal.1.12"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ []) "Gal.1.12"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Gal.1.12"
∷ word (ἐ ∷ δ ∷ ι ∷ δ ∷ ά ∷ χ ∷ θ ∷ η ∷ ν ∷ []) "Gal.1.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.1.12"
∷ word (δ ∷ ι ∷ []) "Gal.1.12"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.1.12"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.1.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.12"
∷ word (Ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.1.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.1.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.1.13"
∷ word (ἐ ∷ μ ∷ ὴ ∷ ν ∷ []) "Gal.1.13"
∷ word (ἀ ∷ ν ∷ α ∷ σ ∷ τ ∷ ρ ∷ ο ∷ φ ∷ ή ∷ ν ∷ []) "Gal.1.13"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Gal.1.13"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.13"
∷ word (τ ∷ ῷ ∷ []) "Gal.1.13"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ϊ ∷ σ ∷ μ ∷ ῷ ∷ []) "Gal.1.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.1.13"
∷ word (κ ∷ α ∷ θ ∷ []) "Gal.1.13"
∷ word (ὑ ∷ π ∷ ε ∷ ρ ∷ β ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Gal.1.13"
∷ word (ἐ ∷ δ ∷ ί ∷ ω ∷ κ ∷ ο ∷ ν ∷ []) "Gal.1.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.1.13"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Gal.1.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.13"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.13"
∷ word (ἐ ∷ π ∷ ό ∷ ρ ∷ θ ∷ ο ∷ υ ∷ ν ∷ []) "Gal.1.13"
∷ word (α ∷ ὐ ∷ τ ∷ ή ∷ ν ∷ []) "Gal.1.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.14"
∷ word (π ∷ ρ ∷ ο ∷ έ ∷ κ ∷ ο ∷ π ∷ τ ∷ ο ∷ ν ∷ []) "Gal.1.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.14"
∷ word (τ ∷ ῷ ∷ []) "Gal.1.14"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ϊ ∷ σ ∷ μ ∷ ῷ ∷ []) "Gal.1.14"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Gal.1.14"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.1.14"
∷ word (σ ∷ υ ∷ ν ∷ η ∷ ∙λ ∷ ι ∷ κ ∷ ι ∷ ώ ∷ τ ∷ α ∷ ς ∷ []) "Gal.1.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.14"
∷ word (τ ∷ ῷ ∷ []) "Gal.1.14"
∷ word (γ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "Gal.1.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.1.14"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ σ ∷ σ ∷ ο ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ς ∷ []) "Gal.1.14"
∷ word (ζ ∷ η ∷ ∙λ ∷ ω ∷ τ ∷ ὴ ∷ ς ∷ []) "Gal.1.14"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "Gal.1.14"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.1.14"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ι ∷ κ ∷ ῶ ∷ ν ∷ []) "Gal.1.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.1.14"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ό ∷ σ ∷ ε ∷ ω ∷ ν ∷ []) "Gal.1.14"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Gal.1.15"
∷ word (δ ∷ ὲ ∷ []) "Gal.1.15"
∷ word (ε ∷ ὐ ∷ δ ∷ ό ∷ κ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Gal.1.15"
∷ word (ὁ ∷ []) "Gal.1.15"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ ί ∷ σ ∷ α ∷ ς ∷ []) "Gal.1.15"
∷ word (μ ∷ ε ∷ []) "Gal.1.15"
∷ word (ἐ ∷ κ ∷ []) "Gal.1.15"
∷ word (κ ∷ ο ∷ ι ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Gal.1.15"
∷ word (μ ∷ η ∷ τ ∷ ρ ∷ ό ∷ ς ∷ []) "Gal.1.15"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.1.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.15"
∷ word (κ ∷ α ∷ ∙λ ∷ έ ∷ σ ∷ α ∷ ς ∷ []) "Gal.1.15"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.1.15"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.1.15"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Gal.1.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.15"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ύ ∷ ψ ∷ α ∷ ι ∷ []) "Gal.1.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.1.16"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "Gal.1.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.1.16"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.16"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.1.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.1.16"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Gal.1.16"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.1.16"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.1.16"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Gal.1.16"
∷ word (ε ∷ ὐ ∷ θ ∷ έ ∷ ω ∷ ς ∷ []) "Gal.1.16"
∷ word (ο ∷ ὐ ∷ []) "Gal.1.16"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ ν ∷ ε ∷ θ ∷ έ ∷ μ ∷ η ∷ ν ∷ []) "Gal.1.16"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Gal.1.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.16"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.1.16"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.1.17"
∷ word (ἀ ∷ ν ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.1.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.17"
∷ word (Ἱ ∷ ε ∷ ρ ∷ ο ∷ σ ∷ ό ∷ ∙λ ∷ υ ∷ μ ∷ α ∷ []) "Gal.1.17"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.1.17"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.1.17"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "Gal.1.17"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Gal.1.17"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.1.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.1.17"
∷ word (ἀ ∷ π ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.1.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.17"
∷ word (Ἀ ∷ ρ ∷ α ∷ β ∷ ί ∷ α ∷ ν ∷ []) "Gal.1.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.17"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.1.17"
∷ word (ὑ ∷ π ∷ έ ∷ σ ∷ τ ∷ ρ ∷ ε ∷ ψ ∷ α ∷ []) "Gal.1.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.17"
∷ word (Δ ∷ α ∷ μ ∷ α ∷ σ ∷ κ ∷ ό ∷ ν ∷ []) "Gal.1.17"
∷ word (Ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "Gal.1.18"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.1.18"
∷ word (ἔ ∷ τ ∷ η ∷ []) "Gal.1.18"
∷ word (τ ∷ ρ ∷ ί ∷ α ∷ []) "Gal.1.18"
∷ word (ἀ ∷ ν ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.1.18"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.18"
∷ word (Ἱ ∷ ε ∷ ρ ∷ ο ∷ σ ∷ ό ∷ ∙λ ∷ υ ∷ μ ∷ α ∷ []) "Gal.1.18"
∷ word (ἱ ∷ σ ∷ τ ∷ ο ∷ ρ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.1.18"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ ν ∷ []) "Gal.1.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.18"
∷ word (ἐ ∷ π ∷ έ ∷ μ ∷ ε ∷ ι ∷ ν ∷ α ∷ []) "Gal.1.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.1.18"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.1.18"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.1.18"
∷ word (δ ∷ ε ∷ κ ∷ α ∷ π ∷ έ ∷ ν ∷ τ ∷ ε ∷ []) "Gal.1.18"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Gal.1.19"
∷ word (δ ∷ ὲ ∷ []) "Gal.1.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.1.19"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ό ∷ ∙λ ∷ ω ∷ ν ∷ []) "Gal.1.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.1.19"
∷ word (ε ∷ ἶ ∷ δ ∷ ο ∷ ν ∷ []) "Gal.1.19"
∷ word (ε ∷ ἰ ∷ []) "Gal.1.19"
∷ word (μ ∷ ὴ ∷ []) "Gal.1.19"
∷ word (Ἰ ∷ ά ∷ κ ∷ ω ∷ β ∷ ο ∷ ν ∷ []) "Gal.1.19"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.1.19"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "Gal.1.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.19"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.1.19"
∷ word (ἃ ∷ []) "Gal.1.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.1.20"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "Gal.1.20"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.1.20"
∷ word (ἰ ∷ δ ∷ ο ∷ ὺ ∷ []) "Gal.1.20"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "Gal.1.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.1.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.1.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.1.20"
∷ word (ο ∷ ὐ ∷ []) "Gal.1.20"
∷ word (ψ ∷ ε ∷ ύ ∷ δ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Gal.1.20"
∷ word (ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "Gal.1.21"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.1.21"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.1.21"
∷ word (τ ∷ ὰ ∷ []) "Gal.1.21"
∷ word (κ ∷ ∙λ ∷ ί ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Gal.1.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.1.21"
∷ word (Σ ∷ υ ∷ ρ ∷ ί ∷ α ∷ ς ∷ []) "Gal.1.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.1.21"
∷ word (Κ ∷ ι ∷ ∙λ ∷ ι ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "Gal.1.21"
∷ word (ἤ ∷ μ ∷ η ∷ ν ∷ []) "Gal.1.22"
∷ word (δ ∷ ὲ ∷ []) "Gal.1.22"
∷ word (ἀ ∷ γ ∷ ν ∷ ο ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.1.22"
∷ word (τ ∷ ῷ ∷ []) "Gal.1.22"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ώ ∷ π ∷ ῳ ∷ []) "Gal.1.22"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Gal.1.22"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Gal.1.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.1.22"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ί ∷ α ∷ ς ∷ []) "Gal.1.22"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Gal.1.22"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.22"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.1.22"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.1.23"
∷ word (δ ∷ ὲ ∷ []) "Gal.1.23"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.1.23"
∷ word (ἦ ∷ σ ∷ α ∷ ν ∷ []) "Gal.1.23"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.1.23"
∷ word (Ὁ ∷ []) "Gal.1.23"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ω ∷ ν ∷ []) "Gal.1.23"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.1.23"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Gal.1.23"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.1.23"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.1.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.1.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.1.23"
∷ word (ἥ ∷ ν ∷ []) "Gal.1.23"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Gal.1.23"
∷ word (ἐ ∷ π ∷ ό ∷ ρ ∷ θ ∷ ε ∷ ι ∷ []) "Gal.1.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.1.24"
∷ word (ἐ ∷ δ ∷ ό ∷ ξ ∷ α ∷ ζ ∷ ο ∷ ν ∷ []) "Gal.1.24"
∷ word (ἐ ∷ ν ∷ []) "Gal.1.24"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.1.24"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.1.24"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Gal.1.24"
∷ word (Ἔ ∷ π ∷ ε ∷ ι ∷ τ ∷ α ∷ []) "Gal.2.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.2.1"
∷ word (δ ∷ ε ∷ κ ∷ α ∷ τ ∷ ε ∷ σ ∷ σ ∷ ά ∷ ρ ∷ ω ∷ ν ∷ []) "Gal.2.1"
∷ word (ἐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Gal.2.1"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.2.1"
∷ word (ἀ ∷ ν ∷ έ ∷ β ∷ η ∷ ν ∷ []) "Gal.2.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.1"
∷ word (Ἱ ∷ ε ∷ ρ ∷ ο ∷ σ ∷ ό ∷ ∙λ ∷ υ ∷ μ ∷ α ∷ []) "Gal.2.1"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.2.1"
∷ word (Β ∷ α ∷ ρ ∷ ν ∷ α ∷ β ∷ ᾶ ∷ []) "Gal.2.1"
∷ word (σ ∷ υ ∷ μ ∷ π ∷ α ∷ ρ ∷ α ∷ ∙λ ∷ α ∷ β ∷ ὼ ∷ ν ∷ []) "Gal.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.1"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ν ∷ []) "Gal.2.1"
∷ word (ἀ ∷ ν ∷ έ ∷ β ∷ η ∷ ν ∷ []) "Gal.2.2"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.2"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.2.2"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ά ∷ ∙λ ∷ υ ∷ ψ ∷ ι ∷ ν ∷ []) "Gal.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.2"
∷ word (ἀ ∷ ν ∷ ε ∷ θ ∷ έ ∷ μ ∷ η ∷ ν ∷ []) "Gal.2.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.2.2"
∷ word (τ ∷ ὸ ∷ []) "Gal.2.2"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.2.2"
∷ word (ὃ ∷ []) "Gal.2.2"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ω ∷ []) "Gal.2.2"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.2.2"
∷ word (ἔ ∷ θ ∷ ν ∷ ε ∷ σ ∷ ι ∷ ν ∷ []) "Gal.2.2"
∷ word (κ ∷ α ∷ τ ∷ []) "Gal.2.2"
∷ word (ἰ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "Gal.2.2"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.2"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.2.2"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.2.2"
∷ word (μ ∷ ή ∷ []) "Gal.2.2"
∷ word (π ∷ ω ∷ ς ∷ []) "Gal.2.2"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.2"
∷ word (κ ∷ ε ∷ ν ∷ ὸ ∷ ν ∷ []) "Gal.2.2"
∷ word (τ ∷ ρ ∷ έ ∷ χ ∷ ω ∷ []) "Gal.2.2"
∷ word (ἢ ∷ []) "Gal.2.2"
∷ word (ἔ ∷ δ ∷ ρ ∷ α ∷ μ ∷ ο ∷ ν ∷ []) "Gal.2.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.2.3"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.2.3"
∷ word (Τ ∷ ί ∷ τ ∷ ο ∷ ς ∷ []) "Gal.2.3"
∷ word (ὁ ∷ []) "Gal.2.3"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Gal.2.3"
∷ word (ἐ ∷ μ ∷ ο ∷ ί ∷ []) "Gal.2.3"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ []) "Gal.2.3"
∷ word (ὤ ∷ ν ∷ []) "Gal.2.3"
∷ word (ἠ ∷ ν ∷ α ∷ γ ∷ κ ∷ ά ∷ σ ∷ θ ∷ η ∷ []) "Gal.2.3"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ μ ∷ η ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Gal.2.3"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.2.4"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.4"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.2.4"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ι ∷ σ ∷ ά ∷ κ ∷ τ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.2.4"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ α ∷ δ ∷ έ ∷ ∙λ ∷ φ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.2.4"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Gal.2.4"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ι ∷ σ ∷ ῆ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.2.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ σ ∷ κ ∷ ο ∷ π ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.2.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.4"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Gal.2.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.2.4"
∷ word (ἣ ∷ ν ∷ []) "Gal.2.4"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.4"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.2.4"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.2.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.4"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.2.4"
∷ word (κ ∷ α ∷ τ ∷ α ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ώ ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.2.4"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Gal.2.5"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.2.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.2.5"
∷ word (ὥ ∷ ρ ∷ α ∷ ν ∷ []) "Gal.2.5"
∷ word (ε ∷ ἴ ∷ ξ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.5"
∷ word (τ ∷ ῇ ∷ []) "Gal.2.5"
∷ word (ὑ ∷ π ∷ ο ∷ τ ∷ α ∷ γ ∷ ῇ ∷ []) "Gal.2.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.5"
∷ word (ἡ ∷ []) "Gal.2.5"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "Gal.2.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.5"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.2.5"
∷ word (δ ∷ ι ∷ α ∷ μ ∷ ε ∷ ί ∷ ν ∷ ῃ ∷ []) "Gal.2.5"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.2.5"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.2.5"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.2.6"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.6"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.2.6"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ύ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Gal.2.6"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ί ∷ []) "Gal.2.6"
∷ word (τ ∷ ι ∷ []) "Gal.2.6"
∷ word (ὁ ∷ π ∷ ο ∷ ῖ ∷ ο ∷ ί ∷ []) "Gal.2.6"
∷ word (π ∷ ο ∷ τ ∷ ε ∷ []) "Gal.2.6"
∷ word (ἦ ∷ σ ∷ α ∷ ν ∷ []) "Gal.2.6"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ []) "Gal.2.6"
∷ word (μ ∷ ο ∷ ι ∷ []) "Gal.2.6"
∷ word (δ ∷ ι ∷ α ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "Gal.2.6"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Gal.2.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.2.6"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Gal.2.6"
∷ word (ο ∷ ὐ ∷ []) "Gal.2.6"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "Gal.2.6"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.2.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.6"
∷ word (ο ∷ ἱ ∷ []) "Gal.2.6"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.6"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Gal.2.6"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ α ∷ ν ∷ έ ∷ θ ∷ ε ∷ ν ∷ τ ∷ ο ∷ []) "Gal.2.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.2.7"
∷ word (τ ∷ ο ∷ ὐ ∷ ν ∷ α ∷ ν ∷ τ ∷ ί ∷ ο ∷ ν ∷ []) "Gal.2.7"
∷ word (ἰ ∷ δ ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.2.7"
∷ word (π ∷ ε ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ μ ∷ α ∷ ι ∷ []) "Gal.2.7"
∷ word (τ ∷ ὸ ∷ []) "Gal.2.7"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.2.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.2.7"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Gal.2.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Gal.2.7"
∷ word (Π ∷ έ ∷ τ ∷ ρ ∷ ο ∷ ς ∷ []) "Gal.2.7"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.2.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Gal.2.7"
∷ word (ὁ ∷ []) "Gal.2.8"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.8"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Gal.2.8"
∷ word (Π ∷ έ ∷ τ ∷ ρ ∷ ῳ ∷ []) "Gal.2.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.8"
∷ word (ἀ ∷ π ∷ ο ∷ σ ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "Gal.2.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.2.8"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Gal.2.8"
∷ word (ἐ ∷ ν ∷ ή ∷ ρ ∷ γ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "Gal.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.8"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.2.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.8"
∷ word (τ ∷ ὰ ∷ []) "Gal.2.8"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.9"
∷ word (γ ∷ ν ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.9"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Gal.2.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.9"
∷ word (δ ∷ ο ∷ θ ∷ ε ∷ ῖ ∷ σ ∷ ά ∷ ν ∷ []) "Gal.2.9"
∷ word (μ ∷ ο ∷ ι ∷ []) "Gal.2.9"
∷ word (Ἰ ∷ ά ∷ κ ∷ ω ∷ β ∷ ο ∷ ς ∷ []) "Gal.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.9"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ ς ∷ []) "Gal.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.9"
∷ word (Ἰ ∷ ω ∷ ά ∷ ν ∷ ν ∷ η ∷ ς ∷ []) "Gal.2.9"
∷ word (ο ∷ ἱ ∷ []) "Gal.2.9"
∷ word (δ ∷ ο ∷ κ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.9"
∷ word (σ ∷ τ ∷ ῦ ∷ ∙λ ∷ ο ∷ ι ∷ []) "Gal.2.9"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Gal.2.9"
∷ word (δ ∷ ε ∷ ξ ∷ ι ∷ ὰ ∷ ς ∷ []) "Gal.2.9"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ α ∷ ν ∷ []) "Gal.2.9"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.9"
∷ word (Β ∷ α ∷ ρ ∷ ν ∷ α ∷ β ∷ ᾷ ∷ []) "Gal.2.9"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ς ∷ []) "Gal.2.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.9"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.2.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.9"
∷ word (τ ∷ ὰ ∷ []) "Gal.2.9"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.2.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Gal.2.9"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.9"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ή ∷ ν ∷ []) "Gal.2.9"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.2.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.2.10"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ῶ ∷ ν ∷ []) "Gal.2.10"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.10"
∷ word (μ ∷ ν ∷ η ∷ μ ∷ ο ∷ ν ∷ ε ∷ ύ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.10"
∷ word (ὃ ∷ []) "Gal.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.10"
∷ word (ἐ ∷ σ ∷ π ∷ ο ∷ ύ ∷ δ ∷ α ∷ σ ∷ α ∷ []) "Gal.2.10"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ []) "Gal.2.10"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Gal.2.10"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.2.10"
∷ word (Ὅ ∷ τ ∷ ε ∷ []) "Gal.2.11"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.11"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Gal.2.11"
∷ word (Κ ∷ η ∷ φ ∷ ᾶ ∷ ς ∷ []) "Gal.2.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.11"
∷ word (Ἀ ∷ ν ∷ τ ∷ ι ∷ ό ∷ χ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Gal.2.11"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.2.11"
∷ word (π ∷ ρ ∷ ό ∷ σ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Gal.2.11"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Gal.2.11"
∷ word (ἀ ∷ ν ∷ τ ∷ έ ∷ σ ∷ τ ∷ η ∷ ν ∷ []) "Gal.2.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.2.11"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ γ ∷ ν ∷ ω ∷ σ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Gal.2.11"
∷ word (ἦ ∷ ν ∷ []) "Gal.2.11"
∷ word (π ∷ ρ ∷ ὸ ∷ []) "Gal.2.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.12"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.12"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Gal.2.12"
∷ word (τ ∷ ι ∷ ν ∷ α ∷ ς ∷ []) "Gal.2.12"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.2.12"
∷ word (Ἰ ∷ α ∷ κ ∷ ώ ∷ β ∷ ο ∷ υ ∷ []) "Gal.2.12"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.2.12"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.2.12"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Gal.2.12"
∷ word (σ ∷ υ ∷ ν ∷ ή ∷ σ ∷ θ ∷ ι ∷ ε ∷ ν ∷ []) "Gal.2.12"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Gal.2.12"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.12"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ο ∷ ν ∷ []) "Gal.2.12"
∷ word (ὑ ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ∙λ ∷ ∙λ ∷ ε ∷ ν ∷ []) "Gal.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.12"
∷ word (ἀ ∷ φ ∷ ώ ∷ ρ ∷ ι ∷ ζ ∷ ε ∷ ν ∷ []) "Gal.2.12"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Gal.2.12"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.2.12"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.2.12"
∷ word (ἐ ∷ κ ∷ []) "Gal.2.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ῆ ∷ ς ∷ []) "Gal.2.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.13"
∷ word (σ ∷ υ ∷ ν ∷ υ ∷ π ∷ ε ∷ κ ∷ ρ ∷ ί ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Gal.2.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Gal.2.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.13"
∷ word (ο ∷ ἱ ∷ []) "Gal.2.13"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ὶ ∷ []) "Gal.2.13"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "Gal.2.13"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.2.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.13"
∷ word (Β ∷ α ∷ ρ ∷ ν ∷ α ∷ β ∷ ᾶ ∷ ς ∷ []) "Gal.2.13"
∷ word (σ ∷ υ ∷ ν ∷ α ∷ π ∷ ή ∷ χ ∷ θ ∷ η ∷ []) "Gal.2.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "Gal.2.13"
∷ word (τ ∷ ῇ ∷ []) "Gal.2.13"
∷ word (ὑ ∷ π ∷ ο ∷ κ ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Gal.2.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.2.14"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Gal.2.14"
∷ word (ε ∷ ἶ ∷ δ ∷ ο ∷ ν ∷ []) "Gal.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.2.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.2.14"
∷ word (ὀ ∷ ρ ∷ θ ∷ ο ∷ π ∷ ο ∷ δ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.2.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.2.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.14"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Gal.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.14"
∷ word (ε ∷ ὐ ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.2.14"
∷ word (ε ∷ ἶ ∷ π ∷ ο ∷ ν ∷ []) "Gal.2.14"
∷ word (τ ∷ ῷ ∷ []) "Gal.2.14"
∷ word (Κ ∷ η ∷ φ ∷ ᾷ ∷ []) "Gal.2.14"
∷ word (ἔ ∷ μ ∷ π ∷ ρ ∷ ο ∷ σ ∷ θ ∷ ε ∷ ν ∷ []) "Gal.2.14"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Gal.2.14"
∷ word (Ε ∷ ἰ ∷ []) "Gal.2.14"
∷ word (σ ∷ ὺ ∷ []) "Gal.2.14"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "Gal.2.14"
∷ word (ὑ ∷ π ∷ ά ∷ ρ ∷ χ ∷ ω ∷ ν ∷ []) "Gal.2.14"
∷ word (ἐ ∷ θ ∷ ν ∷ ι ∷ κ ∷ ῶ ∷ ς ∷ []) "Gal.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.2.14"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ϊ ∷ κ ∷ ῶ ∷ ς ∷ []) "Gal.2.14"
∷ word (ζ ∷ ῇ ∷ ς ∷ []) "Gal.2.14"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Gal.2.14"
∷ word (τ ∷ ὰ ∷ []) "Gal.2.14"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.2.14"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ κ ∷ ά ∷ ζ ∷ ε ∷ ι ∷ ς ∷ []) "Gal.2.14"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ΐ ∷ ζ ∷ ε ∷ ι ∷ ν ∷ []) "Gal.2.14"
∷ word (Ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.2.15"
∷ word (φ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Gal.2.15"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ι ∷ []) "Gal.2.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.2.15"
∷ word (ἐ ∷ ξ ∷ []) "Gal.2.15"
∷ word (ἐ ∷ θ ∷ ν ∷ ῶ ∷ ν ∷ []) "Gal.2.15"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ο ∷ ί ∷ []) "Gal.2.15"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.16"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.2.16"
∷ word (ο ∷ ὐ ∷ []) "Gal.2.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "Gal.2.16"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Gal.2.16"
∷ word (ἐ ∷ ξ ∷ []) "Gal.2.16"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.2.16"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.2.16"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.2.16"
∷ word (μ ∷ ὴ ∷ []) "Gal.2.16"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.2.16"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.2.16"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.2.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.2.16"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.2.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.2.16"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Gal.2.16"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.16"
∷ word (ἐ ∷ κ ∷ []) "Gal.2.16"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.2.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.2.16"
∷ word (ἐ ∷ ξ ∷ []) "Gal.2.16"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.2.16"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.2.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.2.16"
∷ word (ἐ ∷ ξ ∷ []) "Gal.2.16"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.2.16"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.2.16"
∷ word (ο ∷ ὐ ∷ []) "Gal.2.16"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.2.16"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "Gal.2.16"
∷ word (σ ∷ ά ∷ ρ ∷ ξ ∷ []) "Gal.2.16"
∷ word (ε ∷ ἰ ∷ []) "Gal.2.17"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.17"
∷ word (ζ ∷ η ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.2.17"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Gal.2.17"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.2.17"
∷ word (ε ∷ ὑ ∷ ρ ∷ έ ∷ θ ∷ η ∷ μ ∷ ε ∷ ν ∷ []) "Gal.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Gal.2.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ω ∷ ∙λ ∷ ο ∷ ί ∷ []) "Gal.2.17"
∷ word (ἆ ∷ ρ ∷ α ∷ []) "Gal.2.17"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.2.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "Gal.2.17"
∷ word (δ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ ο ∷ ς ∷ []) "Gal.2.17"
∷ word (μ ∷ ὴ ∷ []) "Gal.2.17"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Gal.2.17"
∷ word (ε ∷ ἰ ∷ []) "Gal.2.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.18"
∷ word (ἃ ∷ []) "Gal.2.18"
∷ word (κ ∷ α ∷ τ ∷ έ ∷ ∙λ ∷ υ ∷ σ ∷ α ∷ []) "Gal.2.18"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Gal.2.18"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.2.18"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ δ ∷ ο ∷ μ ∷ ῶ ∷ []) "Gal.2.18"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ τ ∷ η ∷ ν ∷ []) "Gal.2.18"
∷ word (ἐ ∷ μ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.2.18"
∷ word (σ ∷ υ ∷ ν ∷ ι ∷ σ ∷ τ ∷ ά ∷ ν ∷ ω ∷ []) "Gal.2.18"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.2.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.19"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.2.19"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.2.19"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Gal.2.19"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ο ∷ ν ∷ []) "Gal.2.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.2.19"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Gal.2.19"
∷ word (ζ ∷ ή ∷ σ ∷ ω ∷ []) "Gal.2.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.2.19"
∷ word (σ ∷ υ ∷ ν ∷ ε ∷ σ ∷ τ ∷ α ∷ ύ ∷ ρ ∷ ω ∷ μ ∷ α ∷ ι ∷ []) "Gal.2.19"
∷ word (ζ ∷ ῶ ∷ []) "Gal.2.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.20"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Gal.2.20"
∷ word (ἐ ∷ γ ∷ ώ ∷ []) "Gal.2.20"
∷ word (ζ ∷ ῇ ∷ []) "Gal.2.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.20"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.20"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.2.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Gal.2.20"
∷ word (ὃ ∷ []) "Gal.2.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.2.20"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.2.20"
∷ word (ζ ∷ ῶ ∷ []) "Gal.2.20"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.20"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Gal.2.20"
∷ word (ἐ ∷ ν ∷ []) "Gal.2.20"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ι ∷ []) "Gal.2.20"
∷ word (ζ ∷ ῶ ∷ []) "Gal.2.20"
∷ word (τ ∷ ῇ ∷ []) "Gal.2.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ ό ∷ ς ∷ []) "Gal.2.20"
∷ word (μ ∷ ε ∷ []) "Gal.2.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.2.20"
∷ word (π ∷ α ∷ ρ ∷ α ∷ δ ∷ ό ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Gal.2.20"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.2.20"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Gal.2.20"
∷ word (ἐ ∷ μ ∷ ο ∷ ῦ ∷ []) "Gal.2.20"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.2.21"
∷ word (ἀ ∷ θ ∷ ε ∷ τ ∷ ῶ ∷ []) "Gal.2.21"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.2.21"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Gal.2.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.2.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.2.21"
∷ word (ε ∷ ἰ ∷ []) "Gal.2.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.2.21"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.2.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.2.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Gal.2.21"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Gal.2.21"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.2.21"
∷ word (δ ∷ ω ∷ ρ ∷ ε ∷ ὰ ∷ ν ∷ []) "Gal.2.21"
∷ word (ἀ ∷ π ∷ έ ∷ θ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Gal.2.21"
∷ word (Ὦ ∷ []) "Gal.3.1"
∷ word (ἀ ∷ ν ∷ ό ∷ η ∷ τ ∷ ο ∷ ι ∷ []) "Gal.3.1"
∷ word (Γ ∷ α ∷ ∙λ ∷ ά ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.1"
∷ word (τ ∷ ί ∷ ς ∷ []) "Gal.3.1"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.3.1"
∷ word (ἐ ∷ β ∷ ά ∷ σ ∷ κ ∷ α ∷ ν ∷ ε ∷ ν ∷ []) "Gal.3.1"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Gal.3.1"
∷ word (κ ∷ α ∷ τ ∷ []) "Gal.3.1"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.3.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "Gal.3.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.3.1"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ γ ∷ ρ ∷ ά ∷ φ ∷ η ∷ []) "Gal.3.1"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "Gal.3.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Gal.3.2"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.3.2"
∷ word (θ ∷ έ ∷ ∙λ ∷ ω ∷ []) "Gal.3.2"
∷ word (μ ∷ α ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Gal.3.2"
∷ word (ἀ ∷ φ ∷ []) "Gal.3.2"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.3.2"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.2"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.3.2"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.2"
∷ word (τ ∷ ὸ ∷ []) "Gal.3.2"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.3.2"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "Gal.3.2"
∷ word (ἢ ∷ []) "Gal.3.2"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.2"
∷ word (ἀ ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Gal.3.2"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.2"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Gal.3.3"
∷ word (ἀ ∷ ν ∷ ό ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "Gal.3.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.3.3"
∷ word (ἐ ∷ ν ∷ α ∷ ρ ∷ ξ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Gal.3.3"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.3.3"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.3.3"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Gal.3.3"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ε ∷ ∙λ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "Gal.3.3"
∷ word (τ ∷ ο ∷ σ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Gal.3.4"
∷ word (ἐ ∷ π ∷ ά ∷ θ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.3.4"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "Gal.3.4"
∷ word (ε ∷ ἴ ∷ []) "Gal.3.4"
∷ word (γ ∷ ε ∷ []) "Gal.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.4"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "Gal.3.4"
∷ word (ὁ ∷ []) "Gal.3.5"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.3.5"
∷ word (ἐ ∷ π ∷ ι ∷ χ ∷ ο ∷ ρ ∷ η ∷ γ ∷ ῶ ∷ ν ∷ []) "Gal.3.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.3.5"
∷ word (τ ∷ ὸ ∷ []) "Gal.3.5"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.3.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.5"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ῶ ∷ ν ∷ []) "Gal.3.5"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ι ∷ ς ∷ []) "Gal.3.5"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.3.5"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.5"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.3.5"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.5"
∷ word (ἢ ∷ []) "Gal.3.5"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.5"
∷ word (ἀ ∷ κ ∷ ο ∷ ῆ ∷ ς ∷ []) "Gal.3.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.5"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Gal.3.6"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.6"
∷ word (ἐ ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ σ ∷ ε ∷ ν ∷ []) "Gal.3.6"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.6"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Gal.3.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.6"
∷ word (ἐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ σ ∷ θ ∷ η ∷ []) "Gal.3.6"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "Gal.3.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.6"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "Gal.3.6"
∷ word (Γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.3.7"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Gal.3.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.7"
∷ word (ο ∷ ἱ ∷ []) "Gal.3.7"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.7"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.7"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Gal.3.7"
∷ word (υ ∷ ἱ ∷ ο ∷ ί ∷ []) "Gal.3.7"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.3.7"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Gal.3.7"
∷ word (π ∷ ρ ∷ ο ∷ ϊ ∷ δ ∷ ο ∷ ῦ ∷ σ ∷ α ∷ []) "Gal.3.8"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.8"
∷ word (ἡ ∷ []) "Gal.3.8"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ὴ ∷ []) "Gal.3.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.8"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.8"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.8"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῖ ∷ []) "Gal.3.8"
∷ word (τ ∷ ὰ ∷ []) "Gal.3.8"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.3.8"
∷ word (ὁ ∷ []) "Gal.3.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.3.8"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ υ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ σ ∷ α ∷ τ ∷ ο ∷ []) "Gal.3.8"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.8"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.8"
∷ word (Ἐ ∷ ν ∷ ε ∷ υ ∷ ∙λ ∷ ο ∷ γ ∷ η ∷ θ ∷ ή ∷ σ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.8"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.8"
∷ word (σ ∷ ο ∷ ὶ ∷ []) "Gal.3.8"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Gal.3.8"
∷ word (τ ∷ ὰ ∷ []) "Gal.3.8"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.3.8"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.3.9"
∷ word (ο ∷ ἱ ∷ []) "Gal.3.9"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.9"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.9"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.9"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Gal.3.9"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.9"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.3.9"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ά ∷ μ ∷ []) "Gal.3.9"
∷ word (Ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Gal.3.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.10"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.10"
∷ word (ἔ ∷ ρ ∷ γ ∷ ω ∷ ν ∷ []) "Gal.3.10"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.10"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "Gal.3.10"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.3.10"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ α ∷ ν ∷ []) "Gal.3.10"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "Gal.3.10"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.10"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.10"
∷ word (Ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.3.10"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Gal.3.10"
∷ word (ὃ ∷ ς ∷ []) "Gal.3.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.10"
∷ word (ἐ ∷ μ ∷ μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "Gal.3.10"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.3.10"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.3.10"
∷ word (γ ∷ ε ∷ γ ∷ ρ ∷ α ∷ μ ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ ς ∷ []) "Gal.3.10"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.10"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.10"
∷ word (β ∷ ι ∷ β ∷ ∙λ ∷ ί ∷ ῳ ∷ []) "Gal.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.10"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.10"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.3.10"
∷ word (α ∷ ὐ ∷ τ ∷ ά ∷ []) "Gal.3.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.11"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.11"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.11"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Gal.3.11"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Gal.3.11"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.11"
∷ word (π ∷ α ∷ ρ ∷ ὰ ∷ []) "Gal.3.11"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.11"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "Gal.3.11"
∷ word (δ ∷ ῆ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.3.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.11"
∷ word (Ὁ ∷ []) "Gal.3.11"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ς ∷ []) "Gal.3.11"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.11"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.11"
∷ word (ζ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.11"
∷ word (ὁ ∷ []) "Gal.3.12"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.12"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.12"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.12"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.12"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.3.12"
∷ word (Ὁ ∷ []) "Gal.3.12"
∷ word (π ∷ ο ∷ ι ∷ ή ∷ σ ∷ α ∷ ς ∷ []) "Gal.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ὰ ∷ []) "Gal.3.12"
∷ word (ζ ∷ ή ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.12"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.3.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.3.13"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.3.13"
∷ word (ἐ ∷ ξ ∷ η ∷ γ ∷ ό ∷ ρ ∷ α ∷ σ ∷ ε ∷ ν ∷ []) "Gal.3.13"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.3.13"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ α ∷ ς ∷ []) "Gal.3.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.13"
∷ word (γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.3.13"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "Gal.3.13"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.3.13"
∷ word (κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ α ∷ []) "Gal.3.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.3.13"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.13"
∷ word (Ἐ ∷ π ∷ ι ∷ κ ∷ α ∷ τ ∷ ά ∷ ρ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.3.13"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Gal.3.13"
∷ word (ὁ ∷ []) "Gal.3.13"
∷ word (κ ∷ ρ ∷ ε ∷ μ ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.3.13"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Gal.3.13"
∷ word (ξ ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "Gal.3.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.3.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.14"
∷ word (τ ∷ ὰ ∷ []) "Gal.3.14"
∷ word (ἔ ∷ θ ∷ ν ∷ η ∷ []) "Gal.3.14"
∷ word (ἡ ∷ []) "Gal.3.14"
∷ word (ε ∷ ὐ ∷ ∙λ ∷ ο ∷ γ ∷ ί ∷ α ∷ []) "Gal.3.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.14"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.14"
∷ word (γ ∷ έ ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.3.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.3.14"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.3.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.3.14"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Gal.3.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.14"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.3.14"
∷ word (∙λ ∷ ά ∷ β ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Gal.3.14"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.3.14"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.3.14"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.14"
∷ word (Ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.3.15"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.3.15"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ν ∷ []) "Gal.3.15"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.3.15"
∷ word (ὅ ∷ μ ∷ ω ∷ ς ∷ []) "Gal.3.15"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ο ∷ υ ∷ []) "Gal.3.15"
∷ word (κ ∷ ε ∷ κ ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Gal.3.15"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ ν ∷ []) "Gal.3.15"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Gal.3.15"
∷ word (ἀ ∷ θ ∷ ε ∷ τ ∷ ε ∷ ῖ ∷ []) "Gal.3.15"
∷ word (ἢ ∷ []) "Gal.3.15"
∷ word (ἐ ∷ π ∷ ι ∷ δ ∷ ι ∷ α ∷ τ ∷ ά ∷ σ ∷ σ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.15"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.16"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.16"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.16"
∷ word (ἐ ∷ ρ ∷ ρ ∷ έ ∷ θ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "Gal.3.16"
∷ word (α ∷ ἱ ∷ []) "Gal.3.16"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ι ∷ []) "Gal.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.16"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.16"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.3.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.3.16"
∷ word (ο ∷ ὐ ∷ []) "Gal.3.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Gal.3.16"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Gal.3.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.3.16"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Gal.3.16"
∷ word (ὡ ∷ ς ∷ []) "Gal.3.16"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Gal.3.16"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ῶ ∷ ν ∷ []) "Gal.3.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.3.16"
∷ word (ὡ ∷ ς ∷ []) "Gal.3.16"
∷ word (ἐ ∷ φ ∷ []) "Gal.3.16"
∷ word (ἑ ∷ ν ∷ ό ∷ ς ∷ []) "Gal.3.16"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "Gal.3.16"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.16"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Gal.3.16"
∷ word (σ ∷ ο ∷ υ ∷ []) "Gal.3.16"
∷ word (ὅ ∷ ς ∷ []) "Gal.3.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.16"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "Gal.3.16"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Gal.3.17"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.17"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.3.17"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ή ∷ κ ∷ η ∷ ν ∷ []) "Gal.3.17"
∷ word (π ∷ ρ ∷ ο ∷ κ ∷ ε ∷ κ ∷ υ ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ η ∷ ν ∷ []) "Gal.3.17"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.3.17"
∷ word (ὁ ∷ []) "Gal.3.17"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.3.17"
∷ word (τ ∷ ε ∷ τ ∷ ρ ∷ α ∷ κ ∷ ό ∷ σ ∷ ι ∷ α ∷ []) "Gal.3.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.17"
∷ word (τ ∷ ρ ∷ ι ∷ ά ∷ κ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "Gal.3.17"
∷ word (ἔ ∷ τ ∷ η ∷ []) "Gal.3.17"
∷ word (γ ∷ ε ∷ γ ∷ ο ∷ ν ∷ ὼ ∷ ς ∷ []) "Gal.3.17"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.17"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.17"
∷ word (ἀ ∷ κ ∷ υ ∷ ρ ∷ ο ∷ ῖ ∷ []) "Gal.3.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.17"
∷ word (τ ∷ ὸ ∷ []) "Gal.3.17"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ γ ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.3.17"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.3.17"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Gal.3.17"
∷ word (ε ∷ ἰ ∷ []) "Gal.3.18"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.18"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.18"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.18"
∷ word (ἡ ∷ []) "Gal.3.18"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ []) "Gal.3.18"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Gal.3.18"
∷ word (ἐ ∷ ξ ∷ []) "Gal.3.18"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Gal.3.18"
∷ word (τ ∷ ῷ ∷ []) "Gal.3.18"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.18"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.18"
∷ word (δ ∷ ι ∷ []) "Gal.3.18"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Gal.3.18"
∷ word (κ ∷ ε ∷ χ ∷ ά ∷ ρ ∷ ι ∷ σ ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.18"
∷ word (ὁ ∷ []) "Gal.3.18"
∷ word (θ ∷ ε ∷ ό ∷ ς ∷ []) "Gal.3.18"
∷ word (Τ ∷ ί ∷ []) "Gal.3.19"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.3.19"
∷ word (ὁ ∷ []) "Gal.3.19"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.19"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.3.19"
∷ word (π ∷ α ∷ ρ ∷ α ∷ β ∷ ά ∷ σ ∷ ε ∷ ω ∷ ν ∷ []) "Gal.3.19"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "Gal.3.19"
∷ word (π ∷ ρ ∷ ο ∷ σ ∷ ε ∷ τ ∷ έ ∷ θ ∷ η ∷ []) "Gal.3.19"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ ς ∷ []) "Gal.3.19"
∷ word (ο ∷ ὗ ∷ []) "Gal.3.19"
∷ word (ἔ ∷ ∙λ ∷ θ ∷ ῃ ∷ []) "Gal.3.19"
∷ word (τ ∷ ὸ ∷ []) "Gal.3.19"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Gal.3.19"
∷ word (ᾧ ∷ []) "Gal.3.19"
∷ word (ἐ ∷ π ∷ ή ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ τ ∷ α ∷ ι ∷ []) "Gal.3.19"
∷ word (δ ∷ ι ∷ α ∷ τ ∷ α ∷ γ ∷ ε ∷ ὶ ∷ ς ∷ []) "Gal.3.19"
∷ word (δ ∷ ι ∷ []) "Gal.3.19"
∷ word (ἀ ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ω ∷ ν ∷ []) "Gal.3.19"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.19"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ὶ ∷ []) "Gal.3.19"
∷ word (μ ∷ ε ∷ σ ∷ ί ∷ τ ∷ ο ∷ υ ∷ []) "Gal.3.19"
∷ word (ὁ ∷ []) "Gal.3.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.20"
∷ word (μ ∷ ε ∷ σ ∷ ί ∷ τ ∷ η ∷ ς ∷ []) "Gal.3.20"
∷ word (ἑ ∷ ν ∷ ὸ ∷ ς ∷ []) "Gal.3.20"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.20"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.20"
∷ word (ὁ ∷ []) "Gal.3.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.3.20"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Gal.3.20"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.20"
∷ word (Ὁ ∷ []) "Gal.3.21"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.3.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.21"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.3.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.3.21"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ ῶ ∷ ν ∷ []) "Gal.3.21"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.3.21"
∷ word (μ ∷ ὴ ∷ []) "Gal.3.21"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Gal.3.21"
∷ word (ε ∷ ἰ ∷ []) "Gal.3.21"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.21"
∷ word (ἐ ∷ δ ∷ ό ∷ θ ∷ η ∷ []) "Gal.3.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.21"
∷ word (ὁ ∷ []) "Gal.3.21"
∷ word (δ ∷ υ ∷ ν ∷ ά ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.3.21"
∷ word (ζ ∷ ῳ ∷ ο ∷ π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.3.21"
∷ word (ὄ ∷ ν ∷ τ ∷ ω ∷ ς ∷ []) "Gal.3.21"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ []) "Gal.3.21"
∷ word (ἂ ∷ ν ∷ []) "Gal.3.21"
∷ word (ἦ ∷ ν ∷ []) "Gal.3.21"
∷ word (ἡ ∷ []) "Gal.3.21"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Gal.3.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.3.22"
∷ word (σ ∷ υ ∷ ν ∷ έ ∷ κ ∷ ∙λ ∷ ε ∷ ι ∷ σ ∷ ε ∷ ν ∷ []) "Gal.3.22"
∷ word (ἡ ∷ []) "Gal.3.22"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ὴ ∷ []) "Gal.3.22"
∷ word (τ ∷ ὰ ∷ []) "Gal.3.22"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "Gal.3.22"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.3.22"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "Gal.3.22"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.3.22"
∷ word (ἡ ∷ []) "Gal.3.22"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "Gal.3.22"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.22"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.22"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.3.22"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.3.22"
∷ word (δ ∷ ο ∷ θ ∷ ῇ ∷ []) "Gal.3.22"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.3.22"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.3.22"
∷ word (Π ∷ ρ ∷ ὸ ∷ []) "Gal.3.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.23"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.23"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ε ∷ ῖ ∷ ν ∷ []) "Gal.3.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.3.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.23"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.3.23"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.3.23"
∷ word (ἐ ∷ φ ∷ ρ ∷ ο ∷ υ ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.3.23"
∷ word (σ ∷ υ ∷ γ ∷ κ ∷ ∙λ ∷ ε ∷ ι ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Gal.3.23"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.23"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.3.23"
∷ word (μ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "Gal.3.23"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.3.23"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ υ ∷ φ ∷ θ ∷ ῆ ∷ ν ∷ α ∷ ι ∷ []) "Gal.3.23"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.3.24"
∷ word (ὁ ∷ []) "Gal.3.24"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.3.24"
∷ word (π ∷ α ∷ ι ∷ δ ∷ α ∷ γ ∷ ω ∷ γ ∷ ὸ ∷ ς ∷ []) "Gal.3.24"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.3.24"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ ε ∷ ν ∷ []) "Gal.3.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ν ∷ []) "Gal.3.24"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.3.24"
∷ word (ἐ ∷ κ ∷ []) "Gal.3.24"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.24"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ω ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.3.24"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "Gal.3.25"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.25"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.3.25"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.25"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Gal.3.25"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.3.25"
∷ word (π ∷ α ∷ ι ∷ δ ∷ α ∷ γ ∷ ω ∷ γ ∷ ό ∷ ν ∷ []) "Gal.3.25"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.3.25"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.3.26"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.26"
∷ word (υ ∷ ἱ ∷ ο ∷ ὶ ∷ []) "Gal.3.26"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.3.26"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.3.26"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.3.26"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.3.26"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.3.26"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.26"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.3.26"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.3.26"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Gal.3.27"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.27"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.3.27"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.3.27"
∷ word (ἐ ∷ β ∷ α ∷ π ∷ τ ∷ ί ∷ σ ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Gal.3.27"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.3.27"
∷ word (ἐ ∷ ν ∷ ε ∷ δ ∷ ύ ∷ σ ∷ α ∷ σ ∷ θ ∷ ε ∷ []) "Gal.3.27"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.28"
∷ word (ἔ ∷ ν ∷ ι ∷ []) "Gal.3.28"
∷ word (Ἰ ∷ ο ∷ υ ∷ δ ∷ α ∷ ῖ ∷ ο ∷ ς ∷ []) "Gal.3.28"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.3.28"
∷ word (Ἕ ∷ ∙λ ∷ ∙λ ∷ η ∷ ν ∷ []) "Gal.3.28"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.28"
∷ word (ἔ ∷ ν ∷ ι ∷ []) "Gal.3.28"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.3.28"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.3.28"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ ύ ∷ θ ∷ ε ∷ ρ ∷ ο ∷ ς ∷ []) "Gal.3.28"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.3.28"
∷ word (ἔ ∷ ν ∷ ι ∷ []) "Gal.3.28"
∷ word (ἄ ∷ ρ ∷ σ ∷ ε ∷ ν ∷ []) "Gal.3.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.3.28"
∷ word (θ ∷ ῆ ∷ ∙λ ∷ υ ∷ []) "Gal.3.28"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.3.28"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.3.28"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.3.28"
∷ word (ε ∷ ἷ ∷ ς ∷ []) "Gal.3.28"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.3.28"
∷ word (ἐ ∷ ν ∷ []) "Gal.3.28"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.3.28"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.3.28"
∷ word (ε ∷ ἰ ∷ []) "Gal.3.29"
∷ word (δ ∷ ὲ ∷ []) "Gal.3.29"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.3.29"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.3.29"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Gal.3.29"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.3.29"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.3.29"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "Gal.3.29"
∷ word (ἐ ∷ σ ∷ τ ∷ έ ∷ []) "Gal.3.29"
∷ word (κ ∷ α ∷ τ ∷ []) "Gal.3.29"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ν ∷ []) "Gal.3.29"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ι ∷ []) "Gal.3.29"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.4.1"
∷ word (δ ∷ έ ∷ []) "Gal.4.1"
∷ word (ἐ ∷ φ ∷ []) "Gal.4.1"
∷ word (ὅ ∷ σ ∷ ο ∷ ν ∷ []) "Gal.4.1"
∷ word (χ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.4.1"
∷ word (ὁ ∷ []) "Gal.4.1"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.4.1"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ό ∷ ς ∷ []) "Gal.4.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.4.1"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Gal.4.1"
∷ word (δ ∷ ι ∷ α ∷ φ ∷ έ ∷ ρ ∷ ε ∷ ι ∷ []) "Gal.4.1"
∷ word (δ ∷ ο ∷ ύ ∷ ∙λ ∷ ο ∷ υ ∷ []) "Gal.4.1"
∷ word (κ ∷ ύ ∷ ρ ∷ ι ∷ ο ∷ ς ∷ []) "Gal.4.1"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "Gal.4.1"
∷ word (ὤ ∷ ν ∷ []) "Gal.4.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.2"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.2"
∷ word (ἐ ∷ π ∷ ι ∷ τ ∷ ρ ∷ ό ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Gal.4.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ []) "Gal.4.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.2"
∷ word (ο ∷ ἰ ∷ κ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.4.2"
∷ word (ἄ ∷ χ ∷ ρ ∷ ι ∷ []) "Gal.4.2"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.2"
∷ word (π ∷ ρ ∷ ο ∷ θ ∷ ε ∷ σ ∷ μ ∷ ί ∷ α ∷ ς ∷ []) "Gal.4.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.4.2"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ό ∷ ς ∷ []) "Gal.4.2"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Gal.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.3"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.4.3"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Gal.4.3"
∷ word (ἦ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.4.3"
∷ word (ν ∷ ή ∷ π ∷ ι ∷ ο ∷ ι ∷ []) "Gal.4.3"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.3"
∷ word (τ ∷ ὰ ∷ []) "Gal.4.3"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ῖ ∷ α ∷ []) "Gal.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.4.3"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "Gal.4.3"
∷ word (ἤ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.4.3"
∷ word (δ ∷ ε ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ω ∷ μ ∷ έ ∷ ν ∷ ο ∷ ι ∷ []) "Gal.4.3"
∷ word (ὅ ∷ τ ∷ ε ∷ []) "Gal.4.4"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.4"
∷ word (ἦ ∷ ∙λ ∷ θ ∷ ε ∷ ν ∷ []) "Gal.4.4"
∷ word (τ ∷ ὸ ∷ []) "Gal.4.4"
∷ word (π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ μ ∷ α ∷ []) "Gal.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.4.4"
∷ word (χ ∷ ρ ∷ ό ∷ ν ∷ ο ∷ υ ∷ []) "Gal.4.4"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ι ∷ ∙λ ∷ ε ∷ ν ∷ []) "Gal.4.4"
∷ word (ὁ ∷ []) "Gal.4.4"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.4.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.4"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "Gal.4.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.4.4"
∷ word (γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Gal.4.4"
∷ word (ἐ ∷ κ ∷ []) "Gal.4.4"
∷ word (γ ∷ υ ∷ ν ∷ α ∷ ι ∷ κ ∷ ό ∷ ς ∷ []) "Gal.4.4"
∷ word (γ ∷ ε ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ν ∷ []) "Gal.4.4"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.4"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.4.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.4.5"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.5"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.5"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.4.5"
∷ word (ἐ ∷ ξ ∷ α ∷ γ ∷ ο ∷ ρ ∷ ά ∷ σ ∷ ῃ ∷ []) "Gal.4.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.4.5"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.4.5"
∷ word (υ ∷ ἱ ∷ ο ∷ θ ∷ ε ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "Gal.4.5"
∷ word (ἀ ∷ π ∷ ο ∷ ∙λ ∷ ά ∷ β ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "Gal.4.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.6"
∷ word (δ ∷ έ ∷ []) "Gal.4.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.4.6"
∷ word (υ ∷ ἱ ∷ ο ∷ ί ∷ []) "Gal.4.6"
∷ word (ἐ ∷ ξ ∷ α ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ι ∷ ∙λ ∷ ε ∷ ν ∷ []) "Gal.4.6"
∷ word (ὁ ∷ []) "Gal.4.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.4.6"
∷ word (τ ∷ ὸ ∷ []) "Gal.4.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.4.6"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Gal.4.6"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.4.6"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.4.6"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "Gal.4.6"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "Gal.4.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.6"
∷ word (κ ∷ ρ ∷ ᾶ ∷ ζ ∷ ο ∷ ν ∷ []) "Gal.4.6"
∷ word (Α ∷ β ∷ β ∷ α ∷ []) "Gal.4.6"
∷ word (ὁ ∷ []) "Gal.4.6"
∷ word (π ∷ α ∷ τ ∷ ή ∷ ρ ∷ []) "Gal.4.6"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.4.7"
∷ word (ο ∷ ὐ ∷ κ ∷ έ ∷ τ ∷ ι ∷ []) "Gal.4.7"
∷ word (ε ∷ ἶ ∷ []) "Gal.4.7"
∷ word (δ ∷ ο ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.4.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.7"
∷ word (υ ∷ ἱ ∷ ό ∷ ς ∷ []) "Gal.4.7"
∷ word (ε ∷ ἰ ∷ []) "Gal.4.7"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.7"
∷ word (υ ∷ ἱ ∷ ό ∷ ς ∷ []) "Gal.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.7"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.4.7"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.4.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.4.7"
∷ word (Ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.8"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "Gal.4.8"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Gal.4.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.4.8"
∷ word (ε ∷ ἰ ∷ δ ∷ ό ∷ τ ∷ ε ∷ ς ∷ []) "Gal.4.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "Gal.4.8"
∷ word (ἐ ∷ δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.4.8"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.4.8"
∷ word (φ ∷ ύ ∷ σ ∷ ε ∷ ι ∷ []) "Gal.4.8"
∷ word (μ ∷ ὴ ∷ []) "Gal.4.8"
∷ word (ο ∷ ὖ ∷ σ ∷ ι ∷ []) "Gal.4.8"
∷ word (θ ∷ ε ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.4.8"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.4.9"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.9"
∷ word (γ ∷ ν ∷ ό ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.4.9"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "Gal.4.9"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.4.9"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.9"
∷ word (γ ∷ ν ∷ ω ∷ σ ∷ θ ∷ έ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.4.9"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.4.9"
∷ word (π ∷ ῶ ∷ ς ∷ []) "Gal.4.9"
∷ word (ἐ ∷ π ∷ ι ∷ σ ∷ τ ∷ ρ ∷ έ ∷ φ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.4.9"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.4.9"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Gal.4.9"
∷ word (τ ∷ ὰ ∷ []) "Gal.4.9"
∷ word (ἀ ∷ σ ∷ θ ∷ ε ∷ ν ∷ ῆ ∷ []) "Gal.4.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.9"
∷ word (π ∷ τ ∷ ω ∷ χ ∷ ὰ ∷ []) "Gal.4.9"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ῖ ∷ α ∷ []) "Gal.4.9"
∷ word (ο ∷ ἷ ∷ ς ∷ []) "Gal.4.9"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.4.9"
∷ word (ἄ ∷ ν ∷ ω ∷ θ ∷ ε ∷ ν ∷ []) "Gal.4.9"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ ν ∷ []) "Gal.4.9"
∷ word (θ ∷ έ ∷ ∙λ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.4.9"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.4.10"
∷ word (π ∷ α ∷ ρ ∷ α ∷ τ ∷ η ∷ ρ ∷ ε ∷ ῖ ∷ σ ∷ θ ∷ ε ∷ []) "Gal.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.10"
∷ word (μ ∷ ῆ ∷ ν ∷ α ∷ ς ∷ []) "Gal.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.10"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.10"
∷ word (ἐ ∷ ν ∷ ι ∷ α ∷ υ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "Gal.4.10"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "Gal.4.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.11"
∷ word (μ ∷ ή ∷ []) "Gal.4.11"
∷ word (π ∷ ω ∷ ς ∷ []) "Gal.4.11"
∷ word (ε ∷ ἰ ∷ κ ∷ ῇ ∷ []) "Gal.4.11"
∷ word (κ ∷ ε ∷ κ ∷ ο ∷ π ∷ ί ∷ α ∷ κ ∷ α ∷ []) "Gal.4.11"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.4.11"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.11"
∷ word (Γ ∷ ί ∷ ν ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Gal.4.12"
∷ word (ὡ ∷ ς ∷ []) "Gal.4.12"
∷ word (ἐ ∷ γ ∷ ώ ∷ []) "Gal.4.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.12"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "Gal.4.12"
∷ word (ὡ ∷ ς ∷ []) "Gal.4.12"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.4.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.4.12"
∷ word (δ ∷ έ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Gal.4.12"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.12"
∷ word (ο ∷ ὐ ∷ δ ∷ έ ∷ ν ∷ []) "Gal.4.12"
∷ word (μ ∷ ε ∷ []) "Gal.4.12"
∷ word (ἠ ∷ δ ∷ ι ∷ κ ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.4.12"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "Gal.4.13"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.13"
∷ word (δ ∷ ι ∷ []) "Gal.4.13"
∷ word (ἀ ∷ σ ∷ θ ∷ έ ∷ ν ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "Gal.4.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.13"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Gal.4.13"
∷ word (ε ∷ ὐ ∷ η ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ι ∷ σ ∷ ά ∷ μ ∷ η ∷ ν ∷ []) "Gal.4.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.4.13"
∷ word (τ ∷ ὸ ∷ []) "Gal.4.13"
∷ word (π ∷ ρ ∷ ό ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Gal.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.14"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "Gal.4.14"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.14"
∷ word (τ ∷ ῇ ∷ []) "Gal.4.14"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Gal.4.14"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.4.14"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.4.14"
∷ word (ἐ ∷ ξ ∷ ο ∷ υ ∷ θ ∷ ε ∷ ν ∷ ή ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.4.14"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.4.14"
∷ word (ἐ ∷ ξ ∷ ε ∷ π ∷ τ ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.4.14"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.14"
∷ word (ὡ ∷ ς ∷ []) "Gal.4.14"
∷ word (ἄ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.4.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.4.14"
∷ word (ἐ ∷ δ ∷ έ ∷ ξ ∷ α ∷ σ ∷ θ ∷ έ ∷ []) "Gal.4.14"
∷ word (μ ∷ ε ∷ []) "Gal.4.14"
∷ word (ὡ ∷ ς ∷ []) "Gal.4.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.4.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "Gal.4.14"
∷ word (π ∷ ο ∷ ῦ ∷ []) "Gal.4.15"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.4.15"
∷ word (ὁ ∷ []) "Gal.4.15"
∷ word (μ ∷ α ∷ κ ∷ α ∷ ρ ∷ ι ∷ σ ∷ μ ∷ ὸ ∷ ς ∷ []) "Gal.4.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.15"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ῶ ∷ []) "Gal.4.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.4.15"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.4.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.15"
∷ word (ε ∷ ἰ ∷ []) "Gal.4.15"
∷ word (δ ∷ υ ∷ ν ∷ α ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.4.15"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.15"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.15"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.15"
∷ word (ἐ ∷ ξ ∷ ο ∷ ρ ∷ ύ ∷ ξ ∷ α ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.4.15"
∷ word (ἐ ∷ δ ∷ ώ ∷ κ ∷ α ∷ τ ∷ έ ∷ []) "Gal.4.15"
∷ word (μ ∷ ο ∷ ι ∷ []) "Gal.4.15"
∷ word (ὥ ∷ σ ∷ τ ∷ ε ∷ []) "Gal.4.16"
∷ word (ἐ ∷ χ ∷ θ ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.4.16"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.16"
∷ word (γ ∷ έ ∷ γ ∷ ο ∷ ν ∷ α ∷ []) "Gal.4.16"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "Gal.4.16"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.4.16"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.4.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.17"
∷ word (ο ∷ ὐ ∷ []) "Gal.4.17"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "Gal.4.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.17"
∷ word (ἐ ∷ κ ∷ κ ∷ ∙λ ∷ ε ∷ ῖ ∷ σ ∷ α ∷ ι ∷ []) "Gal.4.17"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.17"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.4.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.4.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.17"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ τ ∷ ε ∷ []) "Gal.4.17"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "Gal.4.18"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.18"
∷ word (ζ ∷ η ∷ ∙λ ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Gal.4.18"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.18"
∷ word (κ ∷ α ∷ ∙λ ∷ ῷ ∷ []) "Gal.4.18"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ο ∷ τ ∷ ε ∷ []) "Gal.4.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.18"
∷ word (μ ∷ ὴ ∷ []) "Gal.4.18"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.4.18"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.18"
∷ word (τ ∷ ῷ ∷ []) "Gal.4.18"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ῖ ∷ ν ∷ α ∷ ί ∷ []) "Gal.4.18"
∷ word (μ ∷ ε ∷ []) "Gal.4.18"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.4.18"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.18"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Gal.4.19"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.4.19"
∷ word (ο ∷ ὓ ∷ ς ∷ []) "Gal.4.19"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.4.19"
∷ word (ὠ ∷ δ ∷ ί ∷ ν ∷ ω ∷ []) "Gal.4.19"
∷ word (μ ∷ έ ∷ χ ∷ ρ ∷ ι ∷ ς ∷ []) "Gal.4.19"
∷ word (ο ∷ ὗ ∷ []) "Gal.4.19"
∷ word (μ ∷ ο ∷ ρ ∷ φ ∷ ω ∷ θ ∷ ῇ ∷ []) "Gal.4.19"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.4.19"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.19"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.4.19"
∷ word (ἤ ∷ θ ∷ ε ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.4.20"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.20"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "Gal.4.20"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.4.20"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.4.20"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "Gal.4.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.20"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ά ∷ ξ ∷ α ∷ ι ∷ []) "Gal.4.20"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.4.20"
∷ word (φ ∷ ω ∷ ν ∷ ή ∷ ν ∷ []) "Gal.4.20"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.4.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.20"
∷ word (ἀ ∷ π ∷ ο ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ α ∷ ι ∷ []) "Gal.4.20"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.20"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.4.20"
∷ word (Λ ∷ έ ∷ γ ∷ ε ∷ τ ∷ έ ∷ []) "Gal.4.21"
∷ word (μ ∷ ο ∷ ι ∷ []) "Gal.4.21"
∷ word (ο ∷ ἱ ∷ []) "Gal.4.21"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.4.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.4.21"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.4.21"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "Gal.4.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.21"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.4.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.4.21"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.4.21"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Gal.4.22"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.4.22"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.22"
∷ word (Ἀ ∷ β ∷ ρ ∷ α ∷ ὰ ∷ μ ∷ []) "Gal.4.22"
∷ word (δ ∷ ύ ∷ ο ∷ []) "Gal.4.22"
∷ word (υ ∷ ἱ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.4.22"
∷ word (ἔ ∷ σ ∷ χ ∷ ε ∷ ν ∷ []) "Gal.4.22"
∷ word (ἕ ∷ ν ∷ α ∷ []) "Gal.4.22"
∷ word (ἐ ∷ κ ∷ []) "Gal.4.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.22"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ σ ∷ κ ∷ η ∷ ς ∷ []) "Gal.4.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.22"
∷ word (ἕ ∷ ν ∷ α ∷ []) "Gal.4.22"
∷ word (ἐ ∷ κ ∷ []) "Gal.4.22"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.22"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.4.22"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.4.23"
∷ word (ὁ ∷ []) "Gal.4.23"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Gal.4.23"
∷ word (ἐ ∷ κ ∷ []) "Gal.4.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.23"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ σ ∷ κ ∷ η ∷ ς ∷ []) "Gal.4.23"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.4.23"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Gal.4.23"
∷ word (γ ∷ ε ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Gal.4.23"
∷ word (ὁ ∷ []) "Gal.4.23"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.23"
∷ word (ἐ ∷ κ ∷ []) "Gal.4.23"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.23"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.4.23"
∷ word (δ ∷ ι ∷ []) "Gal.4.23"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Gal.4.23"
∷ word (ἅ ∷ τ ∷ ι ∷ ν ∷ ά ∷ []) "Gal.4.24"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.4.24"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ η ∷ γ ∷ ο ∷ ρ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ α ∷ []) "Gal.4.24"
∷ word (α ∷ ὗ ∷ τ ∷ α ∷ ι ∷ []) "Gal.4.24"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Gal.4.24"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.4.24"
∷ word (δ ∷ ύ ∷ ο ∷ []) "Gal.4.24"
∷ word (δ ∷ ι ∷ α ∷ θ ∷ ῆ ∷ κ ∷ α ∷ ι ∷ []) "Gal.4.24"
∷ word (μ ∷ ί ∷ α ∷ []) "Gal.4.24"
∷ word (μ ∷ ὲ ∷ ν ∷ []) "Gal.4.24"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.4.24"
∷ word (ὄ ∷ ρ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.4.24"
∷ word (Σ ∷ ι ∷ ν ∷ ᾶ ∷ []) "Gal.4.24"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.4.24"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Gal.4.24"
∷ word (γ ∷ ε ∷ ν ∷ ν ∷ ῶ ∷ σ ∷ α ∷ []) "Gal.4.24"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Gal.4.24"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Gal.4.24"
∷ word (Ἁ ∷ γ ∷ ά ∷ ρ ∷ []) "Gal.4.24"
∷ word (τ ∷ ὸ ∷ []) "Gal.4.25"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.25"
∷ word (Ἁ ∷ γ ∷ ὰ ∷ ρ ∷ []) "Gal.4.25"
∷ word (Σ ∷ ι ∷ ν ∷ ᾶ ∷ []) "Gal.4.25"
∷ word (ὄ ∷ ρ ∷ ο ∷ ς ∷ []) "Gal.4.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Gal.4.25"
∷ word (ἐ ∷ ν ∷ []) "Gal.4.25"
∷ word (τ ∷ ῇ ∷ []) "Gal.4.25"
∷ word (Ἀ ∷ ρ ∷ α ∷ β ∷ ί ∷ ᾳ ∷ []) "Gal.4.25"
∷ word (σ ∷ υ ∷ σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ε ∷ ῖ ∷ []) "Gal.4.25"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.25"
∷ word (τ ∷ ῇ ∷ []) "Gal.4.25"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.4.25"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ή ∷ μ ∷ []) "Gal.4.25"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ ι ∷ []) "Gal.4.25"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.4.25"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.4.25"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.4.25"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ ω ∷ ν ∷ []) "Gal.4.25"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Gal.4.25"
∷ word (ἡ ∷ []) "Gal.4.26"
∷ word (δ ∷ ὲ ∷ []) "Gal.4.26"
∷ word (ἄ ∷ ν ∷ ω ∷ []) "Gal.4.26"
∷ word (Ἰ ∷ ε ∷ ρ ∷ ο ∷ υ ∷ σ ∷ α ∷ ∙λ ∷ ὴ ∷ μ ∷ []) "Gal.4.26"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ []) "Gal.4.26"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "Gal.4.26"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "Gal.4.26"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Gal.4.26"
∷ word (μ ∷ ή ∷ τ ∷ η ∷ ρ ∷ []) "Gal.4.26"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.4.26"
∷ word (γ ∷ έ ∷ γ ∷ ρ ∷ α ∷ π ∷ τ ∷ α ∷ ι ∷ []) "Gal.4.27"
∷ word (γ ∷ ά ∷ ρ ∷ []) "Gal.4.27"
∷ word (Ε ∷ ὐ ∷ φ ∷ ρ ∷ ά ∷ ν ∷ θ ∷ η ∷ τ ∷ ι ∷ []) "Gal.4.27"
∷ word (σ ∷ τ ∷ ε ∷ ῖ ∷ ρ ∷ α ∷ []) "Gal.4.27"
∷ word (ἡ ∷ []) "Gal.4.27"
∷ word (ο ∷ ὐ ∷ []) "Gal.4.27"
∷ word (τ ∷ ί ∷ κ ∷ τ ∷ ο ∷ υ ∷ σ ∷ α ∷ []) "Gal.4.27"
∷ word (ῥ ∷ ῆ ∷ ξ ∷ ο ∷ ν ∷ []) "Gal.4.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.27"
∷ word (β ∷ ό ∷ η ∷ σ ∷ ο ∷ ν ∷ []) "Gal.4.27"
∷ word (ἡ ∷ []) "Gal.4.27"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.4.27"
∷ word (ὠ ∷ δ ∷ ί ∷ ν ∷ ο ∷ υ ∷ σ ∷ α ∷ []) "Gal.4.27"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.4.27"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.27"
∷ word (τ ∷ ὰ ∷ []) "Gal.4.27"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Gal.4.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.27"
∷ word (ἐ ∷ ρ ∷ ή ∷ μ ∷ ο ∷ υ ∷ []) "Gal.4.27"
∷ word (μ ∷ ᾶ ∷ ∙λ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.4.27"
∷ word (ἢ ∷ []) "Gal.4.27"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.27"
∷ word (ἐ ∷ χ ∷ ο ∷ ύ ∷ σ ∷ η ∷ ς ∷ []) "Gal.4.27"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.27"
∷ word (ἄ ∷ ν ∷ δ ∷ ρ ∷ α ∷ []) "Gal.4.27"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.4.28"
∷ word (δ ∷ έ ∷ []) "Gal.4.28"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.4.28"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.4.28"
∷ word (Ἰ ∷ σ ∷ α ∷ ὰ ∷ κ ∷ []) "Gal.4.28"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ ς ∷ []) "Gal.4.28"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Gal.4.28"
∷ word (ἐ ∷ σ ∷ τ ∷ έ ∷ []) "Gal.4.28"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "Gal.4.29"
∷ word (ὥ ∷ σ ∷ π ∷ ε ∷ ρ ∷ []) "Gal.4.29"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "Gal.4.29"
∷ word (ὁ ∷ []) "Gal.4.29"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.4.29"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Gal.4.29"
∷ word (γ ∷ ε ∷ ν ∷ ν ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "Gal.4.29"
∷ word (ἐ ∷ δ ∷ ί ∷ ω ∷ κ ∷ ε ∷ []) "Gal.4.29"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.29"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.4.29"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.4.29"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Gal.4.29"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.29"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "Gal.4.29"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.30"
∷ word (τ ∷ ί ∷ []) "Gal.4.30"
∷ word (∙λ ∷ έ ∷ γ ∷ ε ∷ ι ∷ []) "Gal.4.30"
∷ word (ἡ ∷ []) "Gal.4.30"
∷ word (γ ∷ ρ ∷ α ∷ φ ∷ ή ∷ []) "Gal.4.30"
∷ word (Ἔ ∷ κ ∷ β ∷ α ∷ ∙λ ∷ ε ∷ []) "Gal.4.30"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.4.30"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ σ ∷ κ ∷ η ∷ ν ∷ []) "Gal.4.30"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.4.30"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.4.30"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "Gal.4.30"
∷ word (α ∷ ὐ ∷ τ ∷ ῆ ∷ ς ∷ []) "Gal.4.30"
∷ word (ο ∷ ὐ ∷ []) "Gal.4.30"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.4.30"
∷ word (μ ∷ ὴ ∷ []) "Gal.4.30"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Gal.4.30"
∷ word (ὁ ∷ []) "Gal.4.30"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "Gal.4.30"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.30"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ σ ∷ κ ∷ η ∷ ς ∷ []) "Gal.4.30"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.4.30"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.4.30"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "Gal.4.30"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.30"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.4.30"
∷ word (δ ∷ ι ∷ ό ∷ []) "Gal.4.31"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.4.31"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.4.31"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "Gal.4.31"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ σ ∷ κ ∷ η ∷ ς ∷ []) "Gal.4.31"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "Gal.4.31"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.4.31"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.4.31"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ α ∷ ς ∷ []) "Gal.4.31"
∷ word (τ ∷ ῇ ∷ []) "Gal.5.1"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ ᾳ ∷ []) "Gal.5.1"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.5.1"
∷ word (ἠ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ έ ∷ ρ ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "Gal.5.1"
∷ word (σ ∷ τ ∷ ή ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.1"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.1"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.1"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.5.1"
∷ word (ζ ∷ υ ∷ γ ∷ ῷ ∷ []) "Gal.5.1"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "Gal.5.1"
∷ word (ἐ ∷ ν ∷ έ ∷ χ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Gal.5.1"
∷ word (Ἴ ∷ δ ∷ ε ∷ []) "Gal.5.2"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.5.2"
∷ word (Π ∷ α ∷ ῦ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.5.2"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.5.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.5.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.5.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.5.2"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ έ ∷ μ ∷ ν ∷ η ∷ σ ∷ θ ∷ ε ∷ []) "Gal.5.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "Gal.5.2"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.2"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Gal.5.2"
∷ word (ὠ ∷ φ ∷ ε ∷ ∙λ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "Gal.5.2"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Gal.5.3"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.3"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "Gal.5.3"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "Gal.5.3"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ῳ ∷ []) "Gal.5.3"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ε ∷ μ ∷ ν ∷ ο ∷ μ ∷ έ ∷ ν ∷ ῳ ∷ []) "Gal.5.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.5.3"
∷ word (ὀ ∷ φ ∷ ε ∷ ι ∷ ∙λ ∷ έ ∷ τ ∷ η ∷ ς ∷ []) "Gal.5.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "Gal.5.3"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.5.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.5.3"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.5.3"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.5.3"
∷ word (κ ∷ α ∷ τ ∷ η ∷ ρ ∷ γ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Gal.5.4"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "Gal.5.4"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.5.4"
∷ word (ο ∷ ἵ ∷ τ ∷ ι ∷ ν ∷ ε ∷ ς ∷ []) "Gal.5.4"
∷ word (ἐ ∷ ν ∷ []) "Gal.5.4"
∷ word (ν ∷ ό ∷ μ ∷ ῳ ∷ []) "Gal.5.4"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ ῦ ∷ σ ∷ θ ∷ ε ∷ []) "Gal.5.4"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.5.4"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ τ ∷ ο ∷ ς ∷ []) "Gal.5.4"
∷ word (ἐ ∷ ξ ∷ ε ∷ π ∷ έ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "Gal.5.4"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.5.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.5"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.5.5"
∷ word (ἐ ∷ κ ∷ []) "Gal.5.5"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.5.5"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "Gal.5.5"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ς ∷ []) "Gal.5.5"
∷ word (ἀ ∷ π ∷ ε ∷ κ ∷ δ ∷ ε ∷ χ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.5.5"
∷ word (ἐ ∷ ν ∷ []) "Gal.5.6"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "Gal.5.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.5.6"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Gal.5.6"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ή ∷ []) "Gal.5.6"
∷ word (τ ∷ ι ∷ []) "Gal.5.6"
∷ word (ἰ ∷ σ ∷ χ ∷ ύ ∷ ε ∷ ι ∷ []) "Gal.5.6"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Gal.5.6"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Gal.5.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.5.6"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Gal.5.6"
∷ word (δ ∷ ι ∷ []) "Gal.5.6"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Gal.5.6"
∷ word (ἐ ∷ ν ∷ ε ∷ ρ ∷ γ ∷ ο ∷ υ ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "Gal.5.6"
∷ word (Ἐ ∷ τ ∷ ρ ∷ έ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.7"
∷ word (κ ∷ α ∷ ∙λ ∷ ῶ ∷ ς ∷ []) "Gal.5.7"
∷ word (τ ∷ ί ∷ ς ∷ []) "Gal.5.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.7"
∷ word (ἐ ∷ ν ∷ έ ∷ κ ∷ ο ∷ ψ ∷ ε ∷ ν ∷ []) "Gal.5.7"
∷ word (τ ∷ ῇ ∷ []) "Gal.5.7"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "Gal.5.7"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.7"
∷ word (π ∷ ε ∷ ί ∷ θ ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Gal.5.7"
∷ word (ἡ ∷ []) "Gal.5.8"
∷ word (π ∷ ε ∷ ι ∷ σ ∷ μ ∷ ο ∷ ν ∷ ὴ ∷ []) "Gal.5.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.5.8"
∷ word (ἐ ∷ κ ∷ []) "Gal.5.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.5.8"
∷ word (κ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ο ∷ ς ∷ []) "Gal.5.8"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.8"
∷ word (μ ∷ ι ∷ κ ∷ ρ ∷ ὰ ∷ []) "Gal.5.9"
∷ word (ζ ∷ ύ ∷ μ ∷ η ∷ []) "Gal.5.9"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.5.9"
∷ word (τ ∷ ὸ ∷ []) "Gal.5.9"
∷ word (φ ∷ ύ ∷ ρ ∷ α ∷ μ ∷ α ∷ []) "Gal.5.9"
∷ word (ζ ∷ υ ∷ μ ∷ ο ∷ ῖ ∷ []) "Gal.5.9"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.5.10"
∷ word (π ∷ έ ∷ π ∷ ο ∷ ι ∷ θ ∷ α ∷ []) "Gal.5.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.5.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.10"
∷ word (ἐ ∷ ν ∷ []) "Gal.5.10"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ῳ ∷ []) "Gal.5.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.5.10"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ ν ∷ []) "Gal.5.10"
∷ word (ἄ ∷ ∙λ ∷ ∙λ ∷ ο ∷ []) "Gal.5.10"
∷ word (φ ∷ ρ ∷ ο ∷ ν ∷ ή ∷ σ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.10"
∷ word (ὁ ∷ []) "Gal.5.10"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.10"
∷ word (τ ∷ α ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ω ∷ ν ∷ []) "Gal.5.10"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.10"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "Gal.5.10"
∷ word (τ ∷ ὸ ∷ []) "Gal.5.10"
∷ word (κ ∷ ρ ∷ ί ∷ μ ∷ α ∷ []) "Gal.5.10"
∷ word (ὅ ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Gal.5.10"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.5.10"
∷ word (ᾖ ∷ []) "Gal.5.10"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.5.11"
∷ word (δ ∷ έ ∷ []) "Gal.5.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.5.11"
∷ word (ε ∷ ἰ ∷ []) "Gal.5.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ὴ ∷ ν ∷ []) "Gal.5.11"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Gal.5.11"
∷ word (κ ∷ η ∷ ρ ∷ ύ ∷ σ ∷ σ ∷ ω ∷ []) "Gal.5.11"
∷ word (τ ∷ ί ∷ []) "Gal.5.11"
∷ word (ἔ ∷ τ ∷ ι ∷ []) "Gal.5.11"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ο ∷ μ ∷ α ∷ ι ∷ []) "Gal.5.11"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Gal.5.11"
∷ word (κ ∷ α ∷ τ ∷ ή ∷ ρ ∷ γ ∷ η ∷ τ ∷ α ∷ ι ∷ []) "Gal.5.11"
∷ word (τ ∷ ὸ ∷ []) "Gal.5.11"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.5.11"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.5.11"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ []) "Gal.5.11"
∷ word (ὄ ∷ φ ∷ ε ∷ ∙λ ∷ ο ∷ ν ∷ []) "Gal.5.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.12"
∷ word (ἀ ∷ π ∷ ο ∷ κ ∷ ό ∷ ψ ∷ ο ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Gal.5.12"
∷ word (ο ∷ ἱ ∷ []) "Gal.5.12"
∷ word (ἀ ∷ ν ∷ α ∷ σ ∷ τ ∷ α ∷ τ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.5.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.5.12"
∷ word (Ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.5.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.13"
∷ word (ἐ ∷ π ∷ []) "Gal.5.13"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ ᾳ ∷ []) "Gal.5.13"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ ή ∷ θ ∷ η ∷ τ ∷ ε ∷ []) "Gal.5.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.5.13"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.5.13"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.13"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.5.13"
∷ word (ἐ ∷ ∙λ ∷ ε ∷ υ ∷ θ ∷ ε ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "Gal.5.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.5.13"
∷ word (ἀ ∷ φ ∷ ο ∷ ρ ∷ μ ∷ ὴ ∷ ν ∷ []) "Gal.5.13"
∷ word (τ ∷ ῇ ∷ []) "Gal.5.13"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Gal.5.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.5.13"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "Gal.5.13"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.5.13"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ς ∷ []) "Gal.5.13"
∷ word (δ ∷ ο ∷ υ ∷ ∙λ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Gal.5.13"
∷ word (ὁ ∷ []) "Gal.5.14"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.14"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "Gal.5.14"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.5.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.5.14"
∷ word (ἑ ∷ ν ∷ ὶ ∷ []) "Gal.5.14"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "Gal.5.14"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ ή ∷ ρ ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "Gal.5.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.5.14"
∷ word (τ ∷ ῷ ∷ []) "Gal.5.14"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ ή ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Gal.5.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.5.14"
∷ word (π ∷ ∙λ ∷ η ∷ σ ∷ ί ∷ ο ∷ ν ∷ []) "Gal.5.14"
∷ word (σ ∷ ο ∷ υ ∷ []) "Gal.5.14"
∷ word (ὡ ∷ ς ∷ []) "Gal.5.14"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Gal.5.14"
∷ word (ε ∷ ἰ ∷ []) "Gal.5.15"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.5.15"
∷ word (δ ∷ ά ∷ κ ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.15"
∷ word (κ ∷ α ∷ τ ∷ ε ∷ σ ∷ θ ∷ ί ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.15"
∷ word (β ∷ ∙λ ∷ έ ∷ π ∷ ε ∷ τ ∷ ε ∷ []) "Gal.5.15"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.15"
∷ word (ὑ ∷ π ∷ []) "Gal.5.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Gal.5.15"
∷ word (ἀ ∷ ν ∷ α ∷ ∙λ ∷ ω ∷ θ ∷ ῆ ∷ τ ∷ ε ∷ []) "Gal.5.15"
∷ word (Λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.5.16"
∷ word (δ ∷ έ ∷ []) "Gal.5.16"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.5.16"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "Gal.5.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.16"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "Gal.5.16"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Gal.5.16"
∷ word (ο ∷ ὐ ∷ []) "Gal.5.16"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.16"
∷ word (τ ∷ ε ∷ ∙λ ∷ έ ∷ σ ∷ η ∷ τ ∷ ε ∷ []) "Gal.5.16"
∷ word (ἡ ∷ []) "Gal.5.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.17"
∷ word (σ ∷ ὰ ∷ ρ ∷ ξ ∷ []) "Gal.5.17"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ε ∷ ῖ ∷ []) "Gal.5.17"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.5.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.5.17"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.5.17"
∷ word (τ ∷ ὸ ∷ []) "Gal.5.17"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.17"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.5.17"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.5.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.5.17"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "Gal.5.17"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Gal.5.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.5.17"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Gal.5.17"
∷ word (ἀ ∷ ν ∷ τ ∷ ί ∷ κ ∷ ε ∷ ι ∷ τ ∷ α ∷ ι ∷ []) "Gal.5.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.5.17"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.17"
∷ word (ἃ ∷ []) "Gal.5.17"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.5.17"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ τ ∷ ε ∷ []) "Gal.5.17"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Gal.5.17"
∷ word (π ∷ ο ∷ ι ∷ ῆ ∷ τ ∷ ε ∷ []) "Gal.5.17"
∷ word (ε ∷ ἰ ∷ []) "Gal.5.18"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.5.18"
∷ word (ἄ ∷ γ ∷ ε ∷ σ ∷ θ ∷ ε ∷ []) "Gal.5.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.5.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ὲ ∷ []) "Gal.5.18"
∷ word (ὑ ∷ π ∷ ὸ ∷ []) "Gal.5.18"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.5.18"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ὰ ∷ []) "Gal.5.19"
∷ word (δ ∷ έ ∷ []) "Gal.5.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.5.19"
∷ word (τ ∷ ὰ ∷ []) "Gal.5.19"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "Gal.5.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.5.19"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ό ∷ ς ∷ []) "Gal.5.19"
∷ word (ἅ ∷ τ ∷ ι ∷ ν ∷ ά ∷ []) "Gal.5.19"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.5.19"
∷ word (π ∷ ο ∷ ρ ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "Gal.5.19"
∷ word (ἀ ∷ κ ∷ α ∷ θ ∷ α ∷ ρ ∷ σ ∷ ί ∷ α ∷ []) "Gal.5.19"
∷ word (ἀ ∷ σ ∷ έ ∷ ∙λ ∷ γ ∷ ε ∷ ι ∷ α ∷ []) "Gal.5.19"
∷ word (ε ∷ ἰ ∷ δ ∷ ω ∷ ∙λ ∷ ο ∷ ∙λ ∷ α ∷ τ ∷ ρ ∷ ί ∷ α ∷ []) "Gal.5.20"
∷ word (φ ∷ α ∷ ρ ∷ μ ∷ α ∷ κ ∷ ε ∷ ί ∷ α ∷ []) "Gal.5.20"
∷ word (ἔ ∷ χ ∷ θ ∷ ρ ∷ α ∷ ι ∷ []) "Gal.5.20"
∷ word (ἔ ∷ ρ ∷ ι ∷ ς ∷ []) "Gal.5.20"
∷ word (ζ ∷ ῆ ∷ ∙λ ∷ ο ∷ ς ∷ []) "Gal.5.20"
∷ word (θ ∷ υ ∷ μ ∷ ο ∷ ί ∷ []) "Gal.5.20"
∷ word (ἐ ∷ ρ ∷ ι ∷ θ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "Gal.5.20"
∷ word (δ ∷ ι ∷ χ ∷ ο ∷ σ ∷ τ ∷ α ∷ σ ∷ ί ∷ α ∷ ι ∷ []) "Gal.5.20"
∷ word (α ∷ ἱ ∷ ρ ∷ έ ∷ σ ∷ ε ∷ ι ∷ ς ∷ []) "Gal.5.20"
∷ word (φ ∷ θ ∷ ό ∷ ν ∷ ο ∷ ι ∷ []) "Gal.5.21"
∷ word (μ ∷ έ ∷ θ ∷ α ∷ ι ∷ []) "Gal.5.21"
∷ word (κ ∷ ῶ ∷ μ ∷ ο ∷ ι ∷ []) "Gal.5.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.21"
∷ word (τ ∷ ὰ ∷ []) "Gal.5.21"
∷ word (ὅ ∷ μ ∷ ο ∷ ι ∷ α ∷ []) "Gal.5.21"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ ι ∷ ς ∷ []) "Gal.5.21"
∷ word (ἃ ∷ []) "Gal.5.21"
∷ word (π ∷ ρ ∷ ο ∷ ∙λ ∷ έ ∷ γ ∷ ω ∷ []) "Gal.5.21"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.5.21"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "Gal.5.21"
∷ word (π ∷ ρ ∷ ο ∷ ε ∷ ῖ ∷ π ∷ ο ∷ ν ∷ []) "Gal.5.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.5.21"
∷ word (ο ∷ ἱ ∷ []) "Gal.5.21"
∷ word (τ ∷ ὰ ∷ []) "Gal.5.21"
∷ word (τ ∷ ο ∷ ι ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "Gal.5.21"
∷ word (π ∷ ρ ∷ ά ∷ σ ∷ σ ∷ ο ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.5.21"
∷ word (β ∷ α ∷ σ ∷ ι ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "Gal.5.21"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.5.21"
∷ word (ο ∷ ὐ ∷ []) "Gal.5.21"
∷ word (κ ∷ ∙λ ∷ η ∷ ρ ∷ ο ∷ ν ∷ ο ∷ μ ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.5.21"
∷ word (Ὁ ∷ []) "Gal.5.22"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.22"
∷ word (κ ∷ α ∷ ρ ∷ π ∷ ὸ ∷ ς ∷ []) "Gal.5.22"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.5.22"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ό ∷ ς ∷ []) "Gal.5.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.5.22"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "Gal.5.22"
∷ word (χ ∷ α ∷ ρ ∷ ά ∷ []) "Gal.5.22"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Gal.5.22"
∷ word (μ ∷ α ∷ κ ∷ ρ ∷ ο ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "Gal.5.22"
∷ word (χ ∷ ρ ∷ η ∷ σ ∷ τ ∷ ό ∷ τ ∷ η ∷ ς ∷ []) "Gal.5.22"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ω ∷ σ ∷ ύ ∷ ν ∷ η ∷ []) "Gal.5.22"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "Gal.5.22"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ ς ∷ []) "Gal.5.23"
∷ word (ἐ ∷ γ ∷ κ ∷ ρ ∷ ά ∷ τ ∷ ε ∷ ι ∷ α ∷ []) "Gal.5.23"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "Gal.5.23"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "Gal.5.23"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ύ ∷ τ ∷ ω ∷ ν ∷ []) "Gal.5.23"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.5.23"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.5.23"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ς ∷ []) "Gal.5.23"
∷ word (ο ∷ ἱ ∷ []) "Gal.5.24"
∷ word (δ ∷ ὲ ∷ []) "Gal.5.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.5.24"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.5.24"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.5.24"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Gal.5.24"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ ύ ∷ ρ ∷ ω ∷ σ ∷ α ∷ ν ∷ []) "Gal.5.24"
∷ word (σ ∷ ὺ ∷ ν ∷ []) "Gal.5.24"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.5.24"
∷ word (π ∷ α ∷ θ ∷ ή ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Gal.5.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.24"
∷ word (τ ∷ α ∷ ῖ ∷ ς ∷ []) "Gal.5.24"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ ι ∷ ς ∷ []) "Gal.5.24"
∷ word (ε ∷ ἰ ∷ []) "Gal.5.25"
∷ word (ζ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.5.25"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.5.25"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.5.25"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.5.25"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.5.25"
∷ word (μ ∷ ὴ ∷ []) "Gal.5.26"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.5.26"
∷ word (κ ∷ ε ∷ ν ∷ ό ∷ δ ∷ ο ∷ ξ ∷ ο ∷ ι ∷ []) "Gal.5.26"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "Gal.5.26"
∷ word (π ∷ ρ ∷ ο ∷ κ ∷ α ∷ ∙λ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Gal.5.26"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ ι ∷ ς ∷ []) "Gal.5.26"
∷ word (φ ∷ θ ∷ ο ∷ ν ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.5.26"
∷ word (Ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.6.1"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.6.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.1"
∷ word (π ∷ ρ ∷ ο ∷ ∙λ ∷ η ∷ μ ∷ φ ∷ θ ∷ ῇ ∷ []) "Gal.6.1"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Gal.6.1"
∷ word (ἔ ∷ ν ∷ []) "Gal.6.1"
∷ word (τ ∷ ι ∷ ν ∷ ι ∷ []) "Gal.6.1"
∷ word (π ∷ α ∷ ρ ∷ α ∷ π ∷ τ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.6.1"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "Gal.6.1"
∷ word (ο ∷ ἱ ∷ []) "Gal.6.1"
∷ word (π ∷ ν ∷ ε ∷ υ ∷ μ ∷ α ∷ τ ∷ ι ∷ κ ∷ ο ∷ ὶ ∷ []) "Gal.6.1"
∷ word (κ ∷ α ∷ τ ∷ α ∷ ρ ∷ τ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.6.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.6.1"
∷ word (τ ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ ν ∷ []) "Gal.6.1"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.1"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "Gal.6.1"
∷ word (π ∷ ρ ∷ α ∷ ΰ ∷ τ ∷ η ∷ τ ∷ ο ∷ ς ∷ []) "Gal.6.1"
∷ word (σ ∷ κ ∷ ο ∷ π ∷ ῶ ∷ ν ∷ []) "Gal.6.1"
∷ word (σ ∷ ε ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Gal.6.1"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.1"
∷ word (σ ∷ ὺ ∷ []) "Gal.6.1"
∷ word (π ∷ ε ∷ ι ∷ ρ ∷ α ∷ σ ∷ θ ∷ ῇ ∷ ς ∷ []) "Gal.6.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "Gal.6.2"
∷ word (τ ∷ ὰ ∷ []) "Gal.6.2"
∷ word (β ∷ ά ∷ ρ ∷ η ∷ []) "Gal.6.2"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.6.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.2"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "Gal.6.2"
∷ word (ἀ ∷ ν ∷ α ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ώ ∷ σ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.6.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.6.2"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.6.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.2"
∷ word (ε ∷ ἰ ∷ []) "Gal.6.3"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.3"
∷ word (δ ∷ ο ∷ κ ∷ ε ∷ ῖ ∷ []) "Gal.6.3"
∷ word (τ ∷ ι ∷ ς ∷ []) "Gal.6.3"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ί ∷ []) "Gal.6.3"
∷ word (τ ∷ ι ∷ []) "Gal.6.3"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ ν ∷ []) "Gal.6.3"
∷ word (ὤ ∷ ν ∷ []) "Gal.6.3"
∷ word (φ ∷ ρ ∷ ε ∷ ν ∷ α ∷ π ∷ α ∷ τ ∷ ᾷ ∷ []) "Gal.6.3"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ό ∷ ν ∷ []) "Gal.6.3"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.4"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.4"
∷ word (ἔ ∷ ρ ∷ γ ∷ ο ∷ ν ∷ []) "Gal.6.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.4"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ α ∷ ζ ∷ έ ∷ τ ∷ ω ∷ []) "Gal.6.4"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Gal.6.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.4"
∷ word (τ ∷ ό ∷ τ ∷ ε ∷ []) "Gal.6.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.6.4"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "Gal.6.4"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.6.4"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.4"
∷ word (κ ∷ α ∷ ύ ∷ χ ∷ η ∷ μ ∷ α ∷ []) "Gal.6.4"
∷ word (ἕ ∷ ξ ∷ ε ∷ ι ∷ []) "Gal.6.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "Gal.6.4"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.6.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.6.4"
∷ word (ἕ ∷ τ ∷ ε ∷ ρ ∷ ο ∷ ν ∷ []) "Gal.6.4"
∷ word (ἕ ∷ κ ∷ α ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "Gal.6.5"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.5"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.5"
∷ word (ἴ ∷ δ ∷ ι ∷ ο ∷ ν ∷ []) "Gal.6.5"
∷ word (φ ∷ ο ∷ ρ ∷ τ ∷ ί ∷ ο ∷ ν ∷ []) "Gal.6.5"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ σ ∷ ε ∷ ι ∷ []) "Gal.6.5"
∷ word (Κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ε ∷ ί ∷ τ ∷ ω ∷ []) "Gal.6.6"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.6"
∷ word (ὁ ∷ []) "Gal.6.6"
∷ word (κ ∷ α ∷ τ ∷ η ∷ χ ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "Gal.6.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.6.6"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "Gal.6.6"
∷ word (τ ∷ ῷ ∷ []) "Gal.6.6"
∷ word (κ ∷ α ∷ τ ∷ η ∷ χ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ι ∷ []) "Gal.6.6"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.6"
∷ word (π ∷ ᾶ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.6"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ο ∷ ῖ ∷ ς ∷ []) "Gal.6.6"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.7"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ᾶ ∷ σ ∷ θ ∷ ε ∷ []) "Gal.6.7"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "Gal.6.7"
∷ word (ο ∷ ὐ ∷ []) "Gal.6.7"
∷ word (μ ∷ υ ∷ κ ∷ τ ∷ η ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "Gal.6.7"
∷ word (ὃ ∷ []) "Gal.6.7"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.7"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "Gal.6.7"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ῃ ∷ []) "Gal.6.7"
∷ word (ἄ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ ς ∷ []) "Gal.6.7"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "Gal.6.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.7"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Gal.6.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "Gal.6.8"
∷ word (ὁ ∷ []) "Gal.6.8"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ω ∷ ν ∷ []) "Gal.6.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.6.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "Gal.6.8"
∷ word (σ ∷ ά ∷ ρ ∷ κ ∷ α ∷ []) "Gal.6.8"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.8"
∷ word (ἐ ∷ κ ∷ []) "Gal.6.8"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.6.8"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "Gal.6.8"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Gal.6.8"
∷ word (φ ∷ θ ∷ ο ∷ ρ ∷ ά ∷ ν ∷ []) "Gal.6.8"
∷ word (ὁ ∷ []) "Gal.6.8"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.8"
∷ word (σ ∷ π ∷ ε ∷ ί ∷ ρ ∷ ω ∷ ν ∷ []) "Gal.6.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "Gal.6.8"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.8"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "Gal.6.8"
∷ word (ἐ ∷ κ ∷ []) "Gal.6.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.8"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.6.8"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ι ∷ []) "Gal.6.8"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "Gal.6.8"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "Gal.6.8"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.9"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.9"
∷ word (κ ∷ α ∷ ∙λ ∷ ὸ ∷ ν ∷ []) "Gal.6.9"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "Gal.6.9"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.9"
∷ word (ἐ ∷ γ ∷ κ ∷ α ∷ κ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "Gal.6.9"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ῷ ∷ []) "Gal.6.9"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.9"
∷ word (ἰ ∷ δ ∷ ί ∷ ῳ ∷ []) "Gal.6.9"
∷ word (θ ∷ ε ∷ ρ ∷ ί ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Gal.6.9"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.9"
∷ word (ἐ ∷ κ ∷ ∙λ ∷ υ ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Gal.6.9"
∷ word (ἄ ∷ ρ ∷ α ∷ []) "Gal.6.10"
∷ word (ο ∷ ὖ ∷ ν ∷ []) "Gal.6.10"
∷ word (ὡ ∷ ς ∷ []) "Gal.6.10"
∷ word (κ ∷ α ∷ ι ∷ ρ ∷ ὸ ∷ ν ∷ []) "Gal.6.10"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "Gal.6.10"
∷ word (ἐ ∷ ρ ∷ γ ∷ α ∷ ζ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "Gal.6.10"
∷ word (τ ∷ ὸ ∷ []) "Gal.6.10"
∷ word (ἀ ∷ γ ∷ α ∷ θ ∷ ὸ ∷ ν ∷ []) "Gal.6.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.6.10"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ ς ∷ []) "Gal.6.10"
∷ word (μ ∷ ά ∷ ∙λ ∷ ι ∷ σ ∷ τ ∷ α ∷ []) "Gal.6.10"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.10"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "Gal.6.10"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.6.10"
∷ word (ο ∷ ἰ ∷ κ ∷ ε ∷ ί ∷ ο ∷ υ ∷ ς ∷ []) "Gal.6.10"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "Gal.6.10"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ε ∷ ω ∷ ς ∷ []) "Gal.6.10"
∷ word (Ἴ ∷ δ ∷ ε ∷ τ ∷ ε ∷ []) "Gal.6.11"
∷ word (π ∷ η ∷ ∙λ ∷ ί ∷ κ ∷ ο ∷ ι ∷ ς ∷ []) "Gal.6.11"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "Gal.6.11"
∷ word (γ ∷ ρ ∷ ά ∷ μ ∷ μ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.11"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "Gal.6.11"
∷ word (τ ∷ ῇ ∷ []) "Gal.6.11"
∷ word (ἐ ∷ μ ∷ ῇ ∷ []) "Gal.6.11"
∷ word (χ ∷ ε ∷ ι ∷ ρ ∷ ί ∷ []) "Gal.6.11"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Gal.6.12"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.12"
∷ word (ε ∷ ὐ ∷ π ∷ ρ ∷ ο ∷ σ ∷ ω ∷ π ∷ ῆ ∷ σ ∷ α ∷ ι ∷ []) "Gal.6.12"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.12"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ί ∷ []) "Gal.6.12"
∷ word (ο ∷ ὗ ∷ τ ∷ ο ∷ ι ∷ []) "Gal.6.12"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ κ ∷ ά ∷ ζ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.12"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.6.12"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ έ ∷ μ ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Gal.6.12"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "Gal.6.12"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.6.12"
∷ word (τ ∷ ῷ ∷ []) "Gal.6.12"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ῷ ∷ []) "Gal.6.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.12"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.12"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.12"
∷ word (δ ∷ ι ∷ ώ ∷ κ ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Gal.6.12"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "Gal.6.13"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.13"
∷ word (ο ∷ ἱ ∷ []) "Gal.6.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ε ∷ μ ∷ ν ∷ ό ∷ μ ∷ ε ∷ ν ∷ ο ∷ ι ∷ []) "Gal.6.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "Gal.6.13"
∷ word (ν ∷ ό ∷ μ ∷ ο ∷ ν ∷ []) "Gal.6.13"
∷ word (φ ∷ υ ∷ ∙λ ∷ ά ∷ σ ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.13"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.6.13"
∷ word (θ ∷ έ ∷ ∙λ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "Gal.6.13"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ έ ∷ μ ∷ ν ∷ ε ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Gal.6.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "Gal.6.13"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.13"
∷ word (τ ∷ ῇ ∷ []) "Gal.6.13"
∷ word (ὑ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "Gal.6.13"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "Gal.6.13"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ή ∷ σ ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "Gal.6.13"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.6.14"
∷ word (δ ∷ ὲ ∷ []) "Gal.6.14"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.14"
∷ word (γ ∷ έ ∷ ν ∷ ο ∷ ι ∷ τ ∷ ο ∷ []) "Gal.6.14"
∷ word (κ ∷ α ∷ υ ∷ χ ∷ ᾶ ∷ σ ∷ θ ∷ α ∷ ι ∷ []) "Gal.6.14"
∷ word (ε ∷ ἰ ∷ []) "Gal.6.14"
∷ word (μ ∷ ὴ ∷ []) "Gal.6.14"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.14"
∷ word (τ ∷ ῷ ∷ []) "Gal.6.14"
∷ word (σ ∷ τ ∷ α ∷ υ ∷ ρ ∷ ῷ ∷ []) "Gal.6.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.14"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.6.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.6.14"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.6.14"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.14"
∷ word (δ ∷ ι ∷ []) "Gal.6.14"
∷ word (ο ∷ ὗ ∷ []) "Gal.6.14"
∷ word (ἐ ∷ μ ∷ ο ∷ ὶ ∷ []) "Gal.6.14"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "Gal.6.14"
∷ word (ἐ ∷ σ ∷ τ ∷ α ∷ ύ ∷ ρ ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "Gal.6.14"
∷ word (κ ∷ ἀ ∷ γ ∷ ὼ ∷ []) "Gal.6.14"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "Gal.6.14"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Gal.6.15"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.15"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ τ ∷ ο ∷ μ ∷ ή ∷ []) "Gal.6.15"
∷ word (τ ∷ ί ∷ []) "Gal.6.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "Gal.6.15"
∷ word (ο ∷ ὔ ∷ τ ∷ ε ∷ []) "Gal.6.15"
∷ word (ἀ ∷ κ ∷ ρ ∷ ο ∷ β ∷ υ ∷ σ ∷ τ ∷ ί ∷ α ∷ []) "Gal.6.15"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "Gal.6.15"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὴ ∷ []) "Gal.6.15"
∷ word (κ ∷ τ ∷ ί ∷ σ ∷ ι ∷ ς ∷ []) "Gal.6.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.16"
∷ word (ὅ ∷ σ ∷ ο ∷ ι ∷ []) "Gal.6.16"
∷ word (τ ∷ ῷ ∷ []) "Gal.6.16"
∷ word (κ ∷ α ∷ ν ∷ ό ∷ ν ∷ ι ∷ []) "Gal.6.16"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "Gal.6.16"
∷ word (σ ∷ τ ∷ ο ∷ ι ∷ χ ∷ ή ∷ σ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "Gal.6.16"
∷ word (ε ∷ ἰ ∷ ρ ∷ ή ∷ ν ∷ η ∷ []) "Gal.6.16"
∷ word (ἐ ∷ π ∷ []) "Gal.6.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "Gal.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.16"
∷ word (ἔ ∷ ∙λ ∷ ε ∷ ο ∷ ς ∷ []) "Gal.6.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "Gal.6.16"
∷ word (ἐ ∷ π ∷ ὶ ∷ []) "Gal.6.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "Gal.6.16"
∷ word (Ἰ ∷ σ ∷ ρ ∷ α ∷ ὴ ∷ ∙λ ∷ []) "Gal.6.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.16"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "Gal.6.16"
∷ word (Τ ∷ ο ∷ ῦ ∷ []) "Gal.6.17"
∷ word (∙λ ∷ ο ∷ ι ∷ π ∷ ο ∷ ῦ ∷ []) "Gal.6.17"
∷ word (κ ∷ ό ∷ π ∷ ο ∷ υ ∷ ς ∷ []) "Gal.6.17"
∷ word (μ ∷ ο ∷ ι ∷ []) "Gal.6.17"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "Gal.6.17"
∷ word (π ∷ α ∷ ρ ∷ ε ∷ χ ∷ έ ∷ τ ∷ ω ∷ []) "Gal.6.17"
∷ word (ἐ ∷ γ ∷ ὼ ∷ []) "Gal.6.17"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "Gal.6.17"
∷ word (τ ∷ ὰ ∷ []) "Gal.6.17"
∷ word (σ ∷ τ ∷ ί ∷ γ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "Gal.6.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.17"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.6.17"
∷ word (ἐ ∷ ν ∷ []) "Gal.6.17"
∷ word (τ ∷ ῷ ∷ []) "Gal.6.17"
∷ word (σ ∷ ώ ∷ μ ∷ α ∷ τ ∷ ί ∷ []) "Gal.6.17"
∷ word (μ ∷ ο ∷ υ ∷ []) "Gal.6.17"
∷ word (β ∷ α ∷ σ ∷ τ ∷ ά ∷ ζ ∷ ω ∷ []) "Gal.6.17"
∷ word (Ἡ ∷ []) "Gal.6.18"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ς ∷ []) "Gal.6.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.18"
∷ word (κ ∷ υ ∷ ρ ∷ ί ∷ ο ∷ υ ∷ []) "Gal.6.18"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.6.18"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "Gal.6.18"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "Gal.6.18"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "Gal.6.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "Gal.6.18"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "Gal.6.18"
∷ word (ὑ ∷ μ ∷ ῶ ∷ ν ∷ []) "Gal.6.18"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "Gal.6.18"
∷ word (ἀ ∷ μ ∷ ή ∷ ν ∷ []) "Gal.6.18"
∷ []
|
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{-# OPTIONS --cubical --safe #-}
module Cubical.Modalities.Modality where
{-
translated from
https://github.com/HoTT/HoTT-Agda/blob/master/core/lib/types/Modality.agda
-}
open import Cubical.Core.Everything
open import Cubical.Foundations.Everything
record Modality ℓ : Type (ℓ-suc ℓ) where
field
isModal : Type ℓ → Type ℓ
isModalIsProp : {A : Type ℓ} → isProp (isModal A)
◯ : Type ℓ → Type ℓ -- \ciO
◯-isModal : {A : Type ℓ} → isModal (◯ A)
η : {A : Type ℓ} → A → ◯ A
◯-elim : {A : Type ℓ} {B : ◯ A → Type ℓ}
(B-modal : (x : ◯ A) → isModal (B x))
→ ((x : A) → (B (η x))) → ((x : ◯ A) → B x)
◯-elim-β : {A : Type ℓ} {B : ◯ A → Type ℓ}
(B-modal : (x : ◯ A) → isModal (B x)) (f : (x : A) → (B (η x)))
→ (a : A) → ◯-elim B-modal f (η a) ≡ f a
◯-=-isModal : {A : Type ℓ} (x y : ◯ A) → isModal (x ≡ y)
◯-Types : Type (ℓ-suc ℓ)
◯-Types = Σ[ A ∈ Type ℓ ] isModal A
{- elimination rules -}
module ◯Elim {A : Type ℓ} {B : ◯ A → Type ℓ}
(B-modal : (x : ◯ A) → isModal (B x)) (η* : (x : A) → (B (η x))) where
f : (x : ◯ A) → B x
f = ◯-elim B-modal η*
η-β : (a : A) → ◯-elim B-modal η* (η a) ≡ η* a
η-β = ◯-elim-β B-modal η*
module ◯Rec {A : Type ℓ} {B : Type ℓ}
(B-modal : isModal B) (η* : A → B)
= ◯Elim (λ _ → B-modal) η*
◯-rec = ◯Rec.f
◯-rec-β = ◯Rec.η-β
{- functoriality -}
module ◯Fmap {A B : Type ℓ} (f : A → B) =
◯Rec ◯-isModal (η ∘ f)
◯-map = ◯Fmap.f
◯-map-β = ◯Fmap.η-β
◯-preservesEquiv : {A B : Type ℓ} (f : A → B) → isEquiv f → isEquiv (◯-map f)
◯-preservesEquiv f f-ise = isoToIsEquiv (iso _ (◯-map inv) to-from from-to) where
open Iso (equivToIso (f , f-ise))
abstract
to-from : ∀ ◯b → ◯-map f (◯-map inv ◯b) ≡ ◯b
to-from = ◯-elim
(λ ◯b → ◯-=-isModal (◯-map f (◯-map inv ◯b)) ◯b)
(λ b → cong (◯-map f) (◯-map-β inv b) ∙ ◯-map-β f (inv b) ∙ cong η (rightInv b))
from-to : ∀ ◯a → ◯-map inv (◯-map f ◯a) ≡ ◯a
from-to = ◯-elim
(λ ◯a → ◯-=-isModal (◯-map inv (◯-map f ◯a)) ◯a)
(λ a → cong (◯-map inv) (◯-map-β f a) ∙ ◯-map-β inv (f a) ∙ cong η (leftInv a))
◯-equiv : {A B : Type ℓ} → A ≃ B → ◯ A ≃ ◯ B
◯-equiv (f , f-ise) = ◯-map f , ◯-preservesEquiv f f-ise
{- equivalences preserve being modal -}
equivPreservesIsModal : {A B : Type ℓ} → A ≃ B → isModal A → isModal B
equivPreservesIsModal eq = fst (pathToEquiv (cong isModal (ua eq)))
{- modal types and [η] being an equivalence -}
isModalToIsEquiv : {A : Type ℓ} → isModal A → isEquiv (η {A})
isModalToIsEquiv {A} w = isoToIsEquiv (iso (η {A}) η-inv inv-l inv-r)
where η-inv : ◯ A → A
η-inv = ◯-rec w (idfun A)
abstract
inv-r : (a : A) → η-inv (η a) ≡ a
inv-r = ◯-rec-β w (idfun A)
inv-l : (a : ◯ A) → η (η-inv a) ≡ a
inv-l = ◯-elim (λ a₀ → ◯-=-isModal _ _)
(λ a₀ → cong η (inv-r a₀))
abstract
isEquivToIsModal : {A : Type ℓ} → isEquiv (η {A}) → isModal A
isEquivToIsModal {A} eq = equivPreservesIsModal (invEquiv (η , eq)) ◯-isModal
retractIsModal : {A B : Type ℓ} (w : isModal A)
(f : A → B) (g : B → A) (r : (b : B) → f (g b) ≡ b) →
isModal B
retractIsModal {A} {B} w f g r =
isEquivToIsModal
(isoToIsEquiv (iso η η-inv inv-l inv-r))
where η-inv : ◯ B → B
η-inv = f ∘ (◯-rec w g)
inv-r : (b : B) → η-inv (η b) ≡ b
inv-r b = cong f (◯-rec-β w g b) ∙ r b
inv-l : (b : ◯ B) → η (η-inv b) ≡ b
inv-l = ◯-elim (λ b → ◯-=-isModal _ _) (λ b → cong η (inv-r b))
{- function types with modal codomain are modal -}
abstract
Π-isModal : {A : Type ℓ} {B : A → Type ℓ}
(w : (a : A) → isModal (B a)) → isModal ((x : A) → B x)
Π-isModal {A} {B} w = retractIsModal {◯ _} {(x : A) → B x} ◯-isModal η-inv η r
where η-inv : ◯ ((x : A) → B x) → (x : A) → B x
η-inv φ' a = ◯-rec (w a) (λ φ → φ a) φ'
r : (φ : (x : A) → B x) → η-inv (η φ) ≡ φ
r φ = funExt (λ a → ◯-rec-β (w a) (λ φ₀ → φ₀ a) φ)
→-isModal : {A B : Type ℓ} → isModal B → isModal (A → B)
→-isModal w = Π-isModal (λ _ → w)
{- sigma types of a modal dependent type with modal base are modal -}
abstract
Σ-isModal : {A : Type ℓ} (B : A → Type ℓ)
→ isModal A → ((a : A) → isModal (B a))
→ isModal (Σ A B)
Σ-isModal {A} B A-modal B-modal =
retractIsModal {◯ (Σ A B)} {Σ A B} ◯-isModal η-inv η r
where h : ◯ (Σ A B) → A
h = ◯-rec A-modal fst
h-β : (x : Σ A B) → h (η x) ≡ fst x
h-β = ◯-rec-β A-modal fst
f : (j : I) → (x : Σ A B) → B (h-β x j)
f j x = transp (λ i → B (h-β x ((~ i) ∨ j))) j (snd x)
k : (y : ◯ (Σ A B)) → B (h y)
k = ◯-elim (B-modal ∘ h) (f i0)
η-inv : ◯ (Σ A B) → Σ A B
η-inv y = h y , k y
p : (x : Σ A B) → k (η x) ≡ f i0 x
p = ◯-elim-β (B-modal ∘ h) (f i0)
almost : (x : Σ A B) → (h (η x) , f i0 x) ≡ x
almost x i = h-β x i , f i x
r : (x : Σ A B) → η-inv (η x) ≡ x
r x = (λ i → h (η x) , p x i) ∙ (almost x)
|
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{- MJ where variable declarations have been hoisted to the top of a block -}
module CF.Transform.Hoist where
open import Level
open import Function using (_∘_)
open import Data.List
open import Data.List.Properties
open import Data.Unit
open import Data.Product
open import Relation.Unary hiding (_⊢_)
open import Relation.Binary.PropositionalEquality hiding ([_])
open import Relation.Ternary.Core
open import Relation.Ternary.Structures
open import Relation.Ternary.Structures.Syntax
open import Relation.Ternary.Monad
open import Relation.Ternary.Monad.Weakening
open import Relation.Ternary.Structures.Syntax
open import CF.Types
open import CF.Contexts.Lexical
open import CF.Syntax as Src hiding (Stmt; Block; Statement; var) public
open import CF.Syntax.Hoisted as Hoisted
open import Relation.Ternary.Construct.List.Overlapping Ty
open import Relation.Ternary.Data.Bigstar
pattern _⍮⟨_⟩_ s σ b = cons (s ∙⟨ σ ⟩ b)
hoist-binder : ∀ {P : Pred Ctx 0ℓ} {Γ} → ∀[ (Γ ⊢ P) ⇒ ◇ (Vars Γ ✴ P) ]
hoist-binder px = pack (⊢-zip (∙-copy _) (binders ∙⟨ ∙-idˡ ⟩ px))
-- A typed hoisting transformation for statement blocks
{-# TERMINATING #-}
mutual
{- Hoist local variables from blocks -}
hoist : ∀[ Src.Block r ⇒ ◇ (Block r) ]
hoist Src.emp = do
return nil
hoist (ss Src.⍮⟨ σ ⟩ b) = do
b ∙⟨ σ ⟩ s ← translate ss &⟨ Src.Block _ # ∙-comm σ ⟩ b
s ∙⟨ σ ⟩ b ← hoist b &⟨ Hoisted.Stmt _ # ∙-comm σ ⟩ s
return (s ⍮⟨ σ ⟩ b)
hoist (e Src.≔⟨ σ ⟩ Γ⊢b) = do
e×v ∙⟨ σ ⟩ b ← ✴-assocₗ ⟨$⟩ (hoist-binder Γ⊢b &⟨ Src.Exp _ # σ ⟩ e)
(e ∙⟨ σ₁ ⟩ v) ∙⟨ σ₂ ⟩ b' ← hoist b &⟨ _ ✴ _ # σ ⟩ e×v
return (Hoisted.asgn (v ∙⟨ ∙-comm σ₁ ⟩ e) ⍮⟨ σ₂ ⟩ b')
{- Hoist local variables from statements -}
translate : ∀[ Src.Stmt r ⇒ ◇ (Stmt r) ]
translate (Src.asgn x) = do
return (Hoisted.asgn x)
translate (Src.run e) = do
return (Hoisted.run e)
translate (Src.while (e ∙⟨ σ ⟩ body)) = do
e ∙⟨ σ ⟩ body' ← translate body &⟨ Src.Exp _ # σ ⟩ e
return (Hoisted.while (e ∙⟨ σ ⟩ body'))
translate (Src.ifthenelse e×s₁×s₂) = do
let (s₁ ∙⟨ σ ⟩ s₂×e) = ✴-rotateₗ e×s₁×s₂
s₂ ∙⟨ σ ⟩ e×s₁ ← ✴-assocᵣ ⟨$⟩ (translate s₁ &⟨ _ ✴ _ # ∙-comm σ ⟩ s₂×e)
e×s₁×s₂ ← ✴-assocᵣ ⟨$⟩ (translate s₂ &⟨ _ ✴ _ # ∙-comm σ ⟩ e×s₁)
return (Hoisted.ifthenelse e×s₁×s₂)
translate (Src.block bl) = do
bl' ← hoist bl
return (Hoisted.block bl')
|
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{-# OPTIONS --cubical --safe #-}
module Cardinality.Infinite.Split where
open import Prelude
open import Data.List.Kleene
open import Data.Fin
import Data.Nat as ℕ
open import Data.Nat using (_+_)
open import Cubical.Data.Sigma.Properties
open import Cubical.Foundations.Prelude using (J)
import Data.List.Kleene.Membership as Kleene
open import Codata.Stream
open import Data.Sigma.Properties
private
variable
u : Level
U : A → Type u
s : Level
S : ℕ → Type s
ℰ! : Type a → Type a
ℰ! A = Σ[ xs ⦂ Stream A ] Π[ x ⦂ A ] x ∈ xs
ℰ!⇔ℕ↠! : ℰ! A ≡ (ℕ ↠! A)
ℰ!⇔ℕ↠! = refl
infixl 6 _*_ _*⋆_[_]
_*_ : Stream A → (∀ x → Stream (U x)) → Stream (Σ A U ⁺)
_*⋆_[_] : Stream A → (∀ x → Stream (U x)) → Stream (Σ A U ⋆)
cantor : Stream A → (∀ x → Stream (U x)) → Stream (Σ A U)
cantor xs ys = concat (xs * ys)
xs *⋆ ys [ zero ] = []
xs *⋆ ys [ suc n ] = ∹ (xs * ys) n
(xs * ys) n .head = x , ys x n where x = xs 0
(xs * ys) n .tail = (xs ∘ suc) *⋆ ys [ n ]
*-cover : ∀ (x : A) xs (y : U x) (ys : ∀ x → Stream (U x)) → x ∈ xs → y ∈ ys x → (x , y) ∈² xs * ys
*-cover {U = U} x xs y ys (n , x∈xs) (m , y∈ys) = (n + m) , lemma xs n x∈xs
where
lemma : ∀ xs n → xs n ≡ x → (x , y) Kleene.∈⁺ (xs * ys) (n + m)
lemma xs zero x∈xs .fst = f0
lemma xs zero x∈xs .snd i .fst = x∈xs i
lemma xs zero x∈xs .snd i .snd = J (λ z z≡ → PathP (λ j → U (sym z≡ j)) (ys z m) y) y∈ys (sym x∈xs) i
lemma xs (suc n) x∈xs = let i , p = lemma (xs ∘ suc) n x∈xs in fs i , p
_|Σ|_ : ℰ! A → (∀ x → ℰ! (U x)) → ℰ! (Σ A U)
(xs |Σ| ys) .fst = cantor (xs .fst) (fst ∘ ys)
(xs |Σ| ys) .snd (x , y) =
concat-∈
(x , y)
(xs .fst * (fst ∘ ys))
(*-cover x (xs .fst) y (fst ∘ ys) (xs .snd x) (ys x .snd y))
open import Data.Nat using (_+_)
infixl 6 _∔_
_∔_ : ℕ → ℕ → ℕ
zero ∔ m = m
suc n ∔ m = n ∔ suc m
∔-suc : ∀ n m → suc n ∔ m ≡ suc (n ∔ m)
∔-suc zero m = refl
∔-suc (suc n) m = ∔-suc n (suc m)
∔0 : ∀ n → n ∔ zero ≡ n
∔0 zero = refl
∔0 (suc n) = ∔-suc n 0 ; cong suc (∔0 n)
module _ (xs : Stream A) where
mutual
star⁺ : (A ⋆ → B) → B ⋆ → Stream (B ⁺)
star⁺ k ks zero = k [] & ks
star⁺ k ks (suc i) = plus⁺ zero (k ∘ ∹_) ks i
plus⋆ : ℕ → (A ⁺ → B) → B ⋆ → Stream (B ⋆)
plus⋆ n k ks zero = ks
plus⋆ n k ks (suc i) = ∹ plus⁺ n k ks i
plus⁺ : ℕ → (A ⁺ → B) → B ⋆ → Stream (B ⁺)
plus⁺ n k ks i = star⁺ (k ∘ (xs n &_)) (plus⋆ (suc n) k ks i) i
star : Stream (A ⋆)
star = concat (star⁺ id [])
plus : Stream (A ⁺)
plus = concat (plus⁺ zero id [])
module _ (cover : ∀ x → x ∈ xs) where
dist : A ⋆ → ℕ
dist = foldr⋆ (λ y ys → suc (cover y .fst + ys)) zero
mutual
star⁺-cover : (k : A ⋆ → B) → (ks : B ⋆) → ∀ x → k x Kleene.∈⁺ star⁺ k ks (dist x)
star⁺-cover k ks [] = f0 , refl
star⁺-cover k ks (∹ x ) = plus⁺-cover (k ∘ ∹_) ks x
plus⁺-cover : ∀ (k : A ⁺ → B) ks → ∀ x → k x Kleene.∈⁺ plus⁺ zero k ks (cover (head x) .fst + dist (tail x))
plus⁺-cover k ks (x & xxs) =
let n , p = cover x
m , q = plus⁺-dist n (k ) ks xxs
z = m , q ; cong (k ∘ (_& xxs)) p
in plus⁺-shift zero (dist xxs) n k ks (x & xxs) (subst (λ s → k (x & xxs) Kleene.∈⁺ plus⁺ s k ks (dist xxs)) (sym (∔0 (cover x .fst))) z)
plus⁺-dist : ∀ n (k : A ⁺ → B) ks → ∀ xxs → k (xs n & xxs) Kleene.∈⁺ plus⁺ n k ks (dist xxs)
plus⁺-dist n k ks xxs = star⁺-cover (k ∘ _&_ (xs n)) (plus⋆ (suc n) k ks (dist xxs)) xxs
plus⁺-run : ∀ n i (k : A ⁺ → B) ks → ∀ xxs → xxs Kleene.∈⋆ ks → xxs Kleene.∈⁺ plus⁺ n k ks i
plus⁺-run n zero k ks xxs (m , p) = fs m , p
plus⁺-run n (suc i) k ks xxs =
plus⁺-run zero i (k ∘ (xs n &_) ∘ ∹_) (plus⋆ (suc n) k ks (suc i)) xxs ∘
plus⁺-run (suc n) i k ks xxs
plus⁺-shift : ∀ i d n (k : A ⁺ → B) (ks : B ⋆) → ∀ xxs → k xxs Kleene.∈⁺ plus⁺ (n ∔ i) k ks d → k xxs Kleene.∈⁺ plus⁺ i k ks (n + d)
plus⁺-shift i d zero k ks xxs p = p
plus⁺-shift i d (suc n) k ks xxs p = plus⁺-run zero (n + d) (λ z → k (xs i & ∹ z)) (∹ plus⁺ (suc i) k ks (n + d)) (k xxs) (plus⁺-shift (suc i) d n k ks xxs p)
star-cover : ∀ x → x ∈ star
star-cover x = concat-∈ x (star⁺ id []) (dist x , star⁺-cover id [] x)
plus-cover : ∀ x → x ∈ plus
plus-cover x = concat-∈ x (plus⁺ zero id []) (cover (head x) .fst + dist (tail x) , plus⁺-cover id [] x)
|star| : ℰ! A → ℰ! (A ⋆)
|star| xs .fst = star (xs .fst)
|star| xs .snd = star-cover (xs .fst) (xs .snd)
|plus| : ℰ! A → ℰ! (A ⁺)
|plus| xs .fst = plus (xs .fst)
|plus| xs .snd = plus-cover (xs .fst) (xs .snd)
open import Data.Bool using (not; bool)
x≢¬x : ∀ x → x ≢ not x
x≢¬x false p = subst (bool ⊤ ⊥) p tt
x≢¬x true p = subst (bool ⊥ ⊤) p tt
cantor-diag : ¬ (ℰ! (Stream Bool))
cantor-diag (sup , cov) = let n , p = cov (λ n → not (sup n n)) in x≢¬x _ (cong (_$ n) p)
ℰ : Type a → Type a
ℰ A = ∥ ℰ! A ∥
open import Function.Surjective.Properties
open import Data.Nat.Properties using (discreteℕ)
open import HITs.PropositionalTruncation
open import Relation.Nullary.Discrete.Properties
ℰ!⇒Discrete : ℰ! A → Discrete A
ℰ!⇒Discrete xs = Discrete-distrib-surj xs discreteℕ
ℰ⇒Discrete : ℰ A → Discrete A
ℰ⇒Discrete = rec isPropDiscrete ℰ!⇒Discrete
|
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{-# OPTIONS --enable-prop #-}
True : Prop
True = {P : Prop} → P → P
|
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module Issue784.Transformer where
open import Data.List using (List; []; _∷_; _++_; [_]; filter) renaming (map to mapL)
import Level
open import Issue784.Values
open import Issue784.Context
Transformer : ∀ {ℓ} → NonRepetitiveTypes ℓ → NonRepetitiveTypes ℓ → Set (Level.suc ℓ)
Transformer {ℓ} (t-in , nr-in) (t-out , nr-out) =
(v : Context ℓ) → NonRepetitiveContext v → t-in ⊆ signature v → NonRepetitive (ctxnames v ∖ names t-in ∪ names t-out) →
Σ[ w ∈ Context ℓ ] signature w ≋ signature v ∖∖ names t-in ∪ t-out
pipe : ∀ {ℓ} {t-in₁ t-out₁ t-in₂ t-out₂ : Types ℓ} {nr-in₁ nr-out₁ nr-in₂ nr-out₂} →
Transformer (t-in₁ , nr-in₁) (t-out₁ , nr-out₁) →
Transformer (t-in₂ , nr-in₂) (t-out₂ , nr-out₂) →
let n-out₁ = names t-out₁
n-in₂ = names t-in₂
t-in = t-in₁ ∪ (t-in₂ ∖∖ n-out₁)
t-out = t-out₁ ∖∖ n-in₂ ∪ t-out₂
in
filter-∈ t-out₁ n-in₂ ≋ filter-∈ t-in₂ n-out₁ →
(nr-in : NonRepetitiveNames t-in) (nr-out : NonRepetitiveNames t-out) →
Transformer (t-in , nr-in) (t-out , nr-out)
pipe {ℓ} {t-in₁} {t-out₁} {t-in₂} {t-out₂} {nr-in₁} {nr-out₁} {nr-in₂} {nr-out₂} tr₁ tr₂ pr-t nr-in nr-out = tr where
n-in₁ = names t-in₁
n-out₁ = names t-out₁
n-in₂ = names t-in₂
n-out₂ = names t-out₂
t-in = t-in₁ ∪ (t-in₂ ∖∖ n-out₁)
t-out = (t-out₁ ∖∖ n-in₂) ∪ t-out₂
tr : Transformer (t-in , nr-in) (t-out , nr-out)
tr ctx nr-v t-ì⊆v nr-ò = context w , w≋out where
v = Context.get ctx
n-in = names t-in
v̀ = filter-∈ v n-in
nr-v̀ : NonRepetitiveNames v̀
nr-v̀ = nr-x⇒nr-x∩y nr-v n-in
v̀≋i : types v̀ ≋ t-in
v̀≋i = ≋-sym $ ≋-trans p₁ p₂ where
p₁ : t-in ≋ filter-∈ (types v) n-in
p₁ = t₁⊆t₂⇒t₁≋f∈-t₂-nt₁ nr-in (≡-elim′ NonRepetitive (≡-sym $ n-types v) nr-v) t-ì⊆v
p₂ : filter-∈ (types v) n-in ≋ types v̀
p₂ = ≡⇒≋ $ ≡-sym $ t-filter-∈ v n-in
-- transformer₁
i₁⊆v̀ : t-in₁ ⊆ types v̀
i₁⊆v̀ = x⊆y≋z (x⊆x∪y t-in₁ (t-in₂ ∖∖ n-out₁)) (≋-sym v̀≋i)
v̀∖i₁∪o₁≋i₂∖o₁∪o₁ : types v̀ ∖∖ n-in₁ ∪ t-out₁ ≋ t-in₂ ∖∖ n-out₁ ∪ t-out₁
v̀∖i₁∪o₁≋i₂∖o₁∪o₁ = x≋x̀⇒x∪y≋x̀∪y p₂ t-out₁ where
p₁ : NonRepetitiveNames (types v̀)
p₁ = nr-x≋y (≡⇒≋ $ ≡-sym $ n-types v̀) nr-v̀
p₂ : types v̀ ∖∖ n-in₁ ≋ t-in₂ ∖∖ n-out₁
p₂ = t≋t₁∪t₂⇒t∖t₁≋t₂ p₁ t-in₁ (t-in₂ ∖∖ n-out₁) v̀≋i
n-v̀∖i₁∪o₁≋i₂∖o₁∪o₁ : names v̀ ∖ n-in₁ ∪ n-out₁ ≋ n-in₂ ∖ n-out₁ ∪ n-out₁
n-v̀∖i₁∪o₁≋i₂∖o₁∪o₁ = x≋x̀⇒x∪y≋x̀∪y p₃ n-out₁ where
p₁ : names v̀ ≋ n-in
p₁ = ≋-trans (≡⇒≋ $ n-filter-∈ v n-in) (y⊆x⇒x∩y≋y nr-in nr-v (≡-elim′ (λ x → n-in ⊆ x) (n-types v) (x⊆y⇒nx⊆ny t-ì⊆v)))
p₂ : n-in ≋ n-in₁ ∪ (n-in₂ ∖ n-out₁)
p₂ = ≡⇒≋ $ ≡-trans (n-x∪y t-in₁ $ t-in₂ ∖∖ n-out₁) (≡-cong (λ x → n-in₁ ∪ x) (n-x∖y t-in₂ n-out₁))
p₃ : names v̀ ∖ n-in₁ ≋ n-in₂ ∖ n-out₁
p₃ = x≋y∪z⇒x∖y≋z (nr-x≋y (≋-sym p₁) nr-in) n-in₁ (n-in₂ ∖ n-out₁) (≋-trans p₁ p₂)
nr-v̀∖i₁∪o₁ : NonRepetitive (names v̀ ∖ n-in₁ ∪ n-out₁)
nr-v̀∖i₁∪o₁ = nr-x≋y (≋-sym n-v̀∖i₁∪o₁≋i₂∖o₁∪o₁) p₁ where
p₁ : NonRepetitive (n-in₂ ∖ n-out₁ ∪ n-out₁)
p₁ = nr-x∖y∪y nr-in₂ nr-out₁
w-all₁ : Σ[ w ∈ Context ℓ ] signature w ≋ signature (context v̀) ∖∖ names t-in₁ ∪ t-out₁
w-all₁ = tr₁ (context v̀) nr-v̀ i₁⊆v̀ nr-v̀∖i₁∪o₁
w₁ = Context.get $ proj₁ w-all₁
w₁≋v̀∖i₁∪o₁ = proj₂ w-all₁
-- transformer₂
n-w₁≋v̀∖i₁∪o₁ : names w₁ ≋ names v̀ ∖ names t-in₁ ∪ names t-out₁
n-w₁≋v̀∖i₁∪o₁ = ≡-elim p₆ p₅ where
p₁ : names (types w₁) ≋ names (types v̀ ∖∖ names t-in₁ ∪ t-out₁)
p₁ = n-x≋y w₁≋v̀∖i₁∪o₁
p₂ : names (types v̀ ∖∖ names t-in₁ ∪ t-out₁) ≡ names (types v̀ ∖∖ names t-in₁) ∪ names t-out₁
p₂ = n-x∪y (types v̀ ∖∖ names t-in₁) t-out₁
p₃ : names (types v̀ ∖∖ names t-in₁) ≡ names (types v̀) ∖ names t-in₁
p₃ = n-x∖y (types v̀) (names t-in₁)
p₄ : names (types w₁) ≋ names (types v̀ ∖∖ names t-in₁) ∪ names t-out₁
p₄ = ≡-elim′ (λ x → names (types w₁) ≋ x) p₂ p₁
p₅ : names (types w₁) ≋ names (types v̀) ∖ names t-in₁ ∪ names t-out₁
p₅ = ≡-elim′ (λ x → names (types w₁) ≋ x ∪ names t-out₁) p₃ p₄
p₆ : (names (types w₁) ≋ names (types v̀) ∖ names t-in₁ ∪ names t-out₁) ≡
(names w₁ ≋ names v̀ ∖ names t-in₁ ∪ names t-out₁)
p₆ = ≡-cong₂ (λ x y → x ≋ y ∖ names t-in₁ ∪ names t-out₁) (n-types w₁) (n-types v̀)
nr-w₁ : NonRepetitiveNames w₁
nr-w₁ = nr-x≋y (≋-sym n-w₁≋v̀∖i₁∪o₁) nr-v̀∖i₁∪o₁
i₂⊆w₁ : t-in₂ ⊆ types w₁
i₂⊆w₁ = x⊆y≋z (x≋y⊆z (≋-sym p₁) p₄) (≋-sym w₁≋v̀∖i₁∪o₁) where
p₁ : t-in₂ ≋ (t-in₂ ∖∖ n-out₁) ∪ filter-∈ t-in₂ n-out₁
p₁ = t≋t∖n∪t∩n t-in₂ n-out₁
p₂ : filter-∈ t-in₂ n-out₁ ⊆ t-out₁
p₂ = x≋y⊆z pr-t $ x∩y⊆x t-out₁ n-in₂
p₃ : types v̀ ∖∖ n-in₁ ≋ t-in₂ ∖∖ n-out₁
p₃ = t≋t₁∪t₂⇒t∖t₁≋t₂ (≡-elim′ NonRepetitive (≡-sym $ n-types v̀) nr-v̀) t-in₁ (t-in₂ ∖∖ n-out₁) v̀≋i
p₄ : (t-in₂ ∖∖ n-out₁) ∪ (filter-∈ t-in₂ n-out₁) ⊆ (types v̀ ∖∖ n-in₁) ∪ t-out₁
p₄ = x∪y⊆x̀∪ỳ (≋⇒⊆ $ ≋-sym p₃) p₂
w₁∖i₂∪o₂≋out : types w₁ ∖∖ n-in₂ ∪ t-out₂ ≋ t-out
w₁∖i₂∪o₂≋out = x≋x̀⇒x∪y≋x̀∪y p₄ t-out₂ where
p₁ : t-in₂ ∖∖ n-out₁ ∪ t-out₁ ≋ t-out₁ ∖∖ n-in₂ ∪ t-in₂
p₁ = ≋-sym $ x∖y∪y≋y∖x∪x t-out₁ t-in₂ pr-t
p₂ : (t-out₁ ∖∖ n-in₂) ∪ t-in₂ ≋ t-in₂ ∪ (t-out₁ ∖∖ n-in₂)
p₂ = ∪-sym (t-out₁ ∖∖ n-in₂) t-in₂
p₃ : types w₁ ≋ t-in₂ ∪ (t-out₁ ∖∖ n-in₂)
p₃ = ≋-trans w₁≋v̀∖i₁∪o₁ $ ≋-trans v̀∖i₁∪o₁≋i₂∖o₁∪o₁ $ ≋-trans p₁ p₂
p₄ : types w₁ ∖∖ n-in₂ ≋ t-out₁ ∖∖ n-in₂
p₄ = t≋t₁∪t₂⇒t∖t₁≋t₂ (nr-x≋y (≡⇒≋ $ ≡-sym $ n-types w₁) nr-w₁) t-in₂ (t-out₁ ∖∖ n-in₂) p₃
nr-w₁∖i₂∪o₂ : NonRepetitive (names w₁ ∖ n-in₂ ∪ n-out₂)
nr-w₁∖i₂∪o₂ = ≡-elim′ NonRepetitive p₄ p₅ where
p₁ : names (types w₁ ∖∖ n-in₂ ∪ t-out₂) ≡ names (types w₁ ∖∖ n-in₂) ∪ n-out₂
p₁ = n-x∪y (types w₁ ∖∖ n-in₂) t-out₂
p₂ : names (types w₁ ∖∖ n-in₂) ∪ n-out₂ ≡ (names (types w₁) ∖ n-in₂) ∪ n-out₂
p₂ = ≡-cong (λ x → x ∪ names t-out₂) (n-x∖y (types w₁) n-in₂)
p₃ : (names (types w₁) ∖ n-in₂) ∪ n-out₂ ≡ names w₁ ∖ n-in₂ ∪ n-out₂
p₃ = ≡-cong (λ x → x ∖ n-in₂ ∪ n-out₂) (n-types w₁)
p₄ : names (types w₁ ∖∖ n-in₂ ∪ t-out₂) ≡ names w₁ ∖ n-in₂ ∪ n-out₂
p₄ = ≡-trans p₁ $ ≡-trans p₂ p₃
p₅ : NonRepetitiveNames (types w₁ ∖∖ n-in₂ ∪ t-out₂)
p₅ = nr-x≋y (≋-sym $ n-x≋y w₁∖i₂∪o₂≋out) nr-out
w-all₂ : Σ[ w ∈ Context ℓ ] signature w ≋ signature (context w₁) ∖∖ names t-in₂ ∪ t-out₂
w-all₂ = tr₂ (context w₁) nr-w₁ i₂⊆w₁ nr-w₁∖i₂∪o₂
w₂ = Context.get $ proj₁ w-all₂
w₂≋out : types w₂ ≋ t-out
w₂≋out = ≋-trans (proj₂ w-all₂) w₁∖i₂∪o₂≋out
w = Values ℓ ∋ v ∖∖ n-in ∪ w₂
w≋out : types w ≋ types v ∖∖ n-in ∪ t-out
w≋out = ≋-trans (≡⇒≋ p₁) p₂ where
p₁ : types (v ∖∖ n-in ∪ w₂) ≡ types v ∖∖ n-in ∪ types w₂
p₁ = ≡-trans (t-x∪y (v ∖∖ n-in) w₂) (≡-cong (λ x → x ∪ types w₂) (t-x∖y v n-in))
p₂ : types v ∖∖ n-in ∪ types w₂ ≋ types v ∖∖ n-in ∪ t-out
p₂ = y≋ỳ⇒x∪y≋x∪ỳ (types v ∖∖ n-in) w₂≋out
Transformer! : ∀ {ℓ} (t-in : Types ℓ) (t-out : Types ℓ) {nr!-in : NonRepetitiveNames! t-in} {nr!-out : NonRepetitiveNames! t-out} → Set (Level.suc ℓ)
Transformer! t-in t-out {nr!-in = nr!-in} {nr!-out} = Transformer (t-in , s-nr!⇒nr nr!-in) (t-out , s-nr!⇒nr nr!-out)
infix 1 _:=_
_:=_ : ∀ {ℓ} {A : String → Set ℓ} (n : String) → ((n : String) → A n) → A n
n := f = f n
infixl 0 _⇒⟦_⟧⇒_
_⇒⟦_⟧⇒_ : ∀ {ℓ} {t-in₁ t-out₁ t-in₂ t-out₂ : Types ℓ}
{nr!-in₁ : NonRepetitiveNames! t-in₁}
{nr!-out₁ : NonRepetitiveNames! t-out₁}
{nr!-in₂ : NonRepetitiveNames! t-in₂}
{nr!-out₂ : NonRepetitiveNames! t-out₂} →
Transformer (t-in₁ , s-nr!⇒nr nr!-in₁) (t-out₁ , s-nr!⇒nr nr!-out₁) →
let n-out₁ = names t-out₁
n-in₂ = names t-in₂
t-in = t-in₁ ∪ (t-in₂ ∖∖ n-out₁)
t-out = t-out₁ ∖∖ n-in₂ ∪ t-out₂
in
filter-∈ t-out₁ n-in₂ ≋ filter-∈ t-in₂ n-out₁ →
{nr!-in : NonRepetitiveNames! t-in} →
{nr!-out : NonRepetitiveNames! t-out} →
Transformer (t-in₂ , s-nr!⇒nr nr!-in₂) (t-out₂ , s-nr!⇒nr nr!-out₂) →
Transformer (t-in , s-nr!⇒nr nr!-in) (t-out , s-nr!⇒nr nr!-out)
_⇒⟦_⟧⇒_ {nr!-in₁ = nr!-in₁} {nr!-out₁ = nr!-out₁} {nr!-in₂ = nr!-in₂} {nr!-out₂ = nr!-out₂}
tr₁ f≋f {nr!-in = nr!-in} {nr!-out = nr!-out} tr₂ =
pipe {nr-in₁ = s-nr!⇒nr nr!-in₁} {nr-out₁ = s-nr!⇒nr nr!-out₁} {nr-in₂ = s-nr!⇒nr nr!-in₂} {nr-out₂ = s-nr!⇒nr nr!-out₂}
tr₁ tr₂ f≋f (s-nr!⇒nr nr!-in) (s-nr!⇒nr nr!-out)
record Pure {ℓ} (A : Set ℓ) : Set ℓ where
constructor pure
field get : A
record Unique {ℓ} (A : Set ℓ) : Set ℓ where
constructor unique
field get : A
extract : ∀ {ℓ} {n : String} {A : Set ℓ} → let t = [ (n , Pure A) ] in {nr!-t : NonRepetitiveNames! t} → Transformer ([] , []) (t , s-nr!⇒nr nr!-t) → A
extract {n = n} {A = A} {nr!-t = nr!-t} tr =
let e = n , Pure A
v , t-v≋t = (Σ[ v ∈ Context _ ] signature v ≋ [ e ]) ∋ tr (context []) [] (≋⇒⊆ refl) (s-nr!⇒nr nr!-t)
in Pure.get ∘ getBySignature $ a∈x⇒x≋y⇒a∈y (e ∈ [ e ] ∋ here refl) (≋-sym t-v≋t)
|
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------------------------------------------------------------------------
-- Nested applications of the defined function can be handled
------------------------------------------------------------------------
module Nested where
open import Codata.Musical.Notation
open import Codata.Musical.Stream
open import Function
import Relation.Binary.PropositionalEquality as P
------------------------------------------------------------------------
-- A definition of φ (x ∷ xs) = x ∷ φ (φ xs)
module φ where
infixr 5 _∷_
data StreamP (A : Set) : Set where
_∷_ : (x : A) (xs : ∞ (StreamP A)) → StreamP A
φP : (xs : StreamP A) → StreamP A
data StreamW (A : Set) : Set where
_∷_ : (x : A) (xs : StreamP A) → StreamW A
φW : {A : Set} → StreamW A → StreamW A
φW (x ∷ xs) = x ∷ φP (φP xs)
whnf : {A : Set} → StreamP A → StreamW A
whnf (x ∷ xs) = x ∷ ♭ xs
whnf (φP xs) = φW (whnf xs)
mutual
⟦_⟧W : {A : Set} → StreamW A → Stream A
⟦ x ∷ xs ⟧W = x ∷ ♯ ⟦ xs ⟧P
⟦_⟧P : {A : Set} → StreamP A → Stream A
⟦ xs ⟧P = ⟦ whnf xs ⟧W
⌈_⌉ : {A : Set} → Stream A → StreamP A
⌈ x ∷ xs ⌉ = x ∷ ♯ ⌈ ♭ xs ⌉
φ : {A : Set} → Stream A → Stream A
φ xs = ⟦ φP ⌈ xs ⌉ ⟧P
open φ using (⟦_⟧P; ⟦_⟧W; φP; φW; φ; _∷_; ⌈_⌉)
------------------------------------------------------------------------
-- An equality proof language
module Equality where
φ-rhs : {A : Set} → (Stream A → Stream A) → Stream A → Stream A
φ-rhs φ (x ∷ xs) = x ∷ ♯ φ (φ (♭ xs))
SatisfiesEquation : {A : Set} → (Stream A → Stream A) → Set
SatisfiesEquation φ = ∀ xs → φ xs ≈ φ-rhs φ xs
infixr 5 _∷_
infix 4 _≈P_ _≈W_
infix 3 _∎
infixr 2 _≈⟨_⟩_
data _≈P_ {A : Set} : Stream A → Stream A → Set where
_∷_ : ∀ (x : A) {xs ys}
(xs≈ys : ∞ (♭ xs ≈P ♭ ys)) → x ∷ xs ≈P x ∷ ys
_≈⟨_⟩_ : ∀ (xs : Stream A) {ys zs}
(xs≈ys : xs ≈P ys) (ys≈zs : ys ≈P zs) → xs ≈P zs
_∎ : (xs : Stream A) → xs ≈P xs
sym : ∀ {xs ys} (xs≈ys : xs ≈P ys) → ys ≈P xs
φP-cong : (xs ys : φ.StreamP A) (xs≈ys : ⟦ xs ⟧P ≈P ⟦ ys ⟧P) →
⟦ φP xs ⟧P ≈P ⟦ φP ys ⟧P
lemma : (φ₁ φ₂ : Stream A → Stream A)
(s₁ : SatisfiesEquation φ₁) (s₂ : SatisfiesEquation φ₂) →
∀ {xs ys} (xs≈ys : xs ≈P ys) → φ-rhs φ₁ xs ≈P φ-rhs φ₂ ys
-- Completeness.
completeP : {A : Set} {xs ys : Stream A} → xs ≈ ys → xs ≈P ys
completeP (P.refl ∷ xs≈ys) = _ ∷ ♯ completeP (♭ xs≈ys)
-- Weak head normal forms.
data _≈W_ {A : Set} : Stream A → Stream A → Set where
_∷_ : ∀ x {xs ys} (xs≈ys : ♭ xs ≈P ♭ ys) → x ∷ xs ≈W x ∷ ys
transW : {A : Set} {xs ys zs : Stream A} →
xs ≈W ys → ys ≈W zs → xs ≈W zs
transW (x ∷ xs≈ys) (.x ∷ ys≈zs) = x ∷ (_ ≈⟨ xs≈ys ⟩ ys≈zs)
reflW : {A : Set} (xs : Stream A) → xs ≈W xs
reflW (x ∷ xs) = x ∷ (♭ xs ∎)
symW : {A : Set} {xs ys : Stream A} → xs ≈W ys → ys ≈W xs
symW (x ∷ xs≈ys) = x ∷ sym xs≈ys
φW-cong : {A : Set} (xs ys : φ.StreamW A) →
⟦ xs ⟧W ≈W ⟦ ys ⟧W → ⟦ φW xs ⟧W ≈W ⟦ φW ys ⟧W
φW-cong (.x ∷ xs) (.x ∷ ys) (x ∷ xs≈ys) =
x ∷ φP-cong (φP xs) (φP ys) (φP-cong xs ys xs≈ys)
lemmaW : {A : Set} (φ₁ φ₂ : Stream A → Stream A)
(s₁ : SatisfiesEquation φ₁) (s₂ : SatisfiesEquation φ₂) →
∀ {xs ys} → xs ≈W ys → φ-rhs φ₁ xs ≈W φ-rhs φ₂ ys
lemmaW φ₁ φ₂ s₁ s₂ {.x ∷ xs} {.x ∷ ys} (x ∷ xs≈ys) = x ∷ (
φ₁ (φ₁ (♭ xs)) ≈⟨ completeP $ s₁ (φ₁ (♭ xs)) ⟩
φ-rhs φ₁ (φ₁ (♭ xs)) ≈⟨ lemma φ₁ φ₂ s₁ s₂ (
φ₁ (♭ xs) ≈⟨ completeP $ s₁ (♭ xs) ⟩
φ-rhs φ₁ (♭ xs) ≈⟨ lemma φ₁ φ₂ s₁ s₂ xs≈ys ⟩
φ-rhs φ₂ (♭ ys) ≈⟨ sym $ completeP $ s₂ (♭ ys) ⟩
φ₂ (♭ ys) ∎) ⟩
φ-rhs φ₂ (φ₂ (♭ ys)) ≈⟨ sym $ completeP $ s₂ (φ₂ (♭ ys)) ⟩
φ₂ (φ₂ (♭ ys)) ∎)
whnf : {A : Set} {xs ys : Stream A} → xs ≈P ys → xs ≈W ys
whnf (x ∷ xs≈ys) = x ∷ ♭ xs≈ys
whnf (xs ≈⟨ xs≈ys ⟩ ys≈zs) = transW (whnf xs≈ys) (whnf ys≈zs)
whnf (xs ∎) = reflW xs
whnf (sym xs≈ys) = symW (whnf xs≈ys)
whnf (lemma φ₁ φ₂ s₁ s₂ xs≈ys) = lemmaW φ₁ φ₂ s₁ s₂ (whnf xs≈ys)
whnf (φP-cong xs ys xs≈ys) =
φW-cong (φ.whnf xs) (φ.whnf ys) (whnf xs≈ys)
-- Soundness.
mutual
soundW : {A : Set} {xs ys : Stream A} → xs ≈W ys → xs ≈ ys
soundW (x ∷ xs≈ys) = P.refl ∷ ♯ soundP xs≈ys
soundP : {A : Set} {xs ys : Stream A} → xs ≈P ys → xs ≈ ys
soundP xs≈ys = soundW (whnf xs≈ys)
open Equality
using (_≈P_; _∷_; _≈⟨_⟩_; _∎; sym; φP-cong; φ-rhs; SatisfiesEquation)
------------------------------------------------------------------------
-- Correctness
module Correctness where
-- Uniqueness of solutions for φ's defining equation.
φ-unique : {A : Set} (φ₁ φ₂ : Stream A → Stream A) →
SatisfiesEquation φ₁ → SatisfiesEquation φ₂ →
∀ xs → φ₁ xs ≈P φ₂ xs
φ-unique φ₁ φ₂ hyp₁ hyp₂ xs =
φ₁ xs ≈⟨ Equality.completeP (hyp₁ xs) ⟩
φ-rhs φ₁ xs ≈⟨ Equality.lemma φ₁ φ₂ hyp₁ hyp₂ (xs ∎) ⟩
φ-rhs φ₂ xs ≈⟨ sym (Equality.completeP (hyp₂ xs)) ⟩
φ₂ xs ∎
-- The remainder of this module establishes the existence of a
-- solution.
⟦⌈_⌉⟧P : {A : Set} (xs : Stream A) → ⟦ ⌈ xs ⌉ ⟧P ≈P xs
⟦⌈ x ∷ xs ⌉⟧P = x ∷ ♯ ⟦⌈ ♭ xs ⌉⟧P
φ-cong : {A : Set} (xs ys : Stream A) → xs ≈P ys → φ xs ≈P φ ys
φ-cong xs ys xs≈ys =
φ xs ≈⟨ φ xs ∎ ⟩
⟦ φP ⌈ xs ⌉ ⟧P ≈⟨ φP-cong ⌈ xs ⌉ ⌈ ys ⌉ lemma ⟩
⟦ φP ⌈ ys ⌉ ⟧P ≈⟨ φ ys ∎ ⟩
φ ys ∎
where
lemma =
⟦ ⌈ xs ⌉ ⟧P ≈⟨ ⟦⌈ xs ⌉⟧P ⟩
xs ≈⟨ xs≈ys ⟩
ys ≈⟨ sym ⟦⌈ ys ⌉⟧P ⟩
⟦ ⌈ ys ⌉ ⟧P ∎
-- ♯′ provides a workaround for Agda's strange definitional
-- equality.
infix 10 ♯′_
♯′_ : {A : Set} → A → ∞ A
♯′ x = ♯ x
φW-hom : {A : Set} (xs : φ.StreamW A) →
⟦ φW xs ⟧W ≈P head ⟦ xs ⟧W ∷ ♯′ φ (φ (tail ⟦ xs ⟧W))
φW-hom (x ∷ xs) = x ∷ ♯ (
⟦ φP (φP xs) ⟧P ≈⟨ φP-cong (φP xs) (φP ⌈ ⟦ xs ⟧P ⌉) $
φP-cong xs (⌈ ⟦ xs ⟧P ⌉)
(sym ⟦⌈ ⟦ xs ⟧P ⌉⟧P) ⟩
⟦ φP (φP ⌈ ⟦ xs ⟧P ⌉) ⟧P ≈⟨ φP-cong (φP ⌈ ⟦ xs ⟧P ⌉)
⌈ ⟦ φP ⌈ ⟦ xs ⟧P ⌉ ⟧P ⌉
(sym ⟦⌈ ⟦ φP ⌈ ⟦ xs ⟧P ⌉ ⟧P ⌉⟧P) ⟩
⟦ φP ⌈ ⟦ φP ⌈ ⟦ xs ⟧P ⌉ ⟧P ⌉ ⟧P ∎)
φ-hom : {A : Set} (xs : φ.StreamP A) →
⟦ φP xs ⟧P ≈P head ⟦ xs ⟧P ∷ ♯′ φ (φ (tail ⟦ xs ⟧P))
φ-hom xs = φW-hom (φ.whnf xs)
φ-correct : {A : Set} (xs : Stream A) →
φ xs ≈P φ-rhs φ xs
φ-correct (x ∷ xs) =
φ (x ∷ xs) ≈⟨ φ (x ∷ xs) ∎ ⟩
⟦ φP ⌈ x ∷ xs ⌉ ⟧P ≈⟨ φ-hom ⌈ x ∷ xs ⌉ ⟩
x ∷ ♯′ φ (φ ⟦ ⌈ ♭ xs ⌉ ⟧P) ≈⟨ x ∷ ♯ φ-cong (φ ⟦ ⌈ ♭ xs ⌉ ⟧P) (φ (♭ xs))
(φ-cong (⟦ ⌈ ♭ xs ⌉ ⟧P) (♭ xs) ⟦⌈ ♭ xs ⌉⟧P) ⟩
φ-rhs φ (x ∷ xs) ∎
|
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{-# OPTIONS --cubical --safe #-}
module Cubical.Homotopy.Connected where
open import Cubical.Core.Everything
open import Cubical.Foundations.Everything
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Path
open import Cubical.Foundations.Univalence
open import Cubical.Functions.Fibration
open import Cubical.Data.Nat
open import Cubical.Data.Sigma
open import Cubical.HITs.Nullification
open import Cubical.HITs.Truncation as Trunc
isHLevelConnected : ∀ {ℓ} (n : ℕ) (A : Type ℓ) → Type ℓ
isHLevelConnected n A = isContr (hLevelTrunc n A)
isHLevelConnectedFun : ∀ {ℓ ℓ'} (n : ℕ) {A : Type ℓ} {B : Type ℓ'} (f : A → B) → Type (ℓ-max ℓ ℓ')
isHLevelConnectedFun n f = ∀ b → isHLevelConnected n (fiber f b)
isEquivPrecomposeConnected : ∀ {ℓ ℓ' ℓ''} (n : ℕ)
{A : Type ℓ} {B : Type ℓ'} (P : B → HLevel ℓ'' n) (f : A → B)
→ isHLevelConnectedFun n f
→ isEquiv (λ(s : (b : B) → P b .fst) → s ∘ f)
isEquivPrecomposeConnected n {A} {B} P f fConn =
isoToIsEquiv
(iso (_∘ f)
(λ t b → inv t b (fConn b .fst))
(λ t → funExt λ a →
cong (inv t (f a)) (fConn (f a) .snd ∣ a , refl ∣)
∙ substRefl {B = fst ∘ P} (t a))
(λ s → funExt λ b →
Trunc.elim
{B = λ d → inv (s ∘ f) b d ≡ s b}
(λ _ → isOfHLevelPath n (P b .snd) _ _)
(λ {(a , p) i → transp (λ j → P (p (j ∨ i)) .fst) i (s (p i))})
(fConn b .fst)))
where
inv : ((a : A) → P (f a) .fst) → (b : B) → hLevelTrunc n (fiber f b) → P b .fst
inv t b =
Trunc.rec
(P b .snd)
(λ {(a , p) → subst (fst ∘ P) p (t a)})
isOfHLevelPrecomposeConnected : ∀ {ℓ ℓ' ℓ''} (k : ℕ) (n : ℕ)
{A : Type ℓ} {B : Type ℓ'} (P : B → HLevel ℓ'' (k + n)) (f : A → B)
→ isHLevelConnectedFun n f
→ isOfHLevelFun k (λ(s : (b : B) → P b .fst) → s ∘ f)
isOfHLevelPrecomposeConnected zero n P f fConn =
isEquivPrecomposeConnected n P f fConn .equiv-proof
isOfHLevelPrecomposeConnected (suc k) n P f fConn t =
isOfHLevelPath'⁻ k
(λ {(s₀ , p₀) (s₁ , p₁) →
subst (isOfHLevel k) (sym (fiber≡ (s₀ , p₀) (s₁ , p₁)))
(isOfHLevelRetract k
(λ {(q , α) → (funExt⁻ q) , (cong funExt⁻ α)})
(λ {(h , β) → (funExt h) , (cong funExt β)})
(λ _ → refl)
(isOfHLevelPrecomposeConnected k n
(λ b → (s₀ b ≡ s₁ b) , isOfHLevelPath' (k + n) (P b .snd) _ _)
f fConn
(funExt⁻ (p₀ ∙∙ refl ∙∙ sym p₁))))})
isHLevelConnectedPath : ∀ {ℓ} (n : ℕ) {A : Type ℓ}
→ isHLevelConnected (suc n) A
→ (a₀ a₁ : A) → isHLevelConnected n (a₀ ≡ a₁)
isHLevelConnectedPath n connA a₀ a₁ =
subst isContr (PathIdTrunc _)
(isContr→isContrPath connA _ _)
isHLevelConnectedRetract : ∀ {ℓ ℓ'} (n : ℕ)
{A : Type ℓ} {B : Type ℓ'}
(f : A → B) (g : B → A)
(h : (x : A) → g (f x) ≡ x)
→ isHLevelConnected n B → isHLevelConnected n A
isHLevelConnectedRetract n f g h =
isContrRetract
(Trunc.map f)
(Trunc.map g)
(Trunc.elim
(λ _ → isOfHLevelPath n (isOfHLevelTrunc n) _ _)
(λ a → cong ∣_∣ (h a)))
isHLevelConnectedPoint : ∀ {ℓ} (n : ℕ) {A : Type ℓ}
→ isHLevelConnected (suc n) A
→ (a : A) → isHLevelConnectedFun n (λ(_ : Unit) → a)
isHLevelConnectedPoint n connA a₀ a =
isHLevelConnectedRetract n
snd (_ ,_) (λ _ → refl)
(isHLevelConnectedPath n connA a₀ a)
|
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module cry.ec where
open import Level
open import Relation.Nullary
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat using () renaming (Nat to ℕ)
open import Agda.Builtin.List
open import cry.gfp
infixr 4 _,_
infixr 2 _×_
record _×_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
constructor _,_
field
proj₁ : A
proj₂ : B
open _×_ public
-- EC: group of points of elliptic curve (in jacobian coordinates, without conversion to affine)
module ec {c ℓ} (gfp : RawField c ℓ) (a b : RawField.Carrier gfp) where
module F = RawField gfp
open F renaming (Carrier to 𝔽; _≈_ to _=F_; _≈?_ to _≟F_)
record Point : Set c where
constructor _∶_∶_
field
x y z : 𝔽
is-point : Point → Bool
-- (y/z³) ² ≡ (x/z²) ³ + a * (x/z²) + b
is-point (x ∶ y ∶ z) =
let
y² = y ²
x² = x ²
x³ = x² * x
z² = z ²
z⁴ = z² ²
z⁶ = z² * z⁴
xz⁴ = x * z⁴
axz⁴ = a * xz⁴
bz⁶ = b * z⁶
x³+axz⁴ = x³ + axz⁴
x³+axz⁴+bz⁶ = x³+axz⁴ + bz⁶
in y² F.?≈ x³+axz⁴+bz⁶
aff : Point → Point
aff (x ∶ y ∶ z) = x′ ∶ y′ ∶ 1# where
z⁻¹ = z ⁻¹
z⁻² = z⁻¹ ²
z⁻³ = z⁻² * z⁻¹
x′ = x * z⁻²
y′ = y * z⁻³
_==_ : Point → Point → Set _
(x₁ ∶ y₁ ∶ z₁) == (x₂ ∶ y₂ ∶ z₂)
-- p₁ == p₂ with norm2 p₁ p₂
-- ... | (x₁z₂² , x₂z₁²) , (y₁z₂³ , y₂z₁³)
= x₁z₂² =F x₂z₁² × y₁z₂³ =F y₂z₁³ where
{-
x₁/z₁² ≡ x₂/z₂²
y₁/z₁³ ≡ y₂/z₂³
-}
z₂² = z₂ ²
z₁² = z₁ ²
x₁z₂² = x₁ * z₂²
x₂z₁² = x₂ * z₁²
z₂³ = z₂ * z₂²
z₁³ = z₁ * z₁²
y₁z₂³ = y₁ * z₂³
y₂z₁³ = y₂ * z₁³
_≟_ : (p₁ p₂ : Point) → Dec (p₁ == p₂)
(x₁ ∶ y₁ ∶ z₁) ≟ (x₂ ∶ y₂ ∶ z₂) = r where
z₂² = z₂ ²
z₁² = z₁ ²
x₁z₂² = x₁ * z₂²
x₂z₁² = x₂ * z₁²
z₂³ = z₂ * z₂²
z₁³ = z₁ * z₁²
y₁z₂³ = y₁ * z₂³
y₂z₁³ = y₂ * z₁³
r : _
r with x₁z₂² ≟F x₂z₁²
... | no x₁≠x₂ = no (λ p₁=p₂ → x₁≠x₂ (proj₁ p₁=p₂))
... | yes x₁=x₂ with y₁z₂³ ≟F y₂z₁³
... | no y₁≠y₂ = no (λ p₁=p₂ → y₁≠y₂ (proj₂ p₁=p₂))
... | yes y₁=y₂ = yes (x₁=x₂ , y₁=y₂)
is-𝕆 : Point → Set _
is-𝕆 (_ ∶ _ ∶ z) = z =F 0#
{-
x₃ = λ² − x₁ − x₂
y₃ = λ(x₁ − x₃) − y₁
λ = (y₂ - y₁) / (x₂ - x₁), p₁ ≠ p₂
λ = (3 x₁ ² + a) / (2 y₁), p₁ = p₂
(x/z²)₃ = λ² − (x/z²)₁ − (x/z²)₂
(y/z³)₃ = λ((x/z²)₁ − (x/z²)₃) − (y/z³)₁
λ = ((y/z³)₂ - (y/z³)₁) / ((x/z²)₂ - (x/z²)₁), p₁ ≠ p₂
λ = (3 (x/z²)₁ ² + a) / (2 (y/z³)₁), p₁ = p₂
x₃/z₃² z₁² z₂² = λ² z₁² z₂² − x₁ z₂² − x₂ z₁²
y₁/z₁³ + y₃/z₃³ = λ(x₁/z₁² − (x₃/z₃²))
y₂/z₂³ + y₃/z₃³ = λ(x₂/z₂² − (x₃/z₃²))
λ z₁ z₂ (x₂ z₁² - x₁ z₂²) = (y₂ z₁³ - y₁ z₂³), p₁ ≠ p₂
λ = (3 (x/z²)₁ ² + a) / (2 (y/z³)₁), p₁ = p₂
p₁ ≠ p₂:
z₃ = z₁ z₂ (x₂ z₁² - x₁ z₂²)
λ z₃ = (y₂ z₁³ - y₁ z₂³)
x₃ = (y₂ z₁³ - y₁ z₂³)² − (x₁ z₂² + x₂ z₁²) (x₂ z₁² - x₁ z₂²)²
y₃ z₁³ = (y₂ z₁³ - y₁ z₂³) (z₃² x₁ − x₃ z₁²) z₁ − y₁ z₃³
(y₁ z₃³ + y₃ z₁³)/(y₂ z₁³ - y₁ z₂³) = (x₁ z₃² − x₃ z₁²) z₁
(y₂ z₃³ + y₃ z₂³)/(y₂ z₁³ - y₁ z₂³) = (x₂ z₃² − x₃ z₂²) z₂
x₁ z₃² − x₃ z₁² =
x₁ (z₁ z₂ (x₂ z₁² - x₁ z₂²))² -
(y₂ z₁³ - y₁ z₂³)² z₁² +
((x₂ z₁²)² - (x₁ z₂²)²) (x₂ z₁² - x₁ z₂²) z₁²
= x₁ z₁² z₂² ((x₂ z₁²)² - 2 x₂ z₁² x₁ z₂² + (x₁ z₂²)²)
+ ((x₂ z₁²)³ - (x₂ z₁²)² x₁ z₂² - (x₁ z₂²)² x₂ z₁² + (x₁ z₂²)³) z₁²
- (y₂ z₁³ - y₁ z₂³)² z₁²
= (- (y₂ z₁³ - y₁ z₂³)²
+ 2 x₁³ z₂⁶ + x₂³ z₁⁶ + x₁² x₂ (-3 z₁² z₂⁴)) z₁²
(y₁ z₃³ + y₃ z₁³) = (y₂ z₁³ - y₁ z₂³) (- (y₂ z₁³ - y₁ z₂³)²
+ 2 x₁³ z₂⁶ + x₂³ z₁⁶ + x₁² x₂ (-3 z₁² z₂⁴)) z₁³
-}
𝕆 : Point
𝕆 = (1# ∶ 1# ∶ 0#)
dbl : Point → Point
dbl (x₁ ∶ y₁ ∶ z₁) = (x₃ ∶ y₃ ∶ z₃) where
{- p₁ = p₂
x₃ = λ² − x₁ − x₁
y₃ = λ(x₁ − x₃) − y₁
λ = (3 x₁ ² + a) / (2 y₁)
x₃ = λ² z₃² − 2 x₁/z₁² z₃²
y₃/z₃³ = λ(x₁/z₁² − x₃/z₃²) − y₁/z₁³
λ = (3 (x₁/z₁²) ² + a) / (2 y₁/z₁³)
λ 2 y₁ z₁ = 3 x₁² + a z₁⁴
z₃ = 2 y₁ z₁
x₃ = (3 x₁² + a z₁⁴)² − 2 x₁ (2 y₁)²
y₃ = (3 x₁² + a z₁⁴) (x₁ (2 y₁)² − x₃) − y₁ (2 y₁)³
-}
2y₁ = y₁ + y₁
z₃ = 2y₁ * z₁
x₁² = x₁ ²
2x₁² = x₁² + x₁²
3x₁² = 2x₁² + x₁²
z₁² = z₁ ²
z₁⁴ = z₁² ²
az₁⁴ = a * z₁⁴
3x₁²+az₁⁴ = 3x₁² + az₁⁴
[3x₁²+az₁⁴]² = 3x₁²+az₁⁴ ²
y₁² = y₁ ²
2y₁² = y₁² + y₁²
4y₁² = 2y₁² + 2y₁²
x₁[2y₁]² = x₁ * 4y₁²
2x₁[2y₁]² = x₁[2y₁]² + x₁[2y₁]²
x₃ = [3x₁²+az₁⁴]² - 2x₁[2y₁]²
4y₁⁴ = 2y₁² ²
8y₁⁴ = 4y₁⁴ + 4y₁⁴
x₁[2y₁]²-x₃ = x₁[2y₁]² - x₃
[3x₁²+az₁⁴][x₁[2y₁]²-x₃] = 3x₁²+az₁⁴ * x₁[2y₁]²-x₃
y₃ = [3x₁²+az₁⁴][x₁[2y₁]²-x₃] - 8y₁⁴
add : Point → Point → Point
add (x₁ ∶ y₁ ∶ z₁) (x₂ ∶ y₂ ∶ z₂) = x₃ ∶ y₃ ∶ z₃ where
{- p₁ ≠ p₂
x₃ = λ² − x₁ − x₂
y₃ = λ(x₁ − x₃) − y₁
λ = (y₂ - y₁) / (x₂ - x₁)
(x/z²)₃ = λ² − (x/z²)₁ − (x/z²)₂
(y/z³)₃ = λ((x/z²)₁ − (x/z²)₃) − (y/z³)₁
λ = ((y/z³)₂ - (y/z³)₁) / ((x/z²)₂ - (x/z²)₁)
λ z₁ z₂ (x₂ z₁² - x₁ z₂²) = (y₂ z₁³ - y₁ z₂³)
x₃/z₃² z₁² z₂² + x₁ z₂² + x₂ z₁² = λ² z₁² z₂²
y₃/z₃³ = λ(x₁/z₁² − x₃/z₃²) − y₁/z₁³
z₃ = z₁ z₂ (x₂ z₁² - x₁ z₂²)
x₃ = (y₂ z₁³ - y₁ z₂³)² - (x₁ z₂² + x₂ z₁²) (x₂ z₁² - x₁ z₂²)²
y₃ = (y₂ z₁³ - y₁ z₂³) (x₁ z₂² (x₂ z₁² - x₁ z₂²)² − x₃) − y₁ z₂³ (x₂ z₁² - x₁ z₂²)³
y₃ = (y₂ z₁³ - y₁ z₂³) ((x₁ z₂²+x₂ z₁²)/2 (x₂ z₁² - x₁ z₂²)² − x₃)
− (x₂ z₁² - x₁ z₂²)³ (y₁ z₂³ + y₂ z₁³)/2
-}
z₁z₂ = z₁ * z₂
z₁² = z₁ ²
z₂² = z₂ ²
x₁z₂² = x₁ * z₂²
x₂z₁² = x₂ * z₁²
z₂³ = z₂ * z₂²
z₁³ = z₁ * z₁²
y₁z₂³ = y₁ * z₂³
y₂z₁³ = y₂ * z₁³
x₂z₁²+x₁z₂² = x₂z₁² + x₁z₂²
x₂z₁²-x₁z₂² = x₂z₁² - x₁z₂²
y₂z₁³-y₁z₂³ = y₂z₁³ - y₁z₂³
z₃ = z₁z₂ * x₂z₁²-x₁z₂²
[x₂z₁²-x₁z₂²]² = x₂z₁²-x₁z₂² ²
[y₂z₁³-y₁z₂³]² = y₂z₁³-y₁z₂³ ²
[x₂z₁²+x₁z₂²][x₂z₁²-x₁z₂²]² = x₂z₁²+x₁z₂² * [x₂z₁²-x₁z₂²]²
x₃ = [y₂z₁³-y₁z₂³]² - [x₂z₁²+x₁z₂²][x₂z₁²-x₁z₂²]²
[x₂z₁²-x₁z₂²]³ = x₂z₁²-x₁z₂² * [x₂z₁²-x₁z₂²]²
y₁z₂³[x₂z₁²-x₁z₂²]³ = y₁z₂³ * [x₂z₁²-x₁z₂²]³
x₁z₂²[x₂z₁²-x₁z₂²]² = x₁z₂² * [x₂z₁²-x₁z₂²]²
x₁z₂²[x₂z₁²-x₁z₂²]²-x₃ = x₁z₂²[x₂z₁²-x₁z₂²]² - x₃
[y₂z₁³-y₁z₂³][x₁z₂²[x₂z₁²-x₁z₂²]²-x₃] = y₂z₁³-y₁z₂³ * x₁z₂²[x₂z₁²-x₁z₂²]²-x₃
y₃ = [y₂z₁³-y₁z₂³][x₁z₂²[x₂z₁²-x₁z₂²]²-x₃] - y₁z₂³[x₂z₁²-x₁z₂²]³
dblAdd : Point → ℕ → Point
dblAdd = times 𝕆 dbl add
module test-ec where
g = cry.gfp.gfp 7
open RawField g renaming (Carrier to 𝔽) public
a b : 𝔽
a = 4
b = 1
xₚ yₚ zₚ : 𝔽
xₚ = 4
yₚ = 2
zₚ = 1
open ec g a b public
P 2P : Point
P = (xₚ ∶ yₚ ∶ zₚ)
2P = dbl P
|
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module io where
open import bool
open import char
open import list
open import string
open import unit
----------------------------------------------------------------------
-- datatypes
----------------------------------------------------------------------
postulate
IO : Set → Set
{-# COMPILE GHC IO = type IO #-}
{-# BUILTIN IO IO #-}
----------------------------------------------------------------------
-- syntax
----------------------------------------------------------------------
infixl 1 _>>=_
infixl 1 _>>_
----------------------------------------------------------------------
-- postulated operations
----------------------------------------------------------------------
postulate
return : ∀ {A : Set} → A → IO A
_>>=_ : ∀ {A B : Set} → IO A → (A → IO B) → IO B
{-# COMPILE GHC return = \ _ -> return #-}
{-# COMPILE GHC _>>=_ = \ _ _ -> (>>=) #-}
postulate
putStr : string -> IO ⊤
-- Reads a file, which is assumed to be finite.
readFiniteFile : string → IO string
writeFile : string → string → IO ⊤
-- set output to UTF-8 for Windows
initializeStdoutToUTF8 : IO ⊤
-- set newline mode for Windows
setStdoutNewlineMode : IO ⊤
getLine : IO string
private
postulate
privGetArgs : IO (𝕃 string)
privDoesFileExist : string → IO 𝔹
privCreateDirectoryIfMissing : 𝔹 → string → IO ⊤
privTakeDirectory : string → string
privTakeFileName : string → string
privCombineFileNames : string → string → string
privForceFileRead : string {- the contents of the file, not the file name -} → IO ⊤
privGetHomeDirectory : IO string
{-# FOREIGN GHC import qualified System.IO #-}
{-# FOREIGN GHC import qualified Control.DeepSeq #-}
{-# FOREIGN GHC import qualified Data.Text.IO #-}
{-# COMPILE GHC putStr = Data.Text.IO.putStr #-}
{-# COMPILE GHC readFiniteFile = (\y -> let x = Data.Text.unpack y in do inh <- System.IO.openFile x System.IO.ReadMode; System.IO.hSetEncoding inh System.IO.utf8; fileAsString <- System.IO.hGetContents inh; Control.DeepSeq.rnf fileAsString `seq` System.IO.hClose inh; return (Data.Text.pack fileAsString)) #-}
{-# COMPILE GHC writeFile = (\path -> (\str -> do outh <- System.IO.openFile (Data.Text.unpack path) System.IO.WriteMode; System.IO.hSetNewlineMode outh System.IO.noNewlineTranslation; System.IO.hSetEncoding outh System.IO.utf8; Data.Text.IO.hPutStr outh str; System.IO.hFlush outh; System.IO.hClose outh; return () )) #-}
{-# COMPILE GHC initializeStdoutToUTF8 = System.IO.hSetEncoding System.IO.stdout System.IO.utf8 #-}
{-# COMPILE GHC setStdoutNewlineMode = System.IO.hSetNewlineMode System.IO.stdout System.IO.universalNewlineMode #-}
{-# FOREIGN GHC import qualified System.Environment #-}
{-# COMPILE GHC privGetArgs = (do l <- System.Environment.getArgs; return (map Data.Text.pack l)) #-}
{-# FOREIGN GHC import qualified System.Directory #-}
{-# COMPILE GHC privForceFileRead = (\ contents -> seq (length (Data.Text.unpack contents)) (return ())) #-}
{-# COMPILE GHC privDoesFileExist = (\ s -> System.Directory.doesFileExist (Data.Text.unpack s)) #-}
{-# COMPILE GHC privCreateDirectoryIfMissing = (\ b s -> System.Directory.createDirectoryIfMissing b (Data.Text.unpack s)) #-}
{-# FOREIGN GHC import qualified System.FilePath #-}
{-# COMPILE GHC privTakeDirectory = (\ s -> Data.Text.pack (System.FilePath.takeDirectory (Data.Text.unpack s))) #-}
{-# COMPILE GHC privTakeFileName = (\ s -> Data.Text.pack (System.FilePath.takeFileName (Data.Text.unpack s))) #-}
{-# COMPILE GHC privCombineFileNames = (\ s1 s2 -> Data.Text.pack (System.FilePath.combine (Data.Text.unpack s1) (Data.Text.unpack s2))) #-}
{-# COMPILE GHC getLine = (Data.Text.IO.hGetLine System.IO.stdin) #-}
{-# COMPILE GHC privGetHomeDirectory = (do x <- System.Directory.getHomeDirectory; return (Data.Text.pack x)) #-}
getArgs : IO (𝕃 string)
getArgs = privGetArgs
doesFileExist : string → IO 𝔹
doesFileExist = privDoesFileExist
createDirectoryIfMissing : 𝔹 → string → IO ⊤
createDirectoryIfMissing = privCreateDirectoryIfMissing
takeDirectory : string → string
takeDirectory = privTakeDirectory
takeFileName : string → string
takeFileName = privTakeFileName
combineFileNames : string → string → string
combineFileNames = privCombineFileNames
forceFileRead : string {- the contents of the file, not the file name -} → IO ⊤
forceFileRead = privForceFileRead
getHomeDirectory : IO string
getHomeDirectory = privGetHomeDirectory
postulate
fileIsOlder : string → string → IO 𝔹
canonicalizePath : string → IO string
{-# COMPILE GHC fileIsOlder = (\ s1 s2 -> (System.Directory.getModificationTime (Data.Text.unpack s1)) >>= \ t1 -> (System.Directory.getModificationTime (Data.Text.unpack s2)) >>= \ t2 -> return (t1 < t2)) #-}
{-# COMPILE GHC canonicalizePath = (\ s -> do x <- System.Directory.canonicalizePath (Data.Text.unpack s); return (Data.Text.pack x)) #-}
----------------------------------------------------------------------
-- defined operations
----------------------------------------------------------------------
_>>_ : ∀ {A B : Set} → IO A → IO B → IO B
x >> y = x >>= (λ q -> y)
base-filenameh : 𝕃 char → 𝕃 char
base-filenameh [] = []
base-filenameh ('.' :: cs) = cs
base-filenameh (_ :: cs) = base-filenameh cs
-- return the part of the string up to the last (rightmost) period ('.'); so for "foo.txt" return "foo"
base-filename : string → string
base-filename s = 𝕃char-to-string (reverse (base-filenameh (reverse (string-to-𝕃char s))))
|
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{-# OPTIONS --without-K #-}
open import lib.Basics -- hiding (_=⟨_⟩_ ; _∎)
open import lib.types.Paths
open import lib.types.Pi
open import lib.types.Unit
open import lib.types.Nat
open import lib.types.TLevel
open import lib.types.Pointed
open import lib.types.Sigma
open import lib.NType2
open import lib.PathGroupoid
open import nicolai.pseudotruncations.Preliminary-definitions
module nicolai.pseudotruncations.Liblemmas where
-- transport along constant family
transport-const-fam : ∀ {i j} {A : Type i} {B : Type j} {a₁ a₂ : A}
→ (p : a₁ == a₂) → (b : B) → transport (λ _ → B) p b == b
transport-const-fam idp b = idp
-- interaction of transport and ap
trans-ap : ∀ {i j} {A : Type i} {B : Type j} {a₁ a₂ : A}
→ (f g : A → B) → (p : a₁ == a₂) → (q : f a₁ == g a₁)
→ transport (λ x → f x == g x) p q == ! (ap f p) ∙ q ∙ (ap g p)
trans-ap f g idp q = ! (∙-unit-r q)
-- special interaction of transport and ap, where the second map is constant at a point
trans-ap₁ : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) {a₁ a₂ : A} (b : B)
(p : a₁ == a₂) (q : f a₁ == b)
→ transport (λ a → f a == b) p q == ! (ap f p) ∙ q
trans-ap₁ f b idp q = idp
-- first map is constant at a point
trans-ap₂ : ∀ {i j} {A : Type i} {B : Type j} (g : A → B) {a₁ a₂ : A} (b : B)
(p : a₁ == a₂) (q : b == g a₁)
→ transport (λ a → b == g a) p q == q ∙ ap g p
trans-ap₂ g b idp q = !( ∙-unit-r _)
-- if f is weakly constant, then so is ap f
ap-const : ∀ {i j} {A : Type i} {B : Type j} (f : A → B)
→ wconst f → {a₁ a₂ : A} → wconst (ap f {x = a₁} {y = a₂})
ap-const {A = A} f wc p q = calc-ap p ∙ ! (calc-ap q) where
calc-ap : {a₁ a₂ : A} → (p : a₁ == a₂) → ap f p == wc a₁ a₂ ∙ ! (wc a₂ a₂)
calc-ap idp = ! (!-inv-r (wc _ _))
-- in particular, if f is weakly constant, then ap f maps loops to 'refl'
ap-const₁ : ∀ {i j} {A : Type i} {B : Type j} (f : A → B)
→ wconst f → {a₁ : A} → (p : a₁ == a₁) → ap f p == idp
ap-const₁ f wc p = ap-const f wc p idp
-- if f is constant at a point, it maps every path to 'refl'
ap-const-at-point : ∀ {i j} {A : Type i} {B : Type j} {a₁ a₂ : A}
(b : B) (p : a₁ == a₂) → ap (λ _ → b) p == idp
ap-const-at-point b idp = idp
{- this lemma is ad-hoc; it could be proved as a concatenation of
many library lemmas, but it would be much more tedious to do -}
adhoc-lemma : ∀ {i} {A : Type i} {x y z : A}
(p : x == y)
(q : z == y)
(r : z == x)
→ p ∙ ! q ∙ r == idp
→ p == ! r ∙ q
adhoc-lemma p idp idp e = ! (∙-unit-r p) ∙ e
{- If f is weakly constant, then so is ap f. This is a lemma from our
old Hedberg article. -}
ap-wconst : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) (w : wconst f)
→ {a₁ a₂ : A} → wconst (ap f {a₁} {a₂})
ap-wconst f w p q = lemma p ∙ ! (lemma q) where
lemma : ∀ {x y} (p : x == y) → ap f {x} {y} p == ! (w x x) ∙ (w x y)
lemma {x} idp = ! (!-inv-l (w x x))
-- Silly little lemma (is it in the library?)
ap-fst : ∀ {i j} {A : Type i} {B : Type j} {a₁ a₂ : A} {b₁ b₂ : B}
(p : a₁ == a₂) (q : b₁ == b₂)
→ ap fst (pair×= p q) == p
ap-fst idp idp = idp
{- An ad-hoc lemma. Whenever this appears, one could (should, to be honest)
use library lemmas, but it's just so much more convenient to formulate it
and pattern match... -}
adhoc-=-eqv : ∀ {i} {A : Type i} {x y : A} (p : y == x) (q : y == x)
→ (! p ∙ q == idp) ≃ (p == q)
adhoc-=-eqv idp q = !-equiv
{- Another ad-hoc equality; it could be proved easily with many nested
library lemmas -}
multi-cancelling : ∀ {i} {A : Type i} {x y z w : A} (p : y == x) (q : y == z) (r : x == w)
→ (! p) ∙ q ∙ (! q) ∙ p ∙ r == r
multi-cancelling idp idp r = idp
|
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-- self-contained notes following https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html and relevant parts of HoTT book
open import Agda.Primitive public
using (Level ; _⊔_)
renaming (Set to Type ; lzero to 𝓾₀ ; lsuc to _⁺)
variable
𝓁 𝓂 𝓃 : Level
Π : {X : Type 𝓁} (A : X → Type 𝓂) → Type (𝓁 ⊔ 𝓂)
Π A = (x : _) → A x
id : {X : Type 𝓁} → X → X
id x = x
_∘_ : {X : Type 𝓁} {Y : Type 𝓂} {Z : Y → Type 𝓃}
→ ((y : Y) → Z y) → (f : X → Y) → (x : X) → Z (f x)
g ∘ f = λ x → g (f x)
{-# INLINE _∘_ #-}
infixr 50 _∘_
data ⊥ : Type where
⊥-induction : (A : ⊥ → Type 𝓁) → Π A
⊥-induction A ()
data ⊤ : Type where
⋆ : ⊤
⊤-induction : (A : ⊤ → Type 𝓁) → A ⋆ → Π A
⊤-induction A a ⋆ = a
data _+_ (X : Type 𝓁) (Y : Type 𝓂) : Type (𝓁 ⊔ 𝓂) where
inl : X → X + Y
inr : Y → X + Y
+-induction : {X : Type 𝓁} {Y : Type 𝓂} (A : X + Y → Type 𝓃)
→ ((x : X) → A (inl x))
→ ((y : Y) → A (inr y))
→ Π A
+-induction A f g (inl x) = f x
+-induction A f g (inr y) = g y
𝟚 : Type
𝟚 = ⊤ + ⊤
pattern ₀ = inl ⋆
pattern ₁ = inr ⋆
record Σ {X : Type 𝓁} (Y : X → Type 𝓂) : Type (𝓁 ⊔ 𝓂) where
constructor _,_
field
fst : X
snd : Y fst
open Σ
Σ-induction : {X : Type 𝓁} {Y : X → Type 𝓂} (A : Σ Y → Type 𝓃)
→ ((x : X) (y : Y x) → A (x , y))
→ Π A
Σ-induction A f (x , y) = f x y
_×_ : Type 𝓁 → Type 𝓂 → Type (𝓁 ⊔ 𝓂)
X × Y = Σ (λ (_ : X) → Y)
data Id (X : Type 𝓁) (x : X) : X → Type 𝓁 where
refl : Id X x x
Id-induction : {X : Type 𝓁} {x : X} (A : (y : X) → Id X x y → Type 𝓂)
→ A x refl
→ (y : X) (p : Id X x y) → A y p
Id-induction A σ _ refl = σ
data Id2 (X : Type 𝓁) : X → X → Type 𝓁 where
refl2 : (x : X) → Id2 X x x
Id2-induction : {X : Type 𝓁} (A : (x y : X) → Id2 X x y → Type 𝓂)
→ ((x : X) → A x x (refl2 x))
→ (x y : X) (p : Id2 X x y) → A x y p
Id2-induction A σ x x (refl2 x) = σ x
Id→Id2 : {X : Type 𝓁} → (x y : X) → Id X x y → Id2 X x y
Id→Id2 x x refl = refl2 x
Id2→Id : {X : Type 𝓁} → (x y : X) → Id2 X x y → Id X x y
Id2→Id x x (refl2 x) = refl
_≡_ : {X : Type 𝓁} (x y : X) → Type 𝓁
x ≡ y = Id _ x y
infix 1 _≡_
sym : {X : Type 𝓁} {x y : X} → x ≡ y → y ≡ x
sym refl = refl
_∙_ : {X : Type 𝓁} {x y z : X} → x ≡ y → y ≡ z → x ≡ z
refl ∙ refl = refl
_∙∙_∙∙_ : {X : Type 𝓁} {x y z t : X} → x ≡ y → y ≡ z → z ≡ t → x ≡ t
refl ∙∙ refl ∙∙ refl = refl
_≡⟨_⟩_ : {X : Type 𝓁 } (x : X) {y z : X} → x ≡ y → y ≡ z → x ≡ z
x ≡⟨ p ⟩ q = p ∙ q
infixr 0 _≡⟨_⟩_
_∎ : {X : Type 𝓁} (x : X) → x ≡ x
x ∎ = refl
infix 1 _∎
transport : {X Y : Type 𝓁} → X ≡ Y → X → Y
transport refl x = x
subst : {X : Type 𝓁} (A : X → Type 𝓂) {x y : X} → x ≡ y → A x → A y
subst A refl ax = ax
module _ {X : Type 𝓁} {x : X} where
refl-left : {y : X} (p : x ≡ y) → refl ∙ p ≡ p
refl-left refl = refl
refl-right : {y : X} (p : x ≡ y) → p ∙ refl ≡ p
refl-right refl = refl
∙-assoc : {y z t : X} (p : x ≡ y) (q : y ≡ z) (r : z ≡ t)
→ (p ∙ q) ∙ r ≡ p ∙ (q ∙ r)
∙-assoc refl refl refl = refl
sym-left : {y : X} (p : x ≡ y) → sym p ∙ p ≡ refl
sym-left refl = refl
sym-right : {y : X} (p : x ≡ y) → p ∙ sym p ≡ refl
sym-right refl = refl
sym-involutive : {y : X} (p : x ≡ y) → sym (sym p) ≡ p
sym-involutive refl = refl
∙-cancel-left : {X : Type 𝓁} {x y z : X} {p : x ≡ y} {q r : y ≡ z}
→ p ∙ q ≡ p ∙ r → q ≡ r
∙-cancel-left {p = refl} {q = q} {r = r} s =
sym (refl-left q) ∙∙ s ∙∙ refl-left r
∙-cancel-right : {X : Type 𝓁} {x y z : X} {p q : x ≡ y} {r : y ≡ z}
→ p ∙ r ≡ q ∙ r → p ≡ q
∙-cancel-right {p = p} {q = q} {r = refl} s =
sym (refl-right p) ∙∙ s ∙∙ refl-right q
module _ {X : Type 𝓁} {Y : Type 𝓂} (f : X → Y) where
cong : {x y : X} → x ≡ y → f x ≡ f y
cong refl = refl
cong-refl : (x : X) → cong (refl {x = x}) ≡ refl
cong-refl x = refl
cong-sym : {x y : X} → (p : x ≡ y) → cong (sym p) ≡ sym (cong p)
cong-sym refl = refl
cong-∙ : {x y z : X} (p : x ≡ y) (q : y ≡ z)
→ cong (p ∙ q) ≡ cong p ∙ cong q
cong-∙ refl refl = refl
cong-id : {X : Type 𝓁} {x y : X} (p : x ≡ y) → cong id p ≡ p
cong-id refl = refl
cong-∘ : {X : Type 𝓁} {Y : Type 𝓂} {Z : Type 𝓃}
(f : X → Y) (g : Y → Z) {x y : X} (p : x ≡ y)
→ cong (g ∘ f) p ≡ cong g (cong f p)
cong-∘ f g refl = refl
congd : {X : Type 𝓁} {Y : X → Type 𝓂} (f : Π Y) {x y : X} (p : x ≡ y)
→ subst Y p (f x) ≡ f y
congd f refl = refl
cong₂ : {X : Type 𝓁} {Y : Type 𝓂} {Z : Type 𝓃} (f : X → Y → Z)
{x x' : X} → x ≡ x' → {y y' : Y} → y ≡ y' → f x y ≡ f x' y'
cong₂ f refl refl = refl
¬_ : Type 𝓁 → Type 𝓁
¬ A = A → ⊥
contrapositive : {X : Type 𝓁} {Y : Type 𝓂} → (X → Y) → ¬ Y → ¬ X
contrapositive f p x = p (f x)
decidable : Type 𝓁 → Type 𝓁
decidable A = A + (¬ A)
onAllPaths : (Type 𝓁 → Type 𝓁) → Type 𝓁 → Type 𝓁
onAllPaths A X = (x y : X) → A (x ≡ y)
discrete : Type 𝓁 → Type 𝓁
discrete = onAllPaths decidable
₁≢₀ : ¬ (₁ ≡ ₀)
₁≢₀ p = subst (λ { ₀ → ⊥ ; ₁ → ⊤ }) p ⋆
𝟚-is-discrete : discrete 𝟚
𝟚-is-discrete ₀ ₀ = inl refl
𝟚-is-discrete ₀ ₁ = inr (contrapositive sym ₁≢₀)
𝟚-is-discrete ₁ ₀ = inr ₁≢₀
𝟚-is-discrete ₁ ₁ = inl refl
isCenter : (X : Type 𝓁) → X → Type 𝓁
isCenter X x = (y : X) → x ≡ y
isContr : (X : Type 𝓁) → Type 𝓁
isContr X = Σ (isCenter X)
⊤-is-contr : isContr ⊤
⊤-is-contr = ⋆ , λ { ⋆ → refl }
surrounding : {X : Type 𝓁} (x : X) → Type 𝓁
surrounding x = Σ λ y → x ≡ y
surrounding-is-contr : {X : Type 𝓁} (x : X) → isContr (surrounding x)
surrounding-is-contr x = (x , refl) , λ { (.x , refl) → refl }
isProp : (X : Type 𝓁) → Type 𝓁
isProp X = Π (isCenter X)
⊥-is-prop : isProp ⊥
⊥-is-prop ()
⊤-is-prop : isProp ⊤
⊤-is-prop ⋆ ⋆ = refl
isContr→isProp : {X : Type 𝓁} → isContr X → isProp X
isContr→isProp (c , φ) x y = sym (φ x) ∙ φ y
no-unicorns : (X : Type 𝓁) → isProp X → ¬ (isContr X) → ¬ ¬ X → ⊥
no-unicorns X φ ns ne = ne empty where
empty : ¬ X
empty x = ns (x , φ x)
isSet : (X : Type 𝓁) → Type 𝓁
isSet = onAllPaths isProp
⊥-is-set : isSet ⊥
⊥-is-set ()
⊤-is-set : isSet ⊤
⊤-is-set ⋆ ⋆ refl refl = refl
⊤-is-set' : isSet ⊤
⊤-is-set' =
⊤-induction
(λ x → (z : ⊤) → isProp (x ≡ z))
(Id-induction
(λ y → isCenter (⋆ ≡ y))
(Id-induction
refl-eq
refl
⋆))
where
refl-eq : (x : ⊤) → ⋆ ≡ x → Type
refl-eq =
⊤-induction
(λ x → ⋆ ≡ x → Type)
(Id (⋆ ≡ ⋆) refl)
wconstant : {X : Type 𝓁} {Y : Type 𝓂} → (X → Y) → Type (𝓁 ⊔ 𝓂)
wconstant {X = X} f = (x y : X) → f x ≡ f y
endo : Type 𝓁 → Type 𝓁
endo X = X → X
wconstant-endo : Type 𝓁 → Type 𝓁
wconstant-endo X = Σ λ (f : endo X) → wconstant f
Hedberg : {X : Type 𝓁} (x : X)
→ ((y : X) → wconstant-endo (x ≡ y))
→ (y : X) → isProp (x ≡ y)
Hedberg {X = X} x c y p q =
p
≡⟨ sym (a y p) ⟩
sym (f x refl) ∙ f y p
≡⟨ cong (λ r → sym (f x refl) ∙ r) (c y .snd p q) ⟩
sym (f x refl) ∙ f y q
≡⟨ a y q ⟩
q ∎
where
f : (z : X) → endo (x ≡ z)
f z = c z .fst
a : (z : X) (r : x ≡ z) → sym (f x refl) ∙ f z r ≡ r
a x refl = sym-left (f x refl)
isProp→wconstant-endos : {X : Type 𝓁}
→ isProp X → onAllPaths wconstant-endo X
isProp→wconstant-endos φ x y = (λ _ → φ x y) , (λ _ _ → refl)
isSet→wconstant-endos : {X : Type 𝓁}
→ isSet X → onAllPaths wconstant-endo X
isSet→wconstant-endos φ x y = id , φ x y
wconstant-endos→isSet : {X : Type 𝓁}
→ onAllPaths wconstant-endo X → isSet X
wconstant-endos→isSet c x = Hedberg x (c x)
isProp→isSet : {X : Type 𝓁} → isProp X → isSet X
isProp→isSet = wconstant-endos→isSet ∘ isProp→wconstant-endos
pointed→wconstant-endo : {X : Type 𝓁} → X → wconstant-endo X
pointed→wconstant-endo x = (λ _ → x) , (λ _ _ → refl)
empty→wconstant-endo : {X : Type 𝓁} → ¬ X → wconstant-endo X
empty→wconstant-endo e = id , λ x → ⊥-induction _ (e x)
decidable→wconstant-endo : {X : Type 𝓁} → decidable X → wconstant-endo X
decidable→wconstant-endo (inl x) = pointed→wconstant-endo x
decidable→wconstant-endo (inr e) = empty→wconstant-endo e
discrete→wconstant-endos : {X : Type 𝓁}
→ discrete X → onAllPaths wconstant-endo X
discrete→wconstant-endos φ x y = decidable→wconstant-endo (φ x y)
discrete→isSet : {X : Type 𝓁} → discrete X → isSet X
discrete→isSet = wconstant-endos→isSet ∘ discrete→wconstant-endos
isContrΣ : {X : Type 𝓁} {Y : X → Type 𝓂}
→ isContr X → ((x : X) → isContr (Y x))
→ isContr (Σ Y)
isContrΣ {X = X} {Y = Y} (x₀ , c) cy =
(x₀ , cy x₀ .fst) , λ { (x , y) → f (c x) (cy x .snd y) }
where
f : {x : X} {y : Y x} → x₀ ≡ x → cy x .fst ≡ y → (x₀ , cy x₀ .fst) ≡ (x , y)
f refl refl = refl
isPropΣ : {X : Type 𝓁} {Y : X → Type 𝓂}
→ isProp X → ((x : X) → isProp (Y x))
→ isProp (Σ Y)
isPropΣ {X = X} {Y = Y} φ ψ (x₀ , y₀) (x₁ , y₁) =
f (φ x₀ x₁) (ψ x₁ (subst Y (φ x₀ x₁) y₀) y₁)
where
f : {x : X} {y : Y x} → (p : x₀ ≡ x) → subst Y p y₀ ≡ y → (x₀ , y₀) ≡ (x , y)
f refl refl = refl
_∼_ : {X : Type 𝓁} {Y : X → Type 𝓂} (f g : Π Y) → Type (𝓁 ⊔ 𝓂)
f ∼ g = (x : _) → f x ≡ g x
infix 2 _∼_
deformation-induces-natural-iso : {X : Type 𝓁}
{f : X → X} (H : f ∼ id)
{x y : X} (p : x ≡ y)
→ H x ∙ p ≡ cong f p ∙ H y
deformation-induces-natural-iso H {x = x} refl =
refl-right (H x) ∙ sym (refl-left (H x))
deformation-induces-iso : {X : Type 𝓁} (f : X → X) (H : f ∼ id)
(x : X) → H (f x) ≡ cong f (H x)
deformation-induces-iso f H x =
∙-cancel-right (deformation-induces-natural-iso H (H x))
retraction : {X : Type 𝓁} {Y : Type 𝓂} → (X → Y) → Type (𝓁 ⊔ 𝓂)
retraction f = Σ λ g → g ∘ f ∼ id
section : {X : Type 𝓁} {Y : Type 𝓂} → (X → Y) → Type (𝓁 ⊔ 𝓂)
section f = Σ λ h → f ∘ h ∼ id
_◁_ : Type 𝓁 → Type 𝓂 → Type (𝓁 ⊔ 𝓂)
X ◁ Y = Σ λ (r : Y → X) → section r
isContrRetract : {X : Type 𝓁} {Y : Type 𝓂}
→ Y ◁ X → isContr X → isContr Y
isContrRetract {Y = Y} (r , (s , η)) (c , φ) = r c , d
where
d : isCenter Y (r c)
d y = r c ≡⟨ cong r (φ (s y)) ⟩ r (s y) ≡⟨ η y ⟩ y ∎
isPropRetract : {X : Type 𝓁} {Y : Type 𝓂}
→ Y ◁ X → isProp X → isProp Y
isPropRetract {Y = Y} (r , (s , η)) φ y₀ y₁ =
y₀
≡⟨ sym (η y₀) ⟩
r (s y₀)
≡⟨ cong r (φ (s y₀) (s y₁)) ⟩
r (s y₁)
≡⟨ η y₁ ⟩
y₁ ∎
Σ-retract : {X : Type 𝓁} (A : X → Type 𝓂) (B : X → Type 𝓃)
→ ((x : X) → A x ◁ B x) → Σ A ◁ Σ B
Σ-retract A B ρ = r , (s , η)
where
r : Σ B → Σ A
r (x , b) = x , (ρ x .fst b)
s : Σ A → Σ B
s (x , a) = x , ρ x .snd .fst a
η : r ∘ s ∼ id
η (x , a) = cong (_,_ x) (ρ x .snd .snd a)
subst-is-retraction : {X : Type 𝓁} (A : X → Type 𝓂) {x y : X} (p : x ≡ y)
→ subst A p ∘ subst A (sym p) ∼ id
subst-is-retraction A refl ay = refl
subst-is-section : {X : Type 𝓁} (A : X → Type 𝓂) {x y : X} (p : x ≡ y)
→ subst A (sym p) ∘ subst A p ∼ id
subst-is-section A refl ax = refl
module _ {X : Type 𝓁} {A : X → Type 𝓃} where
to-Σ≡ : {σ τ : Σ A}
→ Σ (λ (p : σ .fst ≡ τ .fst) → subst A p (σ .snd) ≡ τ .snd)
→ σ ≡ τ
to-Σ≡ (refl , refl) = refl
from-Σ≡ : {σ τ : Σ A}
→ σ ≡ τ
→ Σ (λ (p : σ .fst ≡ τ .fst) → subst A p (σ .snd) ≡ τ .snd)
from-Σ≡ refl = (refl , refl)
to-Σ≡-is-retraction : {σ τ : Σ A} → to-Σ≡ {σ} {τ} ∘ from-Σ≡ {σ} {τ} ∼ id
to-Σ≡-is-retraction refl = refl
to-Σ≡-is-section : {σ τ : Σ A} → from-Σ≡ {σ} {τ} ∘ to-Σ≡ {σ} {τ} ∼ id
to-Σ≡-is-section (refl , refl) = refl
isSetΣ : isSet X → ((x : X) → isSet (A x)) → isSet (Σ A)
isSetΣ φ ψ (x₀ , y₀) (x₁ , y₁) =
isPropRetract
(to-Σ≡ , (from-Σ≡ , to-Σ≡-is-retraction))
(isPropΣ (φ x₀ x₁) (λ x → ψ x₁ (subst A x y₀) y₁))
Σ-reindexing-retract : {X : Type 𝓁} {Y : Type 𝓂} (A : X → Type 𝓃) (r : Y → X)
→ section r
→ Σ A ◁ Σ (A ∘ r)
Σ-reindexing-retract A r (s , η) = r' , (s' , η')
where
r' : Σ (A ∘ r) → Σ A
r' (y , a) = r y , a
s' : Σ A → Σ (A ∘ r)
s' (x , a) = s x , subst A (sym (η x)) a
η' : r' ∘ s' ∼ id
η' (x , a) = to-Σ≡ (η x , subst-is-retraction A (η x) a)
module Equiv {X : Type 𝓁} {Y : Type 𝓂} (f : X → Y) where
fiber : Y → Type (𝓁 ⊔ 𝓂)
fiber y = Σ λ x → f x ≡ y
isEquiv : Type (𝓁 ⊔ 𝓂)
isEquiv = (y : Y) → isContr (fiber y)
inverse : isEquiv → Y → X
inverse eq y = eq y .fst .fst
inverse-is-section : (eq : isEquiv) → f ∘ inverse eq ∼ id
inverse-is-section eq y = eq y .fst .snd
inverse-is-retraction : (eq : isEquiv) → inverse eq ∘ f ∼ id
inverse-is-retraction eq x = cong fst p where
p : Id (fiber (f x)) (eq (f x) .fst) (x , refl)
p = eq (f x) .snd (x , refl)
isInvertible : Type (𝓁 ⊔ 𝓂)
isInvertible = retraction f × section f
isEquiv→isInvertible : isEquiv → isInvertible
isEquiv→isInvertible eq =
(inverse eq , inverse-is-retraction eq)
, (inverse eq , inverse-is-section eq)
toFiberEq : {y : Y} {σ : fiber y} (τ : fiber y)
→ Σ (λ (γ : σ .fst ≡ τ .fst) → (cong f γ ∙ τ .snd ≡ σ .snd))
→ σ ≡ τ
toFiberEq τ (refl , refl) = cong (λ p → (τ .fst , p)) (refl-left (τ .snd))
record isHAEquiv : Type (𝓁 ⊔ 𝓂) where
field
g : Y → X
η : g ∘ f ∼ id
ε : f ∘ g ∼ id
ha : (x : X) → cong f (η x) ≡ ε (f x)
open isHAEquiv
isHAEquiv→isInvertible : isHAEquiv → isInvertible
isHAEquiv→isInvertible eq = (eq .g , eq .η) , (eq .g , eq .ε)
isInvertible→isHAEquiv : isInvertible → isHAEquiv
isInvertible→isHAEquiv ((g₀ , η₀) , (h₀ , ε₀)) = record {
g = g₀
; η = η₀
; ε = ε₂
; ha = λ x → sym (ha₀ x)
} where
ε₁ : f ∘ g₀ ∼ id
ε₁ y = sym (cong (f ∘ g₀) (ε₀ y)) ∙ (cong f (η₀ (h₀ y)) ∙ ε₀ y)
ε₂ : f ∘ g₀ ∼ id
ε₂ y = sym (ε₁ (f (g₀ y))) ∙ (cong f (η₀ (g₀ y)) ∙ ε₁ y)
ha₀ : (x : X) → ε₂ (f x) ≡ cong f (η₀ x)
ha₀ x =
sym (ε₁ (f (g₀ (f x)))) ∙ (cong f (η₀ (g₀ (f x))) ∙ ε₁ (f x))
≡⟨ cong (λ p → sym (ε₁ (f (g₀ (f x)))) ∙ p)
(
cong f (η₀ (g₀ (f x))) ∙ ε₁ (f x)
≡⟨ cong (λ p → cong f p ∙ ε₁ (f x))
(deformation-induces-iso (g₀ ∘ f) η₀ x) ⟩
cong f (cong (g₀ ∘ f) (η₀ x)) ∙ ε₁ (f x)
≡⟨ cong (λ p → p ∙ ε₁ (f x))
(
cong f (cong (g₀ ∘ f) (η₀ x))
≡⟨ sym (cong-∘ (g₀ ∘ f) f (η₀ x)) ⟩
cong (f ∘ g₀ ∘ f) (η₀ x)
≡⟨ cong-∘ f (f ∘ g₀) (η₀ x) ⟩
cong (f ∘ g₀) (cong f (η₀ x)) ∎ ) ⟩
cong (f ∘ g₀) (cong f (η₀ x)) ∙ ε₁ (f x)
≡⟨ sym (deformation-induces-natural-iso ε₁ (cong f (η₀ x))) ⟩
ε₁ (f (g₀ (f x))) ∙ cong f (η₀ x) ∎ )
⟩
sym (ε₁ (f (g₀ (f x)))) ∙ (ε₁ (f (g₀ (f x))) ∙ cong f (η₀ x))
≡⟨ sym (∙-assoc _ _ _) ⟩
(sym (ε₁ (f (g₀ (f x)))) ∙ ε₁ (f (g₀ (f x)))) ∙ cong f (η₀ x)
≡⟨ cong (λ p → p ∙ cong f (η₀ x)) (sym-left _) ⟩
refl ∙ cong f (η₀ x)
≡⟨ refl-left _ ⟩
cong f (η₀ x) ∎
isHAEquiv→isEquiv : isHAEquiv → isEquiv
isHAEquiv→isEquiv eq y =
(eq .g y , eq .ε y)
, λ τ → toFiberEq τ (γ τ , lem τ)
where
γ : (τ : fiber y) → eq .g y ≡ τ .fst
γ (x , p) = cong (eq .g) (sym p) ∙ eq .η x
natural : {h : Y → Y} (e : h ∼ id) {z z' : Y} (q : z ≡ z')
→ (sym (cong h q) ∙ e z) ∙ q ≡ e z'
natural e {z = z} refl = refl-right (refl ∙ e z) ∙ refl-left (e z)
lem : (τ : fiber y) → cong f (γ τ) ∙ τ .snd ≡ eq .ε y
lem (x , p) =
cong f (cong (eq .g) (sym p) ∙ eq .η x) ∙ p
≡⟨ cong (λ q → q ∙ p)
(
cong f (cong (eq .g) (sym p) ∙ eq .η x)
≡⟨ cong-∙ f (cong (eq .g) (sym p)) (eq .η x) ⟩
cong f (cong (eq .g) (sym p)) ∙ cong f (eq .η x)
≡⟨ cong₂ _∙_
(sym (cong-∘ (eq .g) f (sym p)) ∙ cong-sym (f ∘ eq .g) p)
(eq .ha x) ⟩
sym (cong (f ∘ eq .g) p) ∙ eq .ε (f x) ∎ ) ⟩
(sym (cong (f ∘ eq .g) p) ∙ eq .ε (f x)) ∙ p
≡⟨ natural (eq .ε) p ⟩
eq .ε y ∎
open Equiv
open isHAEquiv
_≃_ : Type 𝓁 → Type 𝓂 → Type (𝓁 ⊔ 𝓂)
X ≃ Y = Σ λ (f : (X → Y)) → isHAEquiv f
idIsHAEquiv : (X : Type 𝓁) → isHAEquiv (id {X = X})
idIsHAEquiv X = record {
g = id
; η = λ x → refl
; ε = λ x → refl
; ha = λ x → refl }
id-≃ : (X : Type 𝓁) → X ≃ X
id-≃ X = (id , idIsHAEquiv X)
∘-≃ : {X : Type 𝓁} {Y : Type 𝓂} {Z : Type 𝓃} → X ≃ Y → Y ≃ Z → X ≃ Z
∘-≃ {X = X} {Y = Y} {Z = Z} (f , eqf) (h , eqh) =
(h ∘ f)
, record { g = g₀ ; η = η₀ ; ε = ε₀ ; ha = ha₀ }
where
g₀ : Z → X
g₀ = eqf .g ∘ eqh .g
η₀ : g₀ ∘ (h ∘ f) ∼ id
η₀ x = cong (eqf .g) (eqh .η (f x)) ∙ eqf .η x
ε₀ : (h ∘ f) ∘ g₀ ∼ id
ε₀ z = cong h (eqf .ε (eqh .g z)) ∙ eqh .ε z
ha₀ : (x : X) → cong (h ∘ f) (η₀ x) ≡ ε₀ (h (f x))
ha₀ x =
cong (h ∘ f) (cong (eqf .g) (eqh .η (f x)) ∙ eqf .η x)
≡⟨ cong-∙ (h ∘ f) _ _ ⟩
cong (h ∘ f) (cong (eqf .g) (eqh .η (f x))) ∙ cong (h ∘ f) (eqf .η x)
≡⟨ cong₂ _∙_
(sym (cong-∘ (eqf .g) (h ∘ f) (eqh .η (f x))))
(cong-∘ f h (eqf .η x)) ⟩
cong (h ∘ f ∘ eqf .g) (eqh .η (f x)) ∙ cong h (cong f (eqf .η x))
≡⟨ cong₂ _∙_
(cong-∘ (f ∘ eqf .g) h (eqh .η (f x)))
(cong (cong h) (eqf .ha x)) ⟩
cong h (cong (f ∘ eqf .g) (eqh .η (f x))) ∙ cong h (eqf .ε (f x))
≡⟨ sym (cong-∙ h _ _) ⟩
cong h (cong (f ∘ eqf .g) (eqh .η (f x)) ∙ eqf .ε (f x))
≡⟨ cong (cong h) (sym (deformation-induces-natural-iso (eqf .ε) (eqh .η (f x)))) ⟩
cong h (eqf .ε (eqh .g (h (f x))) ∙ eqh .η (f x))
≡⟨ cong-∙ h _ _ ⟩
cong h (eqf .ε (eqh .g (h (f x)))) ∙ cong h (eqh .η (f x))
≡⟨ cong (λ p → _ ∙ p) (eqh .ha (f x)) ⟩
cong h (eqf .ε (eqh .g (h (f x)))) ∙ eqh .ε (h (f x)) ∎
sym-≃ : {X : Type 𝓁} {Y : Type 𝓂} → X ≃ Y → Y ≃ X
sym-≃ {X = X} {Y = Y} (f , eq) =
eq .g , record { g = f ; η = eq .ε ; ε = eq .η ; ha = ha₀ }
where
p : (y : Y)
→ cong (eq .g ∘ f ∘ eq .g) (eq .ε y) ∙ eq .η (eq .g y)
≡ cong (eq .g ∘ f ∘ eq .g) (eq .ε y) ∙ cong (eq .g) (eq .ε y)
p y =
cong (eq .g ∘ f ∘ eq .g) (eq .ε y) ∙ eq .η (eq .g y)
≡⟨ cong (λ p → p ∙ _) (cong-∘ (eq .g) (eq .g ∘ f) (eq .ε y)) ⟩
cong (eq .g ∘ f) (cong (eq .g) (eq .ε y)) ∙ eq .η (eq .g y)
≡⟨ sym (deformation-induces-natural-iso (eq .η) (cong (eq .g) (eq .ε y))) ⟩
eq .η (eq .g (f (eq .g y))) ∙ cong (eq .g) (eq .ε y)
≡⟨ cong (λ p → p ∙ _) (deformation-induces-iso (eq .g ∘ f) (eq .η) (eq .g y)) ⟩
cong (eq .g ∘ f) (eq .η (eq .g y)) ∙ cong (eq .g) (eq .ε y)
≡⟨ cong (λ p → p ∙ cong (eq .g) (eq .ε y)) (cong-∘ f (eq .g) (eq .η (eq .g y))) ⟩
cong (eq .g) (cong f (eq .η (eq .g y))) ∙ cong (eq .g) (eq .ε y)
≡⟨ cong (λ p → cong (eq .g) p ∙ cong (eq .g) (eq .ε y)) (eq .ha (eq .g y)) ⟩
cong (eq .g) (eq .ε (f (eq .g y))) ∙ cong (eq .g) (eq .ε y)
≡⟨ sym (cong-∙ (eq .g) (eq .ε (f (eq .g y))) (eq .ε y)) ⟩
cong (eq .g) (eq .ε (f (eq .g y)) ∙ eq .ε y)
≡⟨ cong (cong (eq .g)) (deformation-induces-natural-iso (eq .ε) (eq .ε y)) ⟩
cong (eq .g) (cong (f ∘ eq .g) (eq .ε y) ∙ eq .ε y)
≡⟨ cong-∙ (eq .g) (cong (f ∘ eq .g) (eq .ε y)) (eq .ε y) ⟩
cong (eq .g) (cong (f ∘ eq .g) (eq .ε y)) ∙ cong (eq .g) (eq .ε y)
≡⟨ cong (λ p → p ∙ cong (eq .g) (eq .ε y)) (sym (cong-∘ (f ∘ eq .g) (eq .g) (eq .ε y))) ⟩
cong (eq .g ∘ f ∘ eq .g) (eq .ε y) ∙ cong (eq .g) (eq .ε y) ∎
ha₀ : (y : Y) → cong (eq .g) (eq .ε y) ≡ eq .η (eq .g y)
ha₀ y = ∙-cancel-left (sym (p y))
cong-const : {X : Type 𝓁} {Y : Type 𝓂} (y : Y) {x x' : X} {p : x ≡ x'}
→ cong (λ _ → y) p ≡ refl
cong-const y {p = refl} = refl
contr-fiber : {X : Type 𝓁} (A : X → Type 𝓂) → ((x : X) → isContr (A x))
→ isHAEquiv (λ (a : Σ A) → a .fst)
contr-fiber {X = X} A c = record { g = g₀ ; η = η₀ ; ε = ε₀ ; ha = ha₀ }
where
g₀ : X → Σ A
g₀ x = x , c x .fst
η₀ : g₀ ∘ fst ∼ id
η₀ (x , a) = cong (_,_ x) (c x .snd a)
ε₀ : fst ∘ g₀ ∼ id
ε₀ x = refl
ha₀ : (a : Σ A) → cong fst (η₀ a) ≡ ε₀ (fst a)
ha₀ (x , a) = cong fst (cong (_,_ x) (c x .snd a))
≡⟨ sym (cong-∘ (_,_ x) fst (c x .snd a)) ⟩
cong (λ _ → x) (c x .snd a)
≡⟨ cong-const x ⟩
refl
≡⟨ refl ⟩
ε₀ x ∎
Id→Eq : (X Y : Type 𝓁) → X ≡ Y → X ≃ Y
Id→Eq X X refl = id-≃ X
isUnivalent : (𝓁 : Level) → Type (𝓁 ⁺)
isUnivalent 𝓁 = (X Y : Type 𝓁) → isHAEquiv (Id→Eq X Y)
|
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module Numeral.Natural.Relation.Properties where
import Lvl
open import Data.Tuple as Tuple using (_⨯_ ; _,_)
open import Functional
open import Logic.Propositional
open import Logic.Propositional.Theorems
open import Logic.Predicate
open import Numeral.Natural
open import Numeral.Natural.Oper
open import Numeral.Natural.Oper.Proofs
open import Numeral.Natural.Induction
open import Numeral.Natural.Relation
open import Relator.Equals
open import Relator.Equals.Proofs
open import Structure.Function.Domain
open import Structure.Operator.Properties
open import Structure.Relator.Ordering
open import Structure.Relator.Properties
open import Type
[ℕ]-zero-or-nonzero : ∀{n : ℕ} → (n ≡ 𝟎)∨(n ≢ 𝟎)
[ℕ]-zero-or-nonzero {𝟎} = [∨]-introₗ [≡]-intro
[ℕ]-zero-or-nonzero {𝐒(_)} = [∨]-introᵣ \()
[≡][ℕ]-excluded-middle : ∀{a b : ℕ} → (a ≡ b)∨(a ≢ b)
[≡][ℕ]-excluded-middle {𝟎} {𝟎} = [∨]-introₗ [≡]-intro
[≡][ℕ]-excluded-middle {𝟎} {𝐒(_)} = [∨]-introᵣ \()
[≡][ℕ]-excluded-middle {𝐒(_)}{𝟎} = [∨]-introᵣ \()
[≡][ℕ]-excluded-middle {𝐒(a)}{𝐒(b)} = [∨]-elim ([∨]-introₗ ∘ [≡]-with(𝐒)) ([∨]-introᵣ ∘ (contrapositiveᵣ(injective(𝐒)))) ([≡][ℕ]-excluded-middle {a}{b})
|
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{-# OPTIONS --safe #-}
open import Definition.Typed.EqualityRelation
module Definition.LogicalRelation.Substitution.Weakening {{eqrel : EqRelSet}} where
open EqRelSet {{...}}
open import Definition.Untyped
open import Definition.Untyped.Properties
open import Definition.LogicalRelation
open import Definition.LogicalRelation.Irrelevance
open import Definition.LogicalRelation.Substitution
open import Definition.LogicalRelation.Substitution.MaybeEmbed
open import Definition.LogicalRelation.Substitution.Introductions.Universe
open import Tools.Product
import Tools.PropositionalEquality as PE
-- Weakening of valid types by one.
wk1ᵛ : ∀ {A F rA rF Γ l l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ])
→ Γ ⊩ᵛ⟨ l ⟩ A ^ rA / [Γ]
→ Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ wk1 A ^ rA / ([Γ] ∙ [F])
wk1ᵛ {A} [Γ] [F] [A] ⊢Δ [σ] =
let [σA] = proj₁ ([A] ⊢Δ (proj₁ [σ]))
[σA]′ = irrelevance′ (PE.sym (subst-wk A)) [σA]
in [σA]′
, (λ [σ′] [σ≡σ′] →
irrelevanceEq″ (PE.sym (subst-wk A))
(PE.sym (subst-wk A)) PE.refl PE.refl
[σA] [σA]′
(proj₂ ([A] ⊢Δ (proj₁ [σ])) (proj₁ [σ′]) (proj₁ [σ≡σ′])))
-- Weakening of valid type equality by one.
wk1Eqᵛ : ∀ {A B F rA rF Γ l l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ])
([A] : Γ ⊩ᵛ⟨ l ⟩ A ^ rA / [Γ])
([A≡B] : Γ ⊩ᵛ⟨ l ⟩ A ≡ B ^ rA / [Γ] / [A])
→ Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ wk1 A ≡ wk1 B ^ rA / [Γ] ∙ [F] / wk1ᵛ {A} {F} [Γ] [F] [A]
wk1Eqᵛ {A} {B} [Γ] [F] [A] [A≡B] ⊢Δ [σ] =
let [σA] = proj₁ ([A] ⊢Δ (proj₁ [σ]))
[σA]′ = irrelevance′ (PE.sym (subst-wk A)) [σA]
in irrelevanceEq″ (PE.sym (subst-wk A))
(PE.sym (subst-wk B)) PE.refl PE.refl
[σA] [σA]′
([A≡B] ⊢Δ (proj₁ [σ]))
-- Weakening of valid term as a type by one.
wk1ᵗᵛ : ∀ {F G rF rG lG Γ l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ]) →
let l = ∞
[UG] = maybeEmbᵛ {A = Univ rG _} [Γ] (Uᵛ (proj₂ (levelBounded lG)) [Γ])
[wUG] = maybeEmbᵛ {A = Univ rG _} (_∙_ {A = F} [Γ] [F]) (λ {Δ} {σ} → Uᵛ (proj₂ (levelBounded lG)) (_∙_ {A = F} [Γ] [F]) {Δ} {σ})
in Γ ⊩ᵛ⟨ l ⟩ G ∷ Univ rG lG ^ [ ! , next lG ] / [Γ] / [UG] →
Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ wk1 G ∷ Univ rG lG ^ [ ! , next lG ] / ([Γ] ∙ [F]) / (λ {Δ} {σ} → [wUG] {Δ} {σ})
wk1ᵗᵛ {F} {G} {rF} {rG} {lG} [Γ] [F] [G]ₜ {Δ} {σ} ⊢Δ [σ] =
let l = ∞
[UG] = maybeEmbᵛ {A = Univ rG _} [Γ] (Uᵛ (proj₂ (levelBounded lG)) [Γ])
[wUG] = maybeEmbᵛ {A = Univ rG _} (_∙_ {A = F} [Γ] [F]) (λ {Δ} {σ} → Uᵛ (proj₂ (levelBounded lG)) (_∙_ {A = F} [Γ] [F]) {Δ} {σ})
[σG] = proj₁ ([G]ₜ ⊢Δ (proj₁ [σ]))
[Geq] = PE.sym (subst-wk G)
[σG]′ = irrelevanceTerm″ PE.refl PE.refl PE.refl [Geq] (proj₁ ([UG] ⊢Δ (proj₁ [σ]))) (proj₁ ([wUG] {Δ} {σ} ⊢Δ [σ])) [σG]
in [σG]′
, (λ [σ′] [σ≡σ′] →
irrelevanceEqTerm″ PE.refl PE.refl
(PE.sym (subst-wk G))
(PE.sym (subst-wk G)) PE.refl
(proj₁ ([UG] ⊢Δ (proj₁ [σ]))) (proj₁ ([wUG] {Δ} {σ} ⊢Δ [σ]))
(proj₂ ([G]ₜ ⊢Δ (proj₁ [σ])) (proj₁ [σ′]) (proj₁ [σ≡σ′])))
wk1Termᵛ : ∀ {F G rF rG t Γ l l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ]) →
([G] : Γ ⊩ᵛ⟨ l ⟩ G ^ rG / [Γ]) →
Γ ⊩ᵛ⟨ l ⟩ t ∷ G ^ rG / [Γ] / [G] →
Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ wk1 t ∷ wk1 G ^ rG / ([Γ] ∙ [F]) / wk1ᵛ {A = G} {F = F} [Γ] [F] [G]
wk1Termᵛ {F} {G} {rF} {rG} {t} [Γ] [F] [G] [t]ₜ {Δ} {σ} ⊢Δ [σ] =
let [σt] = proj₁ ([t]ₜ ⊢Δ (proj₁ [σ]))
[σG] = proj₁ ([G] ⊢Δ (proj₁ [σ]))
[teq] = PE.sym (subst-wk {step id} {σ} t)
[Geq] = PE.sym (subst-wk {step id} {σ} G)
[σG]' = irrelevance′ [Geq] [σG]
in irrelevanceTerm″ [Geq] PE.refl PE.refl [teq] [σG] [σG]' [σt] ,
λ [σ′] [σ≡σ′] → irrelevanceEqTerm″ PE.refl PE.refl
(PE.sym (subst-wk t)) (PE.sym (subst-wk t)) (PE.sym (subst-wk G))
[σG] [σG]' (proj₂ ([t]ₜ ⊢Δ (proj₁ [σ])) (proj₁ [σ′]) (proj₁ [σ≡σ′]))
wk1dᵛ : ∀ {F F' G rF rF' lG Γ l l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ]) →
([F'] : Γ ⊩ᵛ⟨ l' ⟩ F' ^ rF' / [Γ]) →
let [ΓF] = _∙_ {A = F} [Γ] [F]
[ΓF'] = _∙_ {A = F'} [Γ] [F']
[ΓF'F] = _∙_ {A = wk1 F} [ΓF'] (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F])
in Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ G ^ [ ! , lG ] / [ΓF] →
Γ ∙ F' ^ rF' ∙ wk1 F ^ rF ⊩ᵛ⟨ l ⟩ wk1d G ^ [ ! , lG ] / [ΓF'F]
wk1dᵛ {F} {F'} {G} [Γ] [F] [F'] [G] {Δ} {σ} ⊢Δ [σ] =
let l = ∞
[ΓF'] = _∙_ {A = F'} [Γ] [F']
[ΓF'F] = _∙_ {A = wk1 F} [ΓF'] (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F])
[wσ] = proj₁ (proj₁ [σ]) , irrelevanceTerm″ (subst-wk F) PE.refl PE.refl PE.refl
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ]))))
(proj₂ [σ])
[σG] = proj₁ ([G] ⊢Δ [wσ])
[Geq] = PE.sym (subst-wk G)
[σG]′ = irrelevance′ [Geq] [σG]
in [σG]′
, (λ {σ′} [σ′] [σ≡σ′] → let [wσ′] = proj₁ (proj₁ [σ′]) ,
irrelevanceTerm″ (subst-wk F) PE.refl PE.refl PE.refl
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ′])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ′]))))
(proj₂ [σ′])
[wσ≡σ′] = (proj₁ (proj₁ [σ≡σ′])),
irrelevanceEqTerm″ PE.refl PE.refl PE.refl PE.refl (subst-wk F)
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ]))))
(proj₂ [σ≡σ′])
in irrelevanceEq″ (PE.sym (subst-wk G)) (PE.sym (subst-wk G)) PE.refl PE.refl
(proj₁ ([G] ⊢Δ [wσ])) [σG]′
(proj₂ ([G] ⊢Δ [wσ]) [wσ′] [wσ≡σ′]))
wk1dᵗᵛ : ∀ {F F' G rF rF' rG lG Γ l l'}
([Γ] : ⊩ᵛ Γ)
([F] : Γ ⊩ᵛ⟨ l' ⟩ F ^ rF / [Γ]) →
([F'] : Γ ⊩ᵛ⟨ l' ⟩ F' ^ rF' / [Γ]) →
let [ΓF] = _∙_ {A = F} [Γ] [F]
[ΓF'] = _∙_ {A = F'} [Γ] [F']
[ΓF'F] = _∙_ {A = wk1 F} [ΓF'] (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F])
in ([UG] : (Γ ∙ F ^ rF) ⊩ᵛ⟨ l ⟩ Univ rG lG ^ [ ! , next lG ] / [ΓF]) →
([wUG] : (Γ ∙ F' ^ rF' ∙ wk1 F ^ rF) ⊩ᵛ⟨ l ⟩ Univ rG lG ^ [ ! , next lG ] / [ΓF'F]) →
Γ ∙ F ^ rF ⊩ᵛ⟨ l ⟩ G ∷ Univ rG lG ^ [ ! , next lG ] / [ΓF] / (λ {Δ} {σ} → [UG] {Δ} {σ}) →
Γ ∙ F' ^ rF' ∙ wk1 F ^ rF ⊩ᵛ⟨ l ⟩ wk1d G ∷ Univ rG lG ^ [ ! , next lG ] / [ΓF'F] / (λ {Δ} {σ} → [wUG] {Δ} {σ})
wk1dᵗᵛ {F} {F'} {G} {rF} {rF'} {rG} {lG} [Γ] [F] [F'] [UG] [wUG] [G]ₜ {Δ} {σ} ⊢Δ [σ] =
let l = ∞
[ΓF'] = _∙_ {A = F'} [Γ] [F']
[ΓF'F] = _∙_ {A = wk1 F} [ΓF'] (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F])
[wσ] = proj₁ (proj₁ [σ]) , irrelevanceTerm″ (subst-wk F) PE.refl PE.refl PE.refl
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ]))))
(proj₂ [σ])
[σG] = proj₁ ([G]ₜ ⊢Δ [wσ])
[Geq] = PE.sym (subst-wk G)
[σG]′ = irrelevanceTerm″ PE.refl PE.refl PE.refl [Geq] (proj₁ ([UG] ⊢Δ [wσ])) (proj₁ ([wUG] {Δ} {σ} ⊢Δ [σ])) [σG]
in [σG]′
, (λ {σ′} [σ′] [σ≡σ′] → let [wσ′] = proj₁ (proj₁ [σ′]) ,
irrelevanceTerm″ (subst-wk F) PE.refl PE.refl PE.refl
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ′])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ′]))))
(proj₂ [σ′])
[wσ≡σ′] = (proj₁ (proj₁ [σ≡σ′])),
irrelevanceEqTerm″ PE.refl PE.refl PE.refl PE.refl (subst-wk F)
(proj₁ (wk1ᵛ {A = F} {F = F'} [Γ] [F'] [F] ⊢Δ (proj₁ [σ])))
(proj₁ ([F] ⊢Δ (proj₁ (proj₁ [σ]))))
(proj₂ [σ≡σ′])
in irrelevanceEqTerm″ PE.refl PE.refl
(PE.sym (subst-wk G))
(PE.sym (subst-wk G)) PE.refl
(proj₁ ([UG] ⊢Δ [wσ])) (proj₁ ([wUG] {Δ} {σ} ⊢Δ [σ]))
(proj₂ ([G]ₜ ⊢Δ [wσ]) [wσ′] [wσ≡σ′]))
|
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module Data.Finitude where
open import Data.Fin as Fin
open import Data.Nat as ℕ
open import Level
open import Function as Fun hiding (id; _∘_)
open import Function.Equality as F using (_⟨$⟩_)
open import Function.Injection as Inj hiding (id; _∘_)
open import Function.Bijection as Bij hiding (id; _∘_)
open import Function.LeftInverse hiding (id; _∘_)
open import Function.Inverse as Inv hiding (id; _∘_)
open import Relation.Binary
import Relation.Binary.PropositionalEquality as P
open import Relation.Binary.PropositionalEquality using (_≡_; _≢_)
Finitude : ∀ {a ℓ} (A : Setoid a ℓ) (n : ℕ) → Set _
Finitude A n = Inverse A (P.setoid (Fin n))
module Subset where
open import Data.Product as Prod hiding (map)
open import Data.Fin.Subset using (Subset; _∈_; outside; inside; ∣_∣) renaming (⊥ to ∅)
open import Data.Vec
subset-finitude : ∀ {n}(s : Subset n) → Finitude (P.setoid (∃ (_∈ s))) ∣ s ∣
subset-finitude {ℕ.zero} [] = record {
to = P.→-to-⟶ (λ ())
; from = P.→-to-⟶ (λ () )
; inverse-of = record {
left-inverse-of = λ ()
; right-inverse-of = λ ()
}
}
subset-finitude {ℕ.suc n} (inside ∷ s) = record {
to = P.→-to-⟶ λ { (_ , here) → Fin.zero
; (_ , there p) → Fin.suc (to ⟨$⟩ (_ , p)) }
; from = P.→-to-⟶ λ { Fin.zero → _ , here
; (Fin.suc i) → _ , there (proj₂ (from ⟨$⟩ i))}
; inverse-of = record {
left-inverse-of = λ { (_ , here) → P.refl
; (_ , there p) → P.cong
(Prod.map Fin.suc there) (linv (_ , p))}
; right-inverse-of = λ { Fin.zero → P.refl
; (Fin.suc i) → P.cong Fin.suc (rinv i)}
}
}
where
open Inverse (subset-finitude s)
open _InverseOf_ inverse-of renaming (left-inverse-of to linv
;right-inverse-of to rinv)
subset-finitude {ℕ.suc n} (outside ∷ s) = record {
to = P.→-to-⟶ (λ { (_ , there p) → to ⟨$⟩ (_ , p)})
; from = P.→-to-⟶ (λ i → Prod.map Fin.suc there (from ⟨$⟩ i) )
; inverse-of = record {
left-inverse-of = λ { (_ , there p) →
P.cong (Prod.map Fin.suc there) (linv (_ , p)) }
; right-inverse-of = rinv
}
}
where
open Inverse (subset-finitude s)
open _InverseOf_ inverse-of renaming ( left-inverse-of to linv
; right-inverse-of to rinv)
|
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{-# OPTIONS --cubical --safe #-}
module Container.List.Syntax where
open import Prelude
open import Container
open import Container.List
open import Data.Fin
record ListSyntax {a b} (A : Type a) (B : Type b) : Type (a ℓ⊔ b) where
field [_] : B → List A
open ListSyntax ⦃ ... ⦄ public
instance
cons : ⦃ _ : ListSyntax A B ⦄ → ListSyntax A (A × B)
[_] ⦃ cons ⦄ (x , xs) .fst = suc ([ xs ] .fst)
[_] ⦃ cons ⦄ (x , xs) .snd f0 = x
[_] ⦃ cons ⦄ (x , xs) .snd (fs n) = [ xs ] .snd n
instance
sing : ListSyntax A A
[_] ⦃ sing ⦄ x = 1 , const x
|
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open import Formalization.PredicateLogic.Signature
module Formalization.PredicateLogic.Syntax.Substitution (𝔏 : Signature) where
open Signature(𝔏)
open import Data.Boolean
open import Data.ListSized
import Data.ListSized.Functions as List
open import Formalization.PredicateLogic.Syntax(𝔏)
open import Functional using (_∘_ ; _∘₂_ ; id ; apply)
open import Numeral.CoordinateVector as Vector using (Vector)
open import Numeral.Finite
open import Numeral.Natural
open import Syntax.Function
open import Type
private variable args n vars vars₁ vars₂ : ℕ
-- Substitutes the variables of a term by mapping every variable index to a term.
substituteTerm : Vector(vars₁)(Term(vars₂)) → Term(vars₁) → Term(vars₂)
substituteTerm₊ : Vector(vars₁)(Term(vars₂)) → List(Term(vars₁))(args) → List(Term(vars₂))(args)
substituteTerm t (var v) = Vector.proj t v
substituteTerm t (func f x) = func f (substituteTerm₊ t x)
substituteTerm₊ t ∅ = ∅
substituteTerm₊ t (x ⊰ xs) = (substituteTerm t x) ⊰ (substituteTerm₊ t xs)
-- Adds a new untouched variable to a term mapper.
-- Example: termMapper𝐒(0 ↦ t0 ; 1 ↦ t1 ; 2 ↦ t2) = (0 ↦ var 0 ; 1 ↦ t0 ; 2 ↦ t1 ; 3 ↦ t2)
termMapper𝐒 : Vector(vars₁)(Term(vars₂)) → Vector(𝐒(vars₁))(Term(𝐒(vars₂)))
termMapper𝐒 = Vector.prepend(var 𝟎) ∘ Vector.map(substituteTerm(var ∘ 𝐒))
-- Substitutes the variables of a formula by mapping every variable index to a term.
substitute : Vector(vars₁)(Term(vars₂)) → Formula(vars₁) → Formula(vars₂)
substitute t (P $ x) = P $ (substituteTerm₊ t x)
substitute t ⊤ = ⊤
substitute t ⊥ = ⊥
substitute t (φ ∧ ψ) = (substitute t φ) ∧ (substitute t ψ)
substitute t (φ ∨ ψ) = (substitute t φ) ∨ (substitute t ψ)
substitute t (φ ⟶ ψ) = (substitute t φ) ⟶ (substitute t ψ)
substitute t (Ɐ φ) = Ɐ(substitute (termMapper𝐒 t) φ)
substitute t (∃ φ) = ∃(substitute (termMapper𝐒 t) φ)
-- Substitutes the most recent variable of a formula by mapping it to a term.
substitute0 : Term(vars) → Formula(𝐒(vars)) → Formula(vars)
substitute0 = substitute ∘ (t ↦ Vector.prepend t var)
-- Substitutes a single arbitrary variable of a formula by mapping it to a term.
-- Note: (substituteN 𝟎) normalizes to substitute0 because of the definition for Vector.insert.
substituteN : 𝕟₌(vars) → Term(vars) → Formula(𝐒(vars)) → Formula(vars)
substituteN n = substitute ∘ (t ↦ Vector.insert₊ n t var)
open import Data
open import Function.Equals
import Function.Names as Names
import Lvl
open import Relator.Equals
open import Relator.Equals.Proofs
open import Structure.Function
open import Structure.Operator
open import Syntax.Number
private variable ℓ : Lvl.Level
private variable A B : Type{ℓ}
private variable f g : A → B
private variable φ : Formula(vars)
termMapper𝐒-identity : (termMapper𝐒{vars₁ = vars} var ⊜ var)
_⊜_.proof termMapper𝐒-identity {x = 𝟎} = [≡]-intro
_⊜_.proof termMapper𝐒-identity {x = 𝐒 v} = [≡]-intro
module _ {f g : 𝕟(vars₁) → Term(vars₂)} (eq : f ⊜ g) where
termMapper𝐒-equal-functions : (termMapper𝐒 f ⊜ termMapper𝐒 g)
_⊜_.proof termMapper𝐒-equal-functions {𝟎} = [≡]-intro
_⊜_.proof termMapper𝐒-equal-functions {𝐒 v} rewrite _⊜_.proof eq{v} = [≡]-intro
substituteTerm-equal-functions-raw : (substituteTerm f Names.⊜ substituteTerm g)
substituteTerm₊-equal-functions-raw : (substituteTerm₊{args = args} f Names.⊜ substituteTerm₊ g)
(substituteTerm-equal-functions-raw) {var x} = _⊜_.proof eq
(substituteTerm-equal-functions-raw) {func f x}
rewrite substituteTerm₊-equal-functions-raw {x = x}
= [≡]-intro
(substituteTerm₊-equal-functions-raw) {x = ∅} = [≡]-intro
(substituteTerm₊-equal-functions-raw) {x = x ⊰ xs}
rewrite substituteTerm-equal-functions-raw {x}
rewrite substituteTerm₊-equal-functions-raw {x = xs}
= [≡]-intro
substituteTerm-equal-functions : (substituteTerm f ⊜ substituteTerm g)
substituteTerm-equal-functions = intro(\{x} → substituteTerm-equal-functions-raw{x})
substituteTerm₊-equal-functions : (substituteTerm₊{args = args} f ⊜ substituteTerm₊ g)
substituteTerm₊-equal-functions = intro substituteTerm₊-equal-functions-raw
substitute-equal-functions : (f ⊜ g) → (substitute f ⊜ substitute g)
substitute-equal-functions = intro ∘ p where
p : (f ⊜ g) → (substitute f Names.⊜ substitute g)
p eq {P $ x}
rewrite _⊜_.proof (substituteTerm₊-equal-functions eq) {x}
= [≡]-intro
p eq {⊤} = [≡]-intro
p eq {⊥} = [≡]-intro
p eq {φ ∧ ψ}
rewrite p eq {φ}
rewrite p eq {ψ}
= [≡]-intro
p eq {φ ∨ ψ}
rewrite p eq {φ}
rewrite p eq {ψ}
= [≡]-intro
p eq {φ ⟶ ψ}
rewrite p eq {φ}
rewrite p eq {ψ}
= [≡]-intro
p eq {Ɐ φ}
rewrite p (termMapper𝐒-equal-functions eq) {φ}
= [≡]-intro
p eq {∃ φ}
rewrite p (termMapper𝐒-equal-functions eq) {φ}
= [≡]-intro
substituteTerm-identity-raw : (substituteTerm{vars₁ = vars} var Names.⊜ id)
substituteTerm₊-identity-raw : (substituteTerm₊{vars₁ = vars}{args = args} var Names.⊜ id)
substituteTerm-identity-raw {x = var x} = [≡]-intro
substituteTerm-identity-raw {x = func f x} rewrite substituteTerm₊-identity-raw{x = x} = [≡]-intro
substituteTerm₊-identity-raw {x = ∅} = [≡]-intro
substituteTerm₊-identity-raw {x = x ⊰ xs}
rewrite substituteTerm-identity-raw{x = x}
rewrite substituteTerm₊-identity-raw{x = xs}
= [≡]-intro
substituteTerm-identity : (substituteTerm{vars₁ = vars} var ⊜ id)
substituteTerm-identity = intro substituteTerm-identity-raw
substituteTerm₊-identity : (substituteTerm₊{vars₁ = vars}{args = args} var ⊜ id)
substituteTerm₊-identity = intro substituteTerm₊-identity-raw
substitute-identity : (substitute{vars₁ = vars} var ⊜ id)
substitute-identity = intro p where
p : (substitute{vars₁ = vars} var Names.⊜ id)
p {x = P $ x} rewrite _⊜_.proof substituteTerm₊-identity {x} = [≡]-intro
p {x = ⊤} = [≡]-intro
p {x = ⊥} = [≡]-intro
p {x = φ ∧ ψ} rewrite p {x = φ} rewrite p {x = ψ} = [≡]-intro
p {x = φ ∨ ψ} rewrite p {x = φ} rewrite p {x = ψ} = [≡]-intro
p {x = φ ⟶ ψ} rewrite p {x = φ} rewrite p {x = ψ} = [≡]-intro
p {x = Ɐ φ}
rewrite _⊜_.proof (substitute-equal-functions termMapper𝐒-identity) {φ}
rewrite p {x = φ}
= [≡]-intro
p {x = ∃ φ}
rewrite _⊜_.proof (substitute-equal-functions termMapper𝐒-identity) {φ}
rewrite p {x = φ}
= [≡]-intro
{-
test1 : ∀{t : Term(vars)}{n : 𝕟(𝐒(𝐒 vars))} → (termMapper𝐒 (introduceVar t) n ≡ introduceVar (termVar𝐒 t) n)
test1 {t = var 𝟎} {𝟎} = {!introduceVar(termVar𝐒{_}(?)) 0!}
test1 {t = var 𝟎} {𝐒 n} = {!termMapper𝐒(introduceVar(?)) 1!}
test1 {t = var (𝐒 v)}{n} = {!!}
test1 {t = func f x}{n} = {!!}
test : ∀{t}{φ : Formula(𝐒 vars)} → (substitute(introduceVar t) φ ≡ substitute0 t φ)
test {vars} {t} {P $ x} = {!!}
test {vars} {t} {⊤} = [≡]-intro
test {vars} {t} {⊥} = [≡]-intro
test {vars} {t} {φ ∧ ψ} rewrite test {vars}{t}{φ} rewrite test{vars}{t}{ψ} = [≡]-intro
test {vars} {t} {φ ∨ ψ} rewrite test {vars}{t}{φ} rewrite test{vars}{t}{ψ} = [≡]-intro
test {vars} {t} {φ ⟶ ψ} rewrite test {vars}{t}{φ} rewrite test{vars}{t}{ψ} = [≡]-intro
test {vars} {t} {Ɐ φ} = {!test{𝐒 vars}{termVar𝐒 t}{φ}!}
test {vars} {t} {∃ φ} = {!!}
-}
|
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{-# OPTIONS --rewriting #-}
module Properties.Subtyping where
open import Agda.Builtin.Equality using (_≡_; refl)
open import FFI.Data.Either using (Either; Left; Right; mapLR; swapLR; cond)
open import FFI.Data.Maybe using (Maybe; just; nothing)
open import Luau.Subtyping using (_<:_; _≮:_; Tree; Language; ¬Language; witness; unknown; never; scalar; function; scalar-function; scalar-function-ok; scalar-function-err; scalar-scalar; function-scalar; function-ok; function-err; left; right; _,_)
open import Luau.Type using (Type; Scalar; nil; number; string; boolean; never; unknown; _⇒_; _∪_; _∩_; skalar)
open import Properties.Contradiction using (CONTRADICTION; ¬; ⊥)
open import Properties.Equality using (_≢_)
open import Properties.Functions using (_∘_)
open import Properties.Product using (_×_; _,_)
-- Language membership is decidable
dec-language : ∀ T t → Either (¬Language T t) (Language T t)
dec-language nil (scalar number) = Left (scalar-scalar number nil (λ ()))
dec-language nil (scalar boolean) = Left (scalar-scalar boolean nil (λ ()))
dec-language nil (scalar string) = Left (scalar-scalar string nil (λ ()))
dec-language nil (scalar nil) = Right (scalar nil)
dec-language nil function = Left (scalar-function nil)
dec-language nil (function-ok t) = Left (scalar-function-ok nil)
dec-language nil (function-err t) = Left (scalar-function-err nil)
dec-language boolean (scalar number) = Left (scalar-scalar number boolean (λ ()))
dec-language boolean (scalar boolean) = Right (scalar boolean)
dec-language boolean (scalar string) = Left (scalar-scalar string boolean (λ ()))
dec-language boolean (scalar nil) = Left (scalar-scalar nil boolean (λ ()))
dec-language boolean function = Left (scalar-function boolean)
dec-language boolean (function-ok t) = Left (scalar-function-ok boolean)
dec-language boolean (function-err t) = Left (scalar-function-err boolean)
dec-language number (scalar number) = Right (scalar number)
dec-language number (scalar boolean) = Left (scalar-scalar boolean number (λ ()))
dec-language number (scalar string) = Left (scalar-scalar string number (λ ()))
dec-language number (scalar nil) = Left (scalar-scalar nil number (λ ()))
dec-language number function = Left (scalar-function number)
dec-language number (function-ok t) = Left (scalar-function-ok number)
dec-language number (function-err t) = Left (scalar-function-err number)
dec-language string (scalar number) = Left (scalar-scalar number string (λ ()))
dec-language string (scalar boolean) = Left (scalar-scalar boolean string (λ ()))
dec-language string (scalar string) = Right (scalar string)
dec-language string (scalar nil) = Left (scalar-scalar nil string (λ ()))
dec-language string function = Left (scalar-function string)
dec-language string (function-ok t) = Left (scalar-function-ok string)
dec-language string (function-err t) = Left (scalar-function-err string)
dec-language (T₁ ⇒ T₂) (scalar s) = Left (function-scalar s)
dec-language (T₁ ⇒ T₂) function = Right function
dec-language (T₁ ⇒ T₂) (function-ok t) = mapLR function-ok function-ok (dec-language T₂ t)
dec-language (T₁ ⇒ T₂) (function-err t) = mapLR function-err function-err (swapLR (dec-language T₁ t))
dec-language never t = Left never
dec-language unknown t = Right unknown
dec-language (T₁ ∪ T₂) t = cond (λ p → cond (Left ∘ _,_ p) (Right ∘ right) (dec-language T₂ t)) (Right ∘ left) (dec-language T₁ t)
dec-language (T₁ ∩ T₂) t = cond (Left ∘ left) (λ p → cond (Left ∘ right) (Right ∘ _,_ p) (dec-language T₂ t)) (dec-language T₁ t)
-- ¬Language T is the complement of Language T
language-comp : ∀ {T} t → ¬Language T t → ¬(Language T t)
language-comp t (p₁ , p₂) (left q) = language-comp t p₁ q
language-comp t (p₁ , p₂) (right q) = language-comp t p₂ q
language-comp t (left p) (q₁ , q₂) = language-comp t p q₁
language-comp t (right p) (q₁ , q₂) = language-comp t p q₂
language-comp (scalar s) (scalar-scalar s p₁ p₂) (scalar s) = p₂ refl
language-comp (scalar s) (function-scalar s) (scalar s) = language-comp function (scalar-function s) function
language-comp (scalar s) never (scalar ())
language-comp function (scalar-function ()) function
language-comp (function-ok t) (scalar-function-ok ()) (function-ok q)
language-comp (function-ok t) (function-ok p) (function-ok q) = language-comp t p q
language-comp (function-err t) (function-err p) (function-err q) = language-comp t q p
-- ≮: is the complement of <:
¬≮:-impl-<: : ∀ {T U} → ¬(T ≮: U) → (T <: U)
¬≮:-impl-<: {T} {U} p t q with dec-language U t
¬≮:-impl-<: {T} {U} p t q | Left r = CONTRADICTION (p (witness t q r))
¬≮:-impl-<: {T} {U} p t q | Right r = r
<:-impl-¬≮: : ∀ {T U} → (T <: U) → ¬(T ≮: U)
<:-impl-¬≮: p (witness t q r) = language-comp t r (p t q)
<:-impl-⊇ : ∀ {T U} → (T <: U) → ∀ t → ¬Language U t → ¬Language T t
<:-impl-⊇ {T} p t q with dec-language T t
<:-impl-⊇ {_} p t q | Left r = r
<:-impl-⊇ {_} p t q | Right r = CONTRADICTION (language-comp t q (p t r))
-- reflexivity
≮:-refl : ∀ {T} → ¬(T ≮: T)
≮:-refl (witness t p q) = language-comp t q p
<:-refl : ∀ {T} → (T <: T)
<:-refl = ¬≮:-impl-<: ≮:-refl
-- transititivity
≮:-trans-≡ : ∀ {S T U} → (S ≮: T) → (T ≡ U) → (S ≮: U)
≮:-trans-≡ p refl = p
≡-trans-≮: : ∀ {S T U} → (S ≡ T) → (T ≮: U) → (S ≮: U)
≡-trans-≮: refl p = p
≮:-trans : ∀ {S T U} → (S ≮: U) → Either (S ≮: T) (T ≮: U)
≮:-trans {T = T} (witness t p q) = mapLR (witness t p) (λ z → witness t z q) (dec-language T t)
<:-trans : ∀ {S T U} → (S <: T) → (T <: U) → (S <: U)
<:-trans p q t r = q t (p t r)
<:-trans-≮: : ∀ {S T U} → (S <: T) → (S ≮: U) → (T ≮: U)
<:-trans-≮: p (witness t q r) = witness t (p t q) r
≮:-trans-<: : ∀ {S T U} → (S ≮: U) → (T <: U) → (S ≮: T)
≮:-trans-<: (witness t p q) r = witness t p (<:-impl-⊇ r t q)
-- Properties of union
<:-union : ∀ {R S T U} → (R <: T) → (S <: U) → ((R ∪ S) <: (T ∪ U))
<:-union p q t (left r) = left (p t r)
<:-union p q t (right r) = right (q t r)
<:-∪-left : ∀ {S T} → S <: (S ∪ T)
<:-∪-left t p = left p
<:-∪-right : ∀ {S T} → T <: (S ∪ T)
<:-∪-right t p = right p
<:-∪-lub : ∀ {S T U} → (S <: U) → (T <: U) → ((S ∪ T) <: U)
<:-∪-lub p q t (left r) = p t r
<:-∪-lub p q t (right r) = q t r
<:-∪-symm : ∀ {T U} → (T ∪ U) <: (U ∪ T)
<:-∪-symm t (left p) = right p
<:-∪-symm t (right p) = left p
<:-∪-assocl : ∀ {S T U} → (S ∪ (T ∪ U)) <: ((S ∪ T) ∪ U)
<:-∪-assocl t (left p) = left (left p)
<:-∪-assocl t (right (left p)) = left (right p)
<:-∪-assocl t (right (right p)) = right p
<:-∪-assocr : ∀ {S T U} → ((S ∪ T) ∪ U) <: (S ∪ (T ∪ U))
<:-∪-assocr t (left (left p)) = left p
<:-∪-assocr t (left (right p)) = right (left p)
<:-∪-assocr t (right p) = right (right p)
≮:-∪-left : ∀ {S T U} → (S ≮: U) → ((S ∪ T) ≮: U)
≮:-∪-left (witness t p q) = witness t (left p) q
≮:-∪-right : ∀ {S T U} → (T ≮: U) → ((S ∪ T) ≮: U)
≮:-∪-right (witness t p q) = witness t (right p) q
-- Properties of intersection
<:-intersect : ∀ {R S T U} → (R <: T) → (S <: U) → ((R ∩ S) <: (T ∩ U))
<:-intersect p q t (r₁ , r₂) = (p t r₁ , q t r₂)
<:-∩-left : ∀ {S T} → (S ∩ T) <: S
<:-∩-left t (p , _) = p
<:-∩-right : ∀ {S T} → (S ∩ T) <: T
<:-∩-right t (_ , p) = p
<:-∩-glb : ∀ {S T U} → (S <: T) → (S <: U) → (S <: (T ∩ U))
<:-∩-glb p q t r = (p t r , q t r)
<:-∩-symm : ∀ {T U} → (T ∩ U) <: (U ∩ T)
<:-∩-symm t (p₁ , p₂) = (p₂ , p₁)
≮:-∩-left : ∀ {S T U} → (S ≮: T) → (S ≮: (T ∩ U))
≮:-∩-left (witness t p q) = witness t p (left q)
≮:-∩-right : ∀ {S T U} → (S ≮: U) → (S ≮: (T ∩ U))
≮:-∩-right (witness t p q) = witness t p (right q)
-- Distribution properties
<:-∩-distl-∪ : ∀ {S T U} → (S ∩ (T ∪ U)) <: ((S ∩ T) ∪ (S ∩ U))
<:-∩-distl-∪ t (p₁ , left p₂) = left (p₁ , p₂)
<:-∩-distl-∪ t (p₁ , right p₂) = right (p₁ , p₂)
∩-distl-∪-<: : ∀ {S T U} → ((S ∩ T) ∪ (S ∩ U)) <: (S ∩ (T ∪ U))
∩-distl-∪-<: t (left (p₁ , p₂)) = (p₁ , left p₂)
∩-distl-∪-<: t (right (p₁ , p₂)) = (p₁ , right p₂)
<:-∩-distr-∪ : ∀ {S T U} → ((S ∪ T) ∩ U) <: ((S ∩ U) ∪ (T ∩ U))
<:-∩-distr-∪ t (left p₁ , p₂) = left (p₁ , p₂)
<:-∩-distr-∪ t (right p₁ , p₂) = right (p₁ , p₂)
∩-distr-∪-<: : ∀ {S T U} → ((S ∩ U) ∪ (T ∩ U)) <: ((S ∪ T) ∩ U)
∩-distr-∪-<: t (left (p₁ , p₂)) = (left p₁ , p₂)
∩-distr-∪-<: t (right (p₁ , p₂)) = (right p₁ , p₂)
<:-∪-distl-∩ : ∀ {S T U} → (S ∪ (T ∩ U)) <: ((S ∪ T) ∩ (S ∪ U))
<:-∪-distl-∩ t (left p) = (left p , left p)
<:-∪-distl-∩ t (right (p₁ , p₂)) = (right p₁ , right p₂)
∪-distl-∩-<: : ∀ {S T U} → ((S ∪ T) ∩ (S ∪ U)) <: (S ∪ (T ∩ U))
∪-distl-∩-<: t (left p₁ , p₂) = left p₁
∪-distl-∩-<: t (right p₁ , left p₂) = left p₂
∪-distl-∩-<: t (right p₁ , right p₂) = right (p₁ , p₂)
<:-∪-distr-∩ : ∀ {S T U} → ((S ∩ T) ∪ U) <: ((S ∪ U) ∩ (T ∪ U))
<:-∪-distr-∩ t (left (p₁ , p₂)) = left p₁ , left p₂
<:-∪-distr-∩ t (right p) = (right p , right p)
∪-distr-∩-<: : ∀ {S T U} → ((S ∪ U) ∩ (T ∪ U)) <: ((S ∩ T) ∪ U)
∪-distr-∩-<: t (left p₁ , left p₂) = left (p₁ , p₂)
∪-distr-∩-<: t (left p₁ , right p₂) = right p₂
∪-distr-∩-<: t (right p₁ , p₂) = right p₁
-- Properties of functions
<:-function : ∀ {R S T U} → (R <: S) → (T <: U) → (S ⇒ T) <: (R ⇒ U)
<:-function p q function function = function
<:-function p q (function-ok t) (function-ok r) = function-ok (q t r)
<:-function p q (function-err s) (function-err r) = function-err (<:-impl-⊇ p s r)
<:-function-∩-∪ : ∀ {R S T U} → ((R ⇒ T) ∩ (S ⇒ U)) <: ((R ∪ S) ⇒ (T ∪ U))
<:-function-∩-∪ function (function , function) = function
<:-function-∩-∪ (function-ok t) (function-ok p₁ , function-ok p₂) = function-ok (right p₂)
<:-function-∩-∪ (function-err _) (function-err p₁ , function-err q₂) = function-err (p₁ , q₂)
<:-function-∩ : ∀ {S T U} → ((S ⇒ T) ∩ (S ⇒ U)) <: (S ⇒ (T ∩ U))
<:-function-∩ function (function , function) = function
<:-function-∩ (function-ok t) (function-ok p₁ , function-ok p₂) = function-ok (p₁ , p₂)
<:-function-∩ (function-err s) (function-err p₁ , function-err p₂) = function-err p₂
<:-function-∪ : ∀ {R S T U} → ((R ⇒ S) ∪ (T ⇒ U)) <: ((R ∩ T) ⇒ (S ∪ U))
<:-function-∪ function (left function) = function
<:-function-∪ (function-ok t) (left (function-ok p)) = function-ok (left p)
<:-function-∪ (function-err s) (left (function-err p)) = function-err (left p)
<:-function-∪ (scalar s) (left (scalar ()))
<:-function-∪ function (right function) = function
<:-function-∪ (function-ok t) (right (function-ok p)) = function-ok (right p)
<:-function-∪ (function-err s) (right (function-err x)) = function-err (right x)
<:-function-∪ (scalar s) (right (scalar ()))
<:-function-∪-∩ : ∀ {R S T U} → ((R ∩ S) ⇒ (T ∪ U)) <: ((R ⇒ T) ∪ (S ⇒ U))
<:-function-∪-∩ function function = left function
<:-function-∪-∩ (function-ok t) (function-ok (left p)) = left (function-ok p)
<:-function-∪-∩ (function-ok t) (function-ok (right p)) = right (function-ok p)
<:-function-∪-∩ (function-err s) (function-err (left p)) = left (function-err p)
<:-function-∪-∩ (function-err s) (function-err (right p)) = right (function-err p)
≮:-function-left : ∀ {R S T U} → (R ≮: S) → (S ⇒ T) ≮: (R ⇒ U)
≮:-function-left (witness t p q) = witness (function-err t) (function-err q) (function-err p)
≮:-function-right : ∀ {R S T U} → (T ≮: U) → (S ⇒ T) ≮: (R ⇒ U)
≮:-function-right (witness t p q) = witness (function-ok t) (function-ok p) (function-ok q)
-- Properties of scalars
skalar-function-ok : ∀ {t} → (¬Language skalar (function-ok t))
skalar-function-ok = (scalar-function-ok number , (scalar-function-ok string , (scalar-function-ok nil , scalar-function-ok boolean)))
scalar-<: : ∀ {S T} → (s : Scalar S) → Language T (scalar s) → (S <: T)
scalar-<: number p (scalar number) (scalar number) = p
scalar-<: boolean p (scalar boolean) (scalar boolean) = p
scalar-<: string p (scalar string) (scalar string) = p
scalar-<: nil p (scalar nil) (scalar nil) = p
scalar-∩-function-<:-never : ∀ {S T U} → (Scalar S) → ((T ⇒ U) ∩ S) <: never
scalar-∩-function-<:-never number .(scalar number) (() , scalar number)
scalar-∩-function-<:-never boolean .(scalar boolean) (() , scalar boolean)
scalar-∩-function-<:-never string .(scalar string) (() , scalar string)
scalar-∩-function-<:-never nil .(scalar nil) (() , scalar nil)
function-≮:-scalar : ∀ {S T U} → (Scalar U) → ((S ⇒ T) ≮: U)
function-≮:-scalar s = witness function function (scalar-function s)
scalar-≮:-function : ∀ {S T U} → (Scalar U) → (U ≮: (S ⇒ T))
scalar-≮:-function s = witness (scalar s) (scalar s) (function-scalar s)
unknown-≮:-scalar : ∀ {U} → (Scalar U) → (unknown ≮: U)
unknown-≮:-scalar s = witness (function-ok (scalar s)) unknown (scalar-function-ok s)
scalar-≮:-never : ∀ {U} → (Scalar U) → (U ≮: never)
scalar-≮:-never s = witness (scalar s) (scalar s) never
scalar-≢-impl-≮: : ∀ {T U} → (Scalar T) → (Scalar U) → (T ≢ U) → (T ≮: U)
scalar-≢-impl-≮: s₁ s₂ p = witness (scalar s₁) (scalar s₁) (scalar-scalar s₁ s₂ p)
scalar-≢-∩-<:-never : ∀ {T U V} → (Scalar T) → (Scalar U) → (T ≢ U) → (T ∩ U) <: V
scalar-≢-∩-<:-never s t p u (scalar s₁ , scalar s₂) = CONTRADICTION (p refl)
skalar-scalar : ∀ {T} (s : Scalar T) → (Language skalar (scalar s))
skalar-scalar number = left (scalar number)
skalar-scalar boolean = right (right (right (scalar boolean)))
skalar-scalar string = right (left (scalar string))
skalar-scalar nil = right (right (left (scalar nil)))
-- Properties of unknown and never
unknown-≮: : ∀ {T U} → (T ≮: U) → (unknown ≮: U)
unknown-≮: (witness t p q) = witness t unknown q
never-≮: : ∀ {T U} → (T ≮: U) → (T ≮: never)
never-≮: (witness t p q) = witness t p never
unknown-≮:-never : (unknown ≮: never)
unknown-≮:-never = witness (scalar nil) unknown never
function-≮:-never : ∀ {T U} → ((T ⇒ U) ≮: never)
function-≮:-never = witness function function never
<:-never : ∀ {T} → (never <: T)
<:-never t (scalar ())
≮:-never-left : ∀ {S T U} → (S <: (T ∪ U)) → (S ≮: T) → (S ∩ U) ≮: never
≮:-never-left p (witness t q₁ q₂) with p t q₁
≮:-never-left p (witness t q₁ q₂) | left r = CONTRADICTION (language-comp t q₂ r)
≮:-never-left p (witness t q₁ q₂) | right r = witness t (q₁ , r) never
≮:-never-right : ∀ {S T U} → (S <: (T ∪ U)) → (S ≮: U) → (S ∩ T) ≮: never
≮:-never-right p (witness t q₁ q₂) with p t q₁
≮:-never-right p (witness t q₁ q₂) | left r = witness t (q₁ , r) never
≮:-never-right p (witness t q₁ q₂) | right r = CONTRADICTION (language-comp t q₂ r)
<:-unknown : ∀ {T} → (T <: unknown)
<:-unknown t p = unknown
<:-everything : unknown <: ((never ⇒ unknown) ∪ skalar)
<:-everything (scalar s) p = right (skalar-scalar s)
<:-everything function p = left function
<:-everything (function-ok t) p = left (function-ok unknown)
<:-everything (function-err s) p = left (function-err never)
-- A Gentle Introduction To Semantic Subtyping (https://www.cduce.org/papers/gentle.pdf)
-- defines a "set-theoretic" model (sec 2.5)
-- Unfortunately we don't quite have this property, due to uninhabited types,
-- for example (never -> T) is equivalent to (never -> U)
-- when types are interpreted as sets of syntactic values.
_⊆_ : ∀ {A : Set} → (A → Set) → (A → Set) → Set
(P ⊆ Q) = ∀ a → (P a) → (Q a)
_⊗_ : ∀ {A B : Set} → (A → Set) → (B → Set) → ((A × B) → Set)
(P ⊗ Q) (a , b) = (P a) × (Q b)
Comp : ∀ {A : Set} → (A → Set) → (A → Set)
Comp P a = ¬(P a)
Lift : ∀ {A : Set} → (A → Set) → (Maybe A → Set)
Lift P nothing = ⊥
Lift P (just a) = P a
set-theoretic-if : ∀ {S₁ T₁ S₂ T₂} →
-- This is the "if" part of being a set-theoretic model
-- though it uses the definition from Frisch's thesis
-- rather than from the Gentle Introduction. The difference
-- being the presence of Lift, (written D_Ω in Defn 4.2 of
-- https://www.cduce.org/papers/frisch_phd.pdf).
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)) →
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Lift(Language T₁))) → Q ⊆ Comp((Language S₂) ⊗ Comp(Lift(Language T₂))))
set-theoretic-if {S₁} {T₁} {S₂} {T₂} p Q q (t , just u) Qtu (S₂t , ¬T₂u) = q (t , just u) Qtu (S₁t , ¬T₁u) where
S₁t : Language S₁ t
S₁t with dec-language S₁ t
S₁t | Left ¬S₁t with p (function-err t) (function-err ¬S₁t)
S₁t | Left ¬S₁t | function-err ¬S₂t = CONTRADICTION (language-comp t ¬S₂t S₂t)
S₁t | Right r = r
¬T₁u : ¬(Language T₁ u)
¬T₁u T₁u with p (function-ok u) (function-ok T₁u)
¬T₁u T₁u | function-ok T₂u = ¬T₂u T₂u
set-theoretic-if {S₁} {T₁} {S₂} {T₂} p Q q (t , nothing) Qt- (S₂t , _) = q (t , nothing) Qt- (S₁t , λ ()) where
S₁t : Language S₁ t
S₁t with dec-language S₁ t
S₁t | Left ¬S₁t with p (function-err t) (function-err ¬S₁t)
S₁t | Left ¬S₁t | function-err ¬S₂t = CONTRADICTION (language-comp t ¬S₂t S₂t)
S₁t | Right r = r
not-quite-set-theoretic-only-if : ∀ {S₁ T₁ S₂ T₂} →
-- We don't quite have that this is a set-theoretic model
-- it's only true when Language T₁ and ¬Language T₂ t₂ are inhabited
-- in particular it's not true when T₁ is never, or T₂ is unknown.
∀ s₂ t₂ → Language S₂ s₂ → ¬Language T₂ t₂ →
-- This is the "only if" part of being a set-theoretic model
(∀ Q → Q ⊆ Comp((Language S₁) ⊗ Comp(Lift(Language T₁))) → Q ⊆ Comp((Language S₂) ⊗ Comp(Lift(Language T₂)))) →
(Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂))
not-quite-set-theoretic-only-if {S₁} {T₁} {S₂} {T₂} s₂ t₂ S₂s₂ ¬T₂t₂ p = r where
Q : (Tree × Maybe Tree) → Set
Q (t , just u) = Either (¬Language S₁ t) (Language T₁ u)
Q (t , nothing) = ¬Language S₁ t
q : Q ⊆ Comp((Language S₁) ⊗ Comp(Lift(Language T₁)))
q (t , just u) (Left ¬S₁t) (S₁t , ¬T₁u) = language-comp t ¬S₁t S₁t
q (t , just u) (Right T₂u) (S₁t , ¬T₁u) = ¬T₁u T₂u
q (t , nothing) ¬S₁t (S₁t , _) = language-comp t ¬S₁t S₁t
r : Language (S₁ ⇒ T₁) ⊆ Language (S₂ ⇒ T₂)
r function function = function
r (function-err s) (function-err ¬S₁s) with dec-language S₂ s
r (function-err s) (function-err ¬S₁s) | Left ¬S₂s = function-err ¬S₂s
r (function-err s) (function-err ¬S₁s) | Right S₂s = CONTRADICTION (p Q q (s , nothing) ¬S₁s (S₂s , λ ()))
r (function-ok t) (function-ok T₁t) with dec-language T₂ t
r (function-ok t) (function-ok T₁t) | Left ¬T₂t = CONTRADICTION (p Q q (s₂ , just t) (Right T₁t) (S₂s₂ , language-comp t ¬T₂t))
r (function-ok t) (function-ok T₁t) | Right T₂t = function-ok T₂t
-- A counterexample when the argument type is empty.
set-theoretic-counterexample-one : (∀ Q → Q ⊆ Comp((Language never) ⊗ Comp(Lift(Language number))) → Q ⊆ Comp((Language never) ⊗ Comp(Lift(Language string))))
set-theoretic-counterexample-one Q q ((scalar s) , u) Qtu (scalar () , p)
set-theoretic-counterexample-two : (never ⇒ number) ≮: (never ⇒ string)
set-theoretic-counterexample-two = witness
(function-ok (scalar number)) (function-ok (scalar number))
(function-ok (scalar-scalar number string (λ ())))
-- At some point we may deal with overloaded function resolution, which should fix this problem...
-- The reason why this is connected to overloaded functions is that currently we have that the type of
-- f(x) is (tgt T) where f:T. Really we should have the type depend on the type of x, that is use (tgt T U),
-- where U is the type of x. In particular (tgt (S => T) (U & V)) should be the same as (tgt ((S&U) => T) V)
-- and tgt(never => T) should be unknown. For example
--
-- tgt((number => string) & (string => bool))(number)
-- is tgt(number => string)(number) & tgt(string => bool)(number)
-- is tgt(number => string)(number) & tgt(string => bool)(number&unknown)
-- is tgt(number => string)(number) & tgt(string&number => bool)(unknown)
-- is tgt(number => string)(number) & tgt(never => bool)(unknown)
-- is string & unknown
-- is string
--
-- there's some discussion of this in the Gentle Introduction paper.
|
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|
module Issue100 where
-- hiding (Nat) goes on the 'open' not on the 'import'.
open import Nat hiding (Nat)
one : Nat.Nat
one = suc zero
|
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|
-- Andreas, 2014-01-08, following Maxime Denes 2014-01-06
-- This file demonstrates then incompatibility of the untyped
-- structural termination ordering with HoTT.
open import Common.Equality
data Empty : Set where
data Box : Set where
wrap : (Empty → Box) → Box
-- Box is inhabited:
gift : Empty → Box
gift ()
box : Box
box = wrap gift
-- wrap has an inverse:
unwrap : Box → (Empty → Box)
unwrap (wrap f) = f
-- Thus, Box is isomorphic to (Empty → Box).
-- However, they cannot be propositionally equal,
-- as this leads to an inconsistency as follows:
postulate iso : (Empty → Box) ≡ Box
module Rewrite where
loop : Box → Empty
loop (wrap x) rewrite iso = loop x
-- rewrite is not to blame, we can do it with with:
module With where
loop : Box → Empty
loop (wrap x) with (Empty → Box) | iso
... | ._ | refl = loop x
-- with is not to be blamed either, we can desugar it:
module Aux where
mutual
loop : Box → Empty
loop (wrap x) = loop' (Empty → Box) iso x
loop' : ∀ A → A ≡ Box → A → Empty
loop' .Box refl x = loop x
open Aux
bug : Empty
bug = loop box
-- Moral of the story: the termination checker should reject `loop'.
-- If the termination checker should be fixed in that way,
-- move this test case to test/fail.
|
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|
{-# OPTIONS --cubical-compatible #-}
module _ where
module M where
data D : Set where
record R₁ : Set where
field
x : let module M′ = M in M′.D
variable
A : Set
record R₂ (A : Set) : Set where
_ : (r : R₂ A) → let open R₂ r in Set₁
_ = λ _ → Set
|
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|
------------------------------------------------------------------------
-- Properties relating All to various list functions
------------------------------------------------------------------------
module Data.List.All.Properties where
open import Data.Bool
open import Data.Bool.Properties
open import Data.Function
open import Data.List as List
import Data.List.Any as Any
open Any.Membership-≡
open import Data.List.All as All using (All; []; _∷_)
open import Data.Product
open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_)
-- Functions can be shifted between the predicate and the list.
All-map : ∀ {A B} {P : Pred B} {f : A → B} {xs} →
All (P ∘₀ f) xs → All P (List.map f xs)
All-map [] = []
All-map (p ∷ ps) = p ∷ All-map ps
map-All : ∀ {A B} {P : Pred B} {f : A → B} {xs} →
All P (List.map f xs) → All (P ∘₀ f) xs
map-All {xs = []} [] = []
map-All {xs = _ ∷ _} (p ∷ ps) = p ∷ map-All ps
-- A variant of All.map.
gmap : ∀ {A B} {P : A → Set} {Q : B → Set} {f : A → B} →
P ⋐ Q ∘₀ f → All P ⋐ All Q ∘₀ List.map f
gmap g = All-map ∘ All.map g
-- All and all are related via T.
All-all : ∀ {A} (p : A → Bool) {xs} →
All (T ∘₀ p) xs → T (all p xs)
All-all p [] = _
All-all p (px ∷ pxs) = proj₂ T-∧ (px , All-all p pxs)
all-All : ∀ {A} (p : A → Bool) xs →
T (all p xs) → All (T ∘₀ p) xs
all-All p [] _ = []
all-All p (x ∷ xs) px∷xs with proj₁ (T-∧ {p x}) px∷xs
all-All p (x ∷ xs) px∷xs | (px , pxs) = px ∷ all-All p xs pxs
-- All is anti-monotone.
anti-mono : ∀ {A} {P : Pred A} {xs ys} → xs ⊆ ys → All P ys → All P xs
anti-mono xs⊆ys pys = All.tabulate (All.lookup pys ∘ xs⊆ys)
-- all is anti-monotone.
all-anti-mono : ∀ {A} (p : A → Bool) {xs ys} →
xs ⊆ ys → T (all p ys) → T (all p xs)
all-anti-mono p xs⊆ys = All-all p ∘ anti-mono xs⊆ys ∘ all-All p _
|
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|
module _ where
open import Agda.Builtin.Equality
open import Agda.Builtin.Nat
module Vars (A : Set) where
variable
x : A
data It {A : Set} : A → Set where
it : ∀ x → It x
module Fixed where
open Vars Nat
ret : It x
ret {x = x} = it x
module Param (A : Set) where
open Vars A
ret : It x
ret {x = x} = it x
open Vars Nat
check : Param.ret Nat ≡ Fixed.ret {x = x}
check = refl
-- Check that you can let open as long as you don't use the variables
foo : (A : Set) (let open Vars A) (x : A) → x ≡ x
foo A x = refl
|
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|
------------------------------------------------------------------------
-- Embeddings with erased "proofs"
------------------------------------------------------------------------
-- Partially following the HoTT book.
{-# OPTIONS --without-K --safe #-}
open import Equality
module Embedding.Erased
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open import Prelude hiding (id; _∘_)
open import Bijection eq using (_↔_)
open Derived-definitions-and-properties eq
open import Embedding eq as Emb using (Is-embedding; Embedding)
open import Equivalence eq as Eq using (_≃_)
open import Equivalence.Erased eq as EEq
using (_≃ᴱ_; Is-equivalenceᴱ)
open import Equivalence.Erased.Contractible-preimages eq as ECP
using (_⁻¹ᴱ_)
open import Erased.Level-1 eq using (Erased; []-cong-axiomatisation)
open import Function-universe eq hiding (id; _∘_; equivalence)
open import H-level.Closure eq
open import Preimage eq using (_⁻¹_)
private
variable
a b ℓ t : Level
A B C : Type a
f k x y : A
------------------------------------------------------------------------
-- Embeddings
-- The property of being an embedding with erased "proofs".
Is-embeddingᴱ : {A : Type a} {B : Type b} → (A → B) → Type (a ⊔ b)
Is-embeddingᴱ f = ∀ x y → Is-equivalenceᴱ (cong {x = x} {y = y} f)
-- Is-embeddingᴱ is propositional in erased contexts (assuming
-- extensionality).
@0 Is-embeddingᴱ-propositional :
{A : Type a} {B : Type b} {f : A → B} →
Extensionality (a ⊔ b) (a ⊔ b) →
Is-proposition (Is-embeddingᴱ f)
Is-embeddingᴱ-propositional {b = b} ext =
Π-closure (lower-extensionality b lzero ext) 1 λ _ →
Π-closure (lower-extensionality b lzero ext) 1 λ _ →
EEq.Is-equivalenceᴱ-propositional ext _
-- Embeddings with erased proofs.
record Embeddingᴱ (From : Type f) (To : Type t) : Type (f ⊔ t) where
field
to : From → To
is-embedding : Is-embeddingᴱ to
equivalence : (x ≡ y) ≃ᴱ (to x ≡ to y)
equivalence = EEq.⟨ _ , is-embedding _ _ ⟩
------------------------------------------------------------------------
-- Some conversion functions
-- The type family above could have been defined using Σ.
Embeddingᴱ-as-Σ : Embeddingᴱ A B ↔ ∃ λ (f : A → B) → Is-embeddingᴱ f
Embeddingᴱ-as-Σ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ emb → Embeddingᴱ.to emb , Embeddingᴱ.is-embedding emb
; from = λ { (f , is) → record { to = f; is-embedding = is } }
}
; right-inverse-of = refl
}
; left-inverse-of = refl
}
-- Conversions between Is-embedding and Is-embeddingᴱ.
Is-embedding→Is-embeddingᴱ : Is-embedding f → Is-embeddingᴱ f
Is-embedding→Is-embeddingᴱ {f = f} =
(∀ x y → Eq.Is-equivalence (cong {x = x} {y = y} f)) ↝⟨ (∀-cong _ λ _ → ∀-cong _ λ _ → EEq.Is-equivalence→Is-equivalenceᴱ) ⟩□
(∀ x y → Is-equivalenceᴱ (cong {x = x} {y = y} f)) □
@0 Is-embedding≃Is-embeddingᴱ :
{A : Type a} {B : Type b} {f : A → B} →
Is-embedding f ↝[ a ∣ a ⊔ b ] Is-embeddingᴱ f
Is-embedding≃Is-embeddingᴱ {f = f} {k = k} ext =
(∀ x y → Eq.Is-equivalence (cong {x = x} {y = y} f)) ↝⟨ (∀-cong ext λ _ → ∀-cong ext λ _ → from-equivalence
EEq.Is-equivalence≃Is-equivalenceᴱ) ⟩□
(∀ x y → Is-equivalenceᴱ (cong {x = x} {y = y} f)) □
@0 Is-embeddingᴱ→Is-embedding : Is-embeddingᴱ f → Is-embedding f
Is-embeddingᴱ→Is-embedding = inverse-ext? Is-embedding≃Is-embeddingᴱ _
-- Conversions between Embedding and Embeddingᴱ.
Embedding→Embeddingᴱ : Embedding A B → Embeddingᴱ A B
Embedding→Embeddingᴱ {A = A} {B = B} =
Embedding A B ↔⟨ Emb.Embedding-as-Σ ⟩
(∃ λ (f : A → B) → Is-embedding f) ↝⟨ (∃-cong λ _ → Is-embedding→Is-embeddingᴱ) ⟩
(∃ λ (f : A → B) → Is-embeddingᴱ f) ↔⟨ inverse Embeddingᴱ-as-Σ ⟩□
Embeddingᴱ A B □
@0 Embedding≃Embeddingᴱ :
{A : Type a} {B : Type b} →
Embedding A B ↝[ a ∣ a ⊔ b ] Embeddingᴱ A B
Embedding≃Embeddingᴱ {A = A} {B = B} ext =
Embedding A B ↔⟨ Emb.Embedding-as-Σ ⟩
(∃ λ (f : A → B) → Is-embedding f) ↝⟨ (∃-cong λ _ → Is-embedding≃Is-embeddingᴱ ext) ⟩
(∃ λ (f : A → B) → Is-embeddingᴱ f) ↔⟨ inverse Embeddingᴱ-as-Σ ⟩□
Embeddingᴱ A B □
@0 Embeddingᴱ→Embedding : Embeddingᴱ A B → Embedding A B
Embeddingᴱ→Embedding = inverse-ext? Embedding≃Embeddingᴱ _
-- Data corresponding to the erased proofs of an embedding with
-- erased proofs.
Erased-proofs :
{A : Type a} {B : Type b} →
(to : A → B) →
(∀ {x y} → to x ≡ to y → x ≡ y) →
Type (a ⊔ b)
Erased-proofs to from =
∀ {x y} → EEq.Erased-proofs (cong {x = x} {y = y} to) from
-- Extracts "erased proofs" from a regular embedding.
[proofs] :
(A↝B : Embedding A B) →
Erased-proofs
(Embedding.to A↝B)
(_≃_.from (Embedding.equivalence A↝B))
[proofs] A↝B = EEq.[proofs] (Embedding.equivalence A↝B)
-- Converts two functions and some erased proofs to an embedding with
-- erased proofs.
--
-- Note that Agda can in many cases infer "to" and "from" from the
-- first explicit argument, see (for instance) _∘_ below.
[Embedding]→Embeddingᴱ :
{to : A → B} {from : ∀ {x y} → to x ≡ to y → x ≡ y} →
@0 Erased-proofs to from →
Embeddingᴱ A B
[Embedding]→Embeddingᴱ {to = to} {from = from} ep = record
{ to = to
; is-embedding = λ _ _ → _≃ᴱ_.is-equivalence (EEq.[≃]→≃ᴱ ep)
}
------------------------------------------------------------------------
-- Preorder
-- Embeddingᴱ is a preorder.
id : Embeddingᴱ A A
id = Embedding→Embeddingᴱ Emb.id
infixr 9 _∘_
_∘_ : Embeddingᴱ B C → Embeddingᴱ A B → Embeddingᴱ A C
f ∘ g =
[Embedding]→Embeddingᴱ
([proofs] (Embeddingᴱ→Embedding f Emb.∘ Embeddingᴱ→Embedding g))
------------------------------------------------------------------------
-- "Preimages"
-- If f is an embedding (with erased proofs), then f ⁻¹ᴱ y is
-- propositional (in an erased context).
--
-- This result is based on the proof of Theorem 4.6.3 in the HoTT book
-- (first edition).
@0 embedding→⁻¹ᴱ-propositional :
Is-embeddingᴱ f →
∀ y → Is-proposition (f ⁻¹ᴱ y)
embedding→⁻¹ᴱ-propositional {f = f} =
Is-embeddingᴱ f ↝⟨ Is-embeddingᴱ→Is-embedding ⟩
Is-embedding f ↝⟨ Emb.embedding→⁻¹-propositional ⟩
(∀ y → Is-proposition (f ⁻¹ y)) ↝⟨ (∀-cong _ λ _ → H-level-cong _ 1 ECP.⁻¹≃⁻¹ᴱ) ⟩□
(∀ y → Is-proposition (f ⁻¹ᴱ y)) □
------------------------------------------------------------------------
-- Results that depend on an axiomatisation of []-cong (for a single
-- universe level)
module []-cong₁ (ax : []-cong-axiomatisation ℓ) where
----------------------------------------------------------------------
-- More conversion functions
-- Equivalences (with erased proofs) from Erased A to B are
-- embeddings (with erased proofs).
Is-equivalenceᴱ→Is-embeddingᴱ-Erased :
{A : Type ℓ} {f : Erased A → B} →
Is-equivalenceᴱ f → Is-embeddingᴱ f
Is-equivalenceᴱ→Is-embeddingᴱ-Erased eq _ _ =
_≃ᴱ_.is-equivalence $ inverse $
EEq.[]-cong₁.to≡to≃ᴱ≡-Erased ax EEq.⟨ _ , eq ⟩
-- Equivalences with erased proofs between Erased A and B can be
-- converted to embeddings with erased proofs.
Erased≃→Embedding :
{A : Type ℓ} →
Erased A ≃ᴱ B → Embeddingᴱ (Erased A) B
Erased≃→Embedding EEq.⟨ f , is-equiv ⟩ = record
{ to = f
; is-embedding = Is-equivalenceᴱ→Is-embeddingᴱ-Erased is-equiv
}
------------------------------------------------------------------------
-- Results that depend on an axiomatisation of []-cong (for all
-- universe levels)
module []-cong (ax : ∀ {ℓ} → []-cong-axiomatisation ℓ) where
private
open module BC₁ {ℓ} =
[]-cong₁ (ax {ℓ = ℓ})
public
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Closure of a unary relation with respect to a preorder
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Relation.Unary.Closure.Preorder {a r e} (P : Preorder a e r) where
open Preorder P
open import Relation.Unary using (Pred)
-- Specialising the results proven generically in `Base`.
import Relation.Unary.Closure.Base _∼_ as Base
open Base public using (□; map; Closed)
module _ {t} {T : Pred Carrier t} where
reindex : ∀ {x y} → x ∼ y → □ T x → □ T y
reindex = Base.reindex trans
extract : ∀ {x} → □ T x → T x
extract = Base.extract refl
duplicate : ∀ {x} → □ T x → □ (□ T) x
duplicate = Base.duplicate trans
□-closed : ∀ {t} {T : Pred Carrier t} → Closed (□ T)
□-closed = Base.□-closed trans
|
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-- Module shadowing using generated modules for records and datatypes
module Issue260d where
data D : Set where
module D where
|
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module PLRTree.Complete.Properties {A : Set} where
open import Data.Empty
open import Data.Sum renaming (_⊎_ to _∨_)
open import PLRTree {A}
open import PLRTree.Complete {A}
open import PLRTree.Equality {A}
open import PLRTree.Equality.Properties {A}
lemma-⋗-≃ : {t t' t'' : PLRTree} → t ⋗ t' → t' ≃ t'' → t ⋗ t''
lemma-⋗-≃ (⋗lf x) ≃lf = ⋗lf x
lemma-⋗-≃ (⋗nd x x' l≃r l'≃r' l⋗l') (≃nd .x' x'' _ l''≃r'' l'≃l'') = ⋗nd x x'' l≃r l''≃r'' (lemma-⋗-≃ l⋗l' l'≃l'')
lemma-≃-⋗ : {t t' t'' : PLRTree} → t ≃ t' → t ⋗ t'' → t' ⋗ t''
lemma-≃-⋗ ≃lf t⋗t'' = t⋗t''
lemma-≃-⋗ (≃nd x x' ≃lf ≃lf ≃lf) (⋗lf .x) = ⋗lf x'
lemma-≃-⋗ (≃nd x _ ≃lf (≃nd _ _ _ _ _) ()) (⋗lf .x)
lemma-≃-⋗ (≃nd x x' l≃r l'≃r' l≃l') (⋗nd .x x'' _ l''≃r'' l⋗l'') = ⋗nd x' x'' l'≃r' l''≃r'' (lemma-≃-⋗ l≃l' l⋗l'')
lemma-⋗* : {t t' t'' : PLRTree} → t ⋗ t' → t ⋗ t'' → t' ≃ t''
lemma-⋗* (⋗lf x) (⋗lf .x) = ≃lf
lemma-⋗* (⋗lf x) (⋗nd .x _ _ _ ())
lemma-⋗* (⋗nd x _ _ _ ()) (⋗lf .x)
lemma-⋗* (⋗nd x x' l≃r l'≃r' l⋗l') (⋗nd .x x'' _ l''≃r'' l'⋗l'') = ≃nd x' x'' l'≃r' l''≃r'' (lemma-⋗* l⋗l' l'⋗l'')
lemma-*⋗ : {t t' t'' : PLRTree} → t ⋗ t' → t'' ⋗ t' → t ≃ t''
lemma-*⋗ (⋗lf x) (⋗lf y) = ≃nd x y ≃lf ≃lf ≃lf
lemma-*⋗ (⋗nd x x' l≃r l'≃r' l⋗l') (⋗nd x'' .x' l''≃r'' _ l''⋗l') = ≃nd x x'' l≃r l''≃r'' (lemma-*⋗ l⋗l' l''⋗l')
lemma-⋗refl-⊥ : {t : PLRTree} → t ⋗ t → ⊥
lemma-⋗refl-⊥ (⋗nd x .x _ _ t⋗t)
with lemma-⋗refl-⊥ t⋗t
... | ()
lemma-⋙-⋗ : {t t' t'' : PLRTree} → t ⋙ t' → t ⋗ t'' → t' ⋘ t'' ∨ t' ≃ t''
lemma-⋙-⋗ (⋙p (⋗lf x)) (⋗lf .x) = inj₂ ≃lf
lemma-⋙-⋗ (⋙p (⋗nd x _ _ _ ())) (⋗lf .x)
lemma-⋙-⋗ (⋙p (⋗lf x)) (⋗nd .x _ _ _ ())
lemma-⋙-⋗ (⋙p (⋗nd x x' l≃r l'≃r' l⋗l')) (⋗nd .x x'' _ l''≃r'' l⋗l'') = inj₂ (≃nd x' x'' l'≃r' l''≃r'' (lemma-⋗* l⋗l' l⋗l''))
lemma-⋙-⋗ (⋙l x _ _ _ ()) (⋗lf .x)
lemma-⋙-⋗ (⋙l x x' l≃r l'⋘r' l⋗r') (⋗nd .x x'' _ l''≃r'' l⋗l'') = inj₁ (l⋘ x' x'' l'⋘r' l''≃r'' (lemma-⋗* l⋗r' l⋗l''))
lemma-⋙-⋗ (⋙r x x' ≃lf (⋙p ()) ≃lf) (⋗lf .x)
lemma-⋙-⋗ (⋙r x x' ≃lf (⋙l _ _ _ _ _) ()) (⋗lf .x)
lemma-⋙-⋗ (⋙r x x' ≃lf (⋙r _ _ _ _ _) ()) (⋗lf .x)
lemma-⋙-⋗ (⋙r x x' l≃r l'⋙r' l≃l') (⋗nd .x x'' _ l''≃r'' l⋗l'') = inj₁ (r⋘ x' x'' l'⋙r' l''≃r'' (lemma-≃-⋗ l≃l' l⋗l''))
|
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-- Basic intuitionistic modal logic S4, without ∨, ⊥, or ◇.
-- Gentzen-style formalisation of syntax with context pairs, after Pfenning-Davies.
-- Normal forms, neutrals, and spines.
module BasicIS4.Syntax.DyadicGentzenSpinalNormalForm where
open import BasicIS4.Syntax.DyadicGentzen public
-- Commuting propositions for neutrals.
data Tyⁿᵉ : Ty → Set where
α_ : (P : Atom) → Tyⁿᵉ (α P)
□_ : (A : Ty) → Tyⁿᵉ (□ A)
-- Derivations.
mutual
-- Normal forms, or introductions.
infix 3 _⊢ⁿᶠ_
data _⊢ⁿᶠ_ : Cx² Ty Ty → Ty → Set where
neⁿᶠ : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A → {{_ : Tyⁿᵉ A}} → Γ ⁏ Δ ⊢ⁿᶠ A
lamⁿᶠ : ∀ {A B Γ Δ} → Γ , A ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ▻ B
boxⁿᶠ : ∀ {A Γ Δ} → ∅ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᶠ □ A
pairⁿᶠ : ∀ {A B Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ⁿᶠ B → Γ ⁏ Δ ⊢ⁿᶠ A ∧ B
unitⁿᶠ : ∀ {Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ ⊤
-- Neutrals, or eliminations.
infix 3 _⊢ⁿᵉ_
data _⊢ⁿᵉ_ : Cx² Ty Ty → Ty → Set where
spⁿᵉ : ∀ {A B C Γ Δ} → A ∈ Γ → Γ ⁏ Δ ⊢ˢᵖ A ⦙ B → Γ ⁏ Δ ⊢ᵗᵖ B ⦙ C → Γ ⁏ Δ ⊢ⁿᵉ C
mspⁿᵉ : ∀ {A B C Γ Δ} → A ∈ Δ → Γ ⁏ Δ ⊢ˢᵖ A ⦙ B → Γ ⁏ Δ ⊢ᵗᵖ B ⦙ C → Γ ⁏ Δ ⊢ⁿᵉ C
-- Spines.
infix 3 _⊢ˢᵖ_⦙_
data _⊢ˢᵖ_⦙_ : Cx² Ty Ty → Ty → Ty → Set where
nilˢᵖ : ∀ {C Γ Δ} → Γ ⁏ Δ ⊢ˢᵖ C ⦙ C
appˢᵖ : ∀ {A B C Γ Δ} → Γ ⁏ Δ ⊢ˢᵖ B ⦙ C → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ˢᵖ A ▻ B ⦙ C
fstˢᵖ : ∀ {A B C Γ Δ} → Γ ⁏ Δ ⊢ˢᵖ A ⦙ C → Γ ⁏ Δ ⊢ˢᵖ A ∧ B ⦙ C
sndˢᵖ : ∀ {A B C Γ Δ} → Γ ⁏ Δ ⊢ˢᵖ B ⦙ C → Γ ⁏ Δ ⊢ˢᵖ A ∧ B ⦙ C
-- Spine tips.
infix 3 _⊢ᵗᵖ_⦙_
data _⊢ᵗᵖ_⦙_ : Cx² Ty Ty → Ty → Ty → Set where
nilᵗᵖ : ∀ {C Γ Δ} → Γ ⁏ Δ ⊢ᵗᵖ C ⦙ C
unboxᵗᵖ : ∀ {A C Γ Δ} → Γ ⁏ Δ , A ⊢ⁿᶠ C → Γ ⁏ Δ ⊢ᵗᵖ □ A ⦙ C
-- Translation to simple terms.
mutual
nf→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ ⊢ A
nf→tm (neⁿᶠ t) = ne→tm t
nf→tm (lamⁿᶠ t) = lam (nf→tm t)
nf→tm (boxⁿᶠ t) = box (nf→tm t)
nf→tm (pairⁿᶠ t u) = pair (nf→tm t) (nf→tm u)
nf→tm unitⁿᶠ = unit
ne→tm : ∀ {A Γ Δ} → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ ⊢ A
ne→tm (spⁿᵉ i xs y) = tp→tm (var i) xs y
ne→tm (mspⁿᵉ i xs y) = tp→tm (mvar i) xs y
sp→tm : ∀ {A C Γ Δ} → Γ ⁏ Δ ⊢ A → Γ ⁏ Δ ⊢ˢᵖ A ⦙ C → Γ ⁏ Δ ⊢ C
sp→tm t nilˢᵖ = t
sp→tm t (appˢᵖ xs u) = sp→tm (app t (nf→tm u)) xs
sp→tm t (fstˢᵖ xs) = sp→tm (fst t) xs
sp→tm t (sndˢᵖ xs) = sp→tm (snd t) xs
tp→tm : ∀ {A B C Γ Δ} → Γ ⁏ Δ ⊢ A → Γ ⁏ Δ ⊢ˢᵖ A ⦙ B → Γ ⁏ Δ ⊢ᵗᵖ B ⦙ C → Γ ⁏ Δ ⊢ C
tp→tm t xs nilᵗᵖ = sp→tm t xs
tp→tm t xs (unboxᵗᵖ u) = unbox (sp→tm t xs) (nf→tm u)
-- Monotonicity with respect to context inclusion.
mutual
mono⊢ⁿᶠ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ′ ⁏ Δ ⊢ⁿᶠ A
mono⊢ⁿᶠ η (neⁿᶠ t) = neⁿᶠ (mono⊢ⁿᵉ η t)
mono⊢ⁿᶠ η (lamⁿᶠ t) = lamⁿᶠ (mono⊢ⁿᶠ (keep η) t)
mono⊢ⁿᶠ η (boxⁿᶠ t) = boxⁿᶠ t
mono⊢ⁿᶠ η (pairⁿᶠ t u) = pairⁿᶠ (mono⊢ⁿᶠ η t) (mono⊢ⁿᶠ η u)
mono⊢ⁿᶠ η unitⁿᶠ = unitⁿᶠ
mono⊢ⁿᵉ : ∀ {A Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ′ ⁏ Δ ⊢ⁿᵉ A
mono⊢ⁿᵉ η (spⁿᵉ i xs y) = spⁿᵉ (mono∈ η i) (mono⊢ˢᵖ η xs) (mono⊢ᵗᵖ η y)
mono⊢ⁿᵉ η (mspⁿᵉ i xs y) = mspⁿᵉ i (mono⊢ˢᵖ η xs) (mono⊢ᵗᵖ η y)
mono⊢ˢᵖ : ∀ {A C Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ˢᵖ A ⦙ C → Γ′ ⁏ Δ ⊢ˢᵖ A ⦙ C
mono⊢ˢᵖ η nilˢᵖ = nilˢᵖ
mono⊢ˢᵖ η (appˢᵖ xs u) = appˢᵖ (mono⊢ˢᵖ η xs) (mono⊢ⁿᶠ η u)
mono⊢ˢᵖ η (fstˢᵖ xs) = fstˢᵖ (mono⊢ˢᵖ η xs)
mono⊢ˢᵖ η (sndˢᵖ xs) = sndˢᵖ (mono⊢ˢᵖ η xs)
mono⊢ᵗᵖ : ∀ {A C Γ Γ′ Δ} → Γ ⊆ Γ′ → Γ ⁏ Δ ⊢ᵗᵖ A ⦙ C → Γ′ ⁏ Δ ⊢ᵗᵖ A ⦙ C
mono⊢ᵗᵖ η nilᵗᵖ = nilᵗᵖ
mono⊢ᵗᵖ η (unboxᵗᵖ u) = unboxᵗᵖ (mono⊢ⁿᶠ η u)
-- Monotonicity with respect to modal context inclusion.
mutual
mmono⊢ⁿᶠ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᶠ A → Γ ⁏ Δ′ ⊢ⁿᶠ A
mmono⊢ⁿᶠ θ (neⁿᶠ t) = neⁿᶠ (mmono⊢ⁿᵉ θ t)
mmono⊢ⁿᶠ θ (lamⁿᶠ t) = lamⁿᶠ (mmono⊢ⁿᶠ θ t)
mmono⊢ⁿᶠ θ (boxⁿᶠ t) = boxⁿᶠ (mmono⊢ⁿᶠ θ t)
mmono⊢ⁿᶠ θ (pairⁿᶠ t u) = pairⁿᶠ (mmono⊢ⁿᶠ θ t) (mmono⊢ⁿᶠ θ u)
mmono⊢ⁿᶠ θ unitⁿᶠ = unitⁿᶠ
mmono⊢ⁿᵉ : ∀ {A Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ⁿᵉ A → Γ ⁏ Δ′ ⊢ⁿᵉ A
mmono⊢ⁿᵉ θ (spⁿᵉ i xs y) = spⁿᵉ i (mmono⊢ˢᵖ θ xs) (mmono⊢ᵗᵖ θ y)
mmono⊢ⁿᵉ θ (mspⁿᵉ i xs y) = mspⁿᵉ (mono∈ θ i) (mmono⊢ˢᵖ θ xs) (mmono⊢ᵗᵖ θ y)
mmono⊢ˢᵖ : ∀ {A C Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ˢᵖ A ⦙ C → Γ ⁏ Δ′ ⊢ˢᵖ A ⦙ C
mmono⊢ˢᵖ θ nilˢᵖ = nilˢᵖ
mmono⊢ˢᵖ θ (appˢᵖ xs u) = appˢᵖ (mmono⊢ˢᵖ θ xs) (mmono⊢ⁿᶠ θ u)
mmono⊢ˢᵖ θ (fstˢᵖ xs) = fstˢᵖ (mmono⊢ˢᵖ θ xs)
mmono⊢ˢᵖ θ (sndˢᵖ xs) = sndˢᵖ (mmono⊢ˢᵖ θ xs)
mmono⊢ᵗᵖ : ∀ {A C Γ Δ Δ′} → Δ ⊆ Δ′ → Γ ⁏ Δ ⊢ᵗᵖ A ⦙ C → Γ ⁏ Δ′ ⊢ᵗᵖ A ⦙ C
mmono⊢ᵗᵖ θ nilᵗᵖ = nilᵗᵖ
mmono⊢ᵗᵖ θ (unboxᵗᵖ u) = unboxᵗᵖ (mmono⊢ⁿᶠ (keep θ) u)
-- Monotonicity using context pairs.
mono²⊢ⁿᶠ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᶠ A → Π′ ⊢ⁿᶠ A
mono²⊢ⁿᶠ (η , θ) = mono⊢ⁿᶠ η ∘ mmono⊢ⁿᶠ θ
mono²⊢ⁿᵉ : ∀ {A Π Π′} → Π ⊆² Π′ → Π ⊢ⁿᵉ A → Π′ ⊢ⁿᵉ A
mono²⊢ⁿᵉ (η , θ) = mono⊢ⁿᵉ η ∘ mmono⊢ⁿᵉ θ
mono²⊢ˢᵖ : ∀ {A C Π Π′} → Π ⊆² Π′ → Π ⊢ˢᵖ A ⦙ C → Π′ ⊢ˢᵖ A ⦙ C
mono²⊢ˢᵖ (η , θ) = mono⊢ˢᵖ η ∘ mmono⊢ˢᵖ θ
mono²⊢ᵗᵖ : ∀ {A C Π Π′} → Π ⊆² Π′ → Π ⊢ᵗᵖ A ⦙ C → Π′ ⊢ᵗᵖ A ⦙ C
mono²⊢ᵗᵖ (η , θ) = mono⊢ᵗᵖ η ∘ mmono⊢ᵗᵖ θ
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module Common.Context where
import Level
open import Data.Nat as Nat
open import Data.List as List
import Level
open import Relation.Binary.PropositionalEquality as PE hiding ([_])
open import Relation.Binary -- using (Setoid; Rel; IsEquivalence)
open ≡-Reasoning
open import Function as Fun hiding (_∘′_)
open import Data.Sum as Sum hiding ([_,_])
open import Categories.Category using (Category)
open import Common.SumProperties
-------------------------
---- Type contexts
Ctx : Set → Set
Ctx Ty = List Ty
-- | De Bruijn variable indexing
data Var {Ty : Set} : (Γ : Ctx Ty) (a : Ty) → Set where
zero : ∀{Γ a} → Var (a ∷ Γ) a
succ : ∀{Γ b} (a : Ty) → (x : Var Γ a) → Var (b ∷ Γ) a
data _≅V_ {Ty} : ∀ {Γ Γ' : Ctx Ty} {a a' : Ty} → Var Γ a → Var Γ' a' → Set where
zero : ∀ {Γ Γ'} {a a'}
→ zero {Γ = Γ} {a} ≅V zero {Γ = Γ'} {a'}
succ : ∀ {Γ Γ'} {a a'}
→ ∀ {x : Var Γ a} {x' : Var Γ' a'} {b b' : Ty}
→ x ≅V x'
→ succ {b = b} a x ≅V succ {b = b'} a' x'
Vrefl : ∀ {Ty} {Γ} {a : Ty} {x : Var Γ a} → x ≅V x
Vrefl {x = zero} = zero
Vrefl {x = succ _ t} = succ Vrefl
Vsym : ∀ {Ty} {Γ Γ'} {a a' : Ty} {x : Var Γ a} {x' : Var Γ' a'}
→ x ≅V x' → x' ≅V x
Vsym zero = zero
Vsym {Ty} (succ [x]) = succ (Vsym [x])
Vtrans : ∀ {Ty} {Γ Γ' Γ''} {a a' a'' : Ty}
{x : Var Γ a} {x' : Var Γ' a'} {x'' : Var Γ'' a''}
→ x ≅V x' → x' ≅V x'' → x ≅V x''
Vtrans zero zero = zero
Vtrans (succ eq) (succ eq') = succ (Vtrans eq eq')
-- Note: makes the equality homogeneous in Γ and a
≅V-setoid : ∀ {Ty} {Γ} {a : Ty} → Setoid _ _
≅V-setoid {Ty} {Γ} {a} = record
{ Carrier = Var Γ a
; _≈_ = _≅V_
; isEquivalence = record
{ refl = Vrefl ; sym = Vsym ; trans = Vtrans }
}
arr : ∀ {Ty} → (Γ Δ : Ctx Ty) → Set
arr {Ty} Γ Δ = ∀ (a : Ty) → Var Γ a → Var Δ a
_►_ = arr
-- _▹_ = arr
infix 4 _≡C_
record _≡C_ {Ty} {Γ Δ : Ctx Ty} (ρ : arr Γ Δ) (γ : arr Γ Δ) : Set where
field
≡C-proof : ∀ {a} {x} → ρ a x ≡ γ a x
open _≡C_
_≈_ = _≡C_
Crefl : ∀ {Ty} {Γ Δ : Ctx Ty} → Reflexive (_≡C_ {Γ = Γ} {Δ})
Crefl = record { ≡C-proof = PE.refl }
Csym : ∀ {Ty} {Γ Δ : Ctx Ty} → Symmetric (_≡C_ {Γ = Γ} {Δ})
Csym p = record { ≡C-proof = PE.sym (≡C-proof p) }
Ctrans : ∀ {Ty} {Γ Δ : Ctx Ty} → Transitive (_≡C_ {Γ = Γ} {Δ})
Ctrans p₁ p₂ = record { ≡C-proof = PE.trans (≡C-proof p₁) (≡C-proof p₂) }
≡C-equiv : ∀ {Ty} {Γ Δ : Ctx Ty} → IsEquivalence (_≡C_ {Γ = Γ} {Δ})
≡C-equiv =
record
{ refl = Crefl
; sym = Csym
; trans = Ctrans
}
≡C-setoid : ∀ {Ty} {Γ Δ : Ctx Ty} → Setoid _ _
≡C-setoid {_} {Γ} {Δ} = record
{ Carrier = arr Γ Δ
; _≈_ = _≡C_
; isEquivalence = ≡C-equiv
}
_∘′_ : ∀ {Ty} {Γ Δ Ξ : Ctx Ty} (ρ : Δ ► Ξ) (γ : Γ ► Δ) → Γ ► Ξ
_∘′_ ρ γ = λ a x → ρ a (γ a x)
_●_ = _∘′_
ctx-id : ∀ {Ty} {Γ : Ctx Ty} → arr Γ Γ
ctx-id = λ _ x → x
comp-resp-≡C : ∀ {Ty} {Γ Δ Ξ : Ctx Ty} {ρ ρ' : arr Δ Ξ} {γ γ' : arr Γ Δ} →
ρ ≡C ρ' → γ ≡C γ' → ρ ∘′ γ ≡C ρ' ∘′ γ'
comp-resp-≡C {_} {Γ} {Δ} {Ξ} {ρ} {ρ'} {γ} {γ'} ρ≡ρ' γ≡γ'
= record { ≡C-proof = p }
where
p : ∀ {a} {x} → (ρ ∘′ γ) a x ≡ (ρ' ∘′ γ') a x
p {a} {x} =
begin
(ρ ∘′ γ) a x
≡⟨ refl ⟩
ρ a (γ a x)
≡⟨ cong (λ u → ρ a u) (≡C-proof γ≡γ') ⟩
ρ a (γ' a x)
≡⟨ ≡C-proof ρ≡ρ' ⟩
ρ' a (γ' a x)
≡⟨ refl ⟩
(ρ' ∘′ γ') a x
∎
-- | Contexts form a category
ctx-cat : Set → Category Level.zero Level.zero Level.zero
ctx-cat Ty = record
{ Obj = Ctx Ty
; _⇒_ = arr
; _≡_ = _≡C_
; _∘_ = _∘′_
; id = ctx-id
; assoc = record { ≡C-proof = refl }
; identityˡ = record { ≡C-proof = refl }
; identityʳ = record { ≡C-proof = refl }
; equiv = ≡C-equiv
; ∘-resp-≡ = comp-resp-≡C
}
-------------------------
---- Coproduct structure on contexts
{-
_⊕_ : Ctx → Ctx → Ctx
Γ₁ ⊕ Γ₂ = Γ₁ ++ Γ₂
in₁ : {Γ₁ Γ₂ : Ctx} → Γ₁ ▹ (Γ₁ ⊕ Γ₂)
in₁ _ zero = zero
in₁ a (succ .a x) = succ a (in₁ a x)
in₂ : {{Γ₁ Γ₂ : Ctx}} → Γ₂ ▹ (Γ₁ ⊕ Γ₂)
in₂ {{[]}} _ x = x
in₂ {{b ∷ Γ₁}} a x = succ a (in₂ a x)
split : {Γ₁ Γ₂ : Ctx} {a : Ty} → Var (Γ₁ ⊕ Γ₂) a → Var Γ₁ a ⊎ Var Γ₂ a
split {[]} {Γ₂} x = inj₂ x
split {a ∷ Γ₁'} {Γ₂} zero = inj₁ zero
split {b ∷ Γ₁'} {Γ₂} {a} (succ .a y) = (Sum.map (succ a) (ctx-id a)) (split {Γ₁'} y)
[_,_] : {Γ₁ Γ₂ Δ : Ctx} (f : Γ₁ ▹ Δ) (g : Γ₂ ▹ Δ)
→ ((Γ₁ ⊕ Γ₂) ▹ Δ)
[_,_] {Γ} {Γ₂} f g a x = ([ f a , g a ]′) (split x)
_-⊕-_ : {Γ Γ₂ Γ' Γ₂' : Ctx} (f : Γ ▹ Γ') (g : Γ₂ ▹ Γ₂')
→ ((Γ ⊕ Γ₂) ▹ (Γ' ⊕ Γ₂'))
_-⊕-_ {Γ} {Γ₂} {Γ'} {Γ₂'} f g = [ in₁ ● f , in₂ {{Γ'}} {{Γ₂'}} ● g ]
succ-distr-lemma : {Γ : Ctx} {a b : Ty} (Γ₂ : Ctx) (x : Var Γ a) →
(in₁ {b ∷ Γ} ● succ {Γ}) a x
≡ (succ {Γ ⊕ Γ₂} ● in₁ {Γ} {Γ₂}) a x
succ-distr-lemma Γ₂ x = refl
split-lemma₁ : {a : Ty} (Γ₁ Γ₂ : Ctx) (x : Var Γ₁ a) →
split {Γ₁} {Γ₂} (in₁ {Γ₁} a x) ≡ inj₁ x
split-lemma₁ (tt ∷ Γ₁) Γ₂ zero = refl
split-lemma₁ (tt ∷ Γ₁) Γ₂ (succ a x) =
begin
split {tt ∷ Γ₁} (in₁ {tt ∷ Γ₁} a (succ a x))
≡⟨ refl ⟩
(Sum.map (succ a) id) (split (in₁ a x))
≡⟨ cong (Sum.map (succ a) id) (split-lemma₁ Γ₁ Γ₂ x) ⟩
(Sum.map (succ a) id) (inj₁ x)
≡⟨ refl ⟩
inj₁ (succ a x)
∎
split-lemma₂ : {a : Ty} (Γ₁ Γ₂ : Ctx) (x : Var Γ₂ a) →
split {Γ₁} {Γ₂} (in₂ a x) ≡ inj₂ x
split-lemma₂ [] Γ₂ x = refl
split-lemma₂ {a} (tt ∷ Γ₁) Γ₂ x =
begin
split {tt ∷ Γ₁} {Γ₂} (in₂ {{tt ∷ Γ₁}} a x)
≡⟨ refl ⟩
Sum.map (succ a) id (split (in₂ {{Γ₁}} a x))
≡⟨ cong (λ u → Sum.map (succ a) id u) (split-lemma₂ Γ₁ Γ₂ x) ⟩
Sum.map (succ a) id (inj₂ x)
≡⟨ refl ⟩
inj₂ x
∎
split-lemma : (Γ₁ Γ₂ : Ctx) (a : Ty) (x : Var (Γ₁ ⊕ Γ₂) a) →
[ in₁ {Γ₁} {Γ₂} a , in₂ a ]′ (split x) ≡ x
split-lemma [] Γ₂ _ x = refl
split-lemma (a ∷ Γ₁) Γ₂ .a zero = refl
split-lemma (b ∷ Γ₁) Γ₂ a (succ .a x) =
begin
[ in₁ {b ∷ Γ₁} a , in₂ {{b ∷ Γ₁}} a ]′ (split (succ a x))
≡⟨ refl ⟩
[ in₁ {b ∷ Γ₁} a , (succ {Γ₁ ⊕ Γ₂} ● in₂ {{Γ₁}} ) a ]′
(Sum.map (succ {Γ₁} a) id (split x))
≡⟨ copair-sum-map-merge {f₁ = Var.succ {Γ₁} {b} a} (split x) ⟩
[ (in₁ {b ∷ Γ₁} ● succ {Γ₁}) a , (succ {Γ₁ ⊕ Γ₂} ● in₂) a ]′ (split x)
≡⟨ copair-cong {f = (in₁ {b ∷ Γ₁} ● succ {Γ₁}) a}
(succ-distr-lemma Γ₂)
(split x) ⟩
[ (succ {Γ₁ ⊕ Γ₂} ● in₁ {Γ₁}) a , (succ {Γ₁ ⊕ Γ₂} ● in₂) a ]′ (split x)
≡⟨ copair-distr {f = in₁ {Γ₁} {Γ₂} a} {h = succ {Γ₁ ⊕ Γ₂} a} (split x)⟩
(Var.succ {Γ₁ ⊕ Γ₂} {b} a ∘ [ in₁ {Γ₁} a , in₂ a ]′) (split x)
≡⟨ cong (succ {Γ₁ ⊕ Γ₂} {b} a) (split-lemma Γ₁ Γ₂ a x) ⟩
succ {Γ₁ ⊕ Γ₂} a x
∎
⊕-is-coprod-arg : ∀{Γ₁ Γ₂ : Ctx} (a : Ty) (x : Var (Γ₁ ⊕ Γ₂) a) →
[ in₁ {Γ₁} {Γ₂} , in₂ ] a x ≡ ctx-id a x
⊕-is-coprod-arg {Γ₁} {Γ₂} = split-lemma Γ₁ Γ₂
⊕-is-coprod : ∀{Γ₁ Γ₂ : Ctx} → [ in₁ {Γ₁} {Γ₂} , in₂ ] ≡C ctx-id
⊕-is-coprod {Γ₁} = {!!}
{-
η-≡ {f₁ = [ in₁ {Γ₁} , in₂ ]}
{f₂ = ctx-id}
(λ (a : Ty) →
η-≡ {f₁ = [ in₁ {Γ₁}, in₂ ] a}
{f₂ = ctx-id a}
(⊕-is-coprod-arg {Γ₁} a)
)
-}
●-distr-copair₁ˡ : ∀{Γ₁ Γ₂ Δ : Ctx}
(h : (Γ₁ ⊕ Γ₂) ▹ Δ) (a : Ty) (x : Var (Γ₁ ⊕ Γ₂) a) →
[ h ● in₁ {Γ₁} {Γ₂} , h ● in₂ {{Γ₁}} {{Γ₂}} ] a x
≡ (h ● [ in₁ {Γ₁} {Γ₂} , in₂ ]) a x
●-distr-copair₁ˡ {Γ₁} {Γ₂} {Δ} h a x =
begin
[ h ● in₁ {Γ₁} , h ● in₂ ] a x
≡⟨ refl ⟩
([ (h ● in₁ {Γ₁}) a , (h ● in₂) a ]′) (split x)
≡⟨ copair-distr {f = in₁ {Γ₁} a} {g = in₂ a} {h = h a} (split x) ⟩
(h ● [ in₁ {Γ₁} , in₂ ]) a x
∎
●-distr-copairˡ : ∀{Γ₁ Γ₂ Δ : Ctx} (h : (Γ₁ ⊕ Γ₂) ▹ Δ) →
[ h ● in₁ {Γ₁} {Γ₂} , h ● in₂ {{Γ₁}} {{Γ₂}} ]
≡ h ● [ in₁ {Γ₁} {Γ₂} , in₂ ]
●-distr-copairˡ {Γ₁} h = {!!}
-- η-≡ (λ a → η-≡ (●-distr-copair₁ˡ {Γ₁} h a))
⊕-is-coprod₁ : ∀{Γ₁ Γ₂ Δ : Ctx} {f : Γ₁ ▹ Δ} {g : Γ₂ ▹ Δ} {h : (Γ₁ ⊕ Γ₂) ▹ Δ} →
h ● in₁ ≡C f → h ● in₂ ≡C g → [ f , g ] ≡C h
⊕-is-coprod₁ {Γ₁} {Γ₂} {Δ} {f} {g} {h} h●in₁≡f h●in₂≡g
= record { ≡C-proof = p }
where
p : ∀ {a} {x} → [ f , g ] a x ≡ h a x
p {a} {x} =
begin
[ f , g ] a x
≡⟨ refl ⟩
([ f a , g a ]′) (split x)
≡⟨ cong (λ u → [ u , g a ]′ (split x)) {!!} ⟩
([ (h ● in₁ {Γ₁}) a , g a ]′) (split x)
≡⟨ {!!} ⟩
h a x
∎
{-
[ h ● in₁ {Γ₁} , g ]
≡⟨ cong (λ u → [ h ● in₁ {Γ₁} , u ]) (sym h●in₂≡g) ⟩
[ h ● in₁ {Γ₁} , h ● in₂ ]
≡⟨ ●-distr-copairˡ {Γ₁} h ⟩
h ● [ in₁ {Γ₁}, in₂ ]
≡⟨ cong (λ u → h ● u) (⊕-is-coprod {Γ₁}) ⟩
h ● ctx-id
≡⟨ refl ⟩
h
-}
commute-in₁-arg : ∀ {Γ₁ Γ₂ Δ : Ctx} {f : Γ₁ ▹ Δ} {g : Γ₂ ▹ Δ}
(a : Ty) (x : Var Γ₁ a) →
([ f , g ] ● in₁) a x ≡ f a x
commute-in₁-arg _ zero = refl
commute-in₁-arg {b ∷ Γ₁} {Γ₂} {Δ} {f} {g} a (succ .a x) =
begin
([ f , g ] ● in₁ {b ∷ Γ₁}) a (succ {Γ₁} a x)
≡⟨ refl ⟩
[ f , g ] a (succ {Γ₁ ⊕ Γ₂} a (in₁ {Γ₁} a x))
≡⟨ refl ⟩
([ f a , g a ]′) (split (succ {Γ₁ ⊕ Γ₂} a (in₁ {Γ₁} a x)))
≡⟨ refl ⟩
[ f a , g a ]′ ((Sum.map (succ a) id) (split {Γ₁} {Γ₂} (in₁ {Γ₁} a x)))
≡⟨ refl ⟩
(([ f a , g a ]′ ∘ (Sum.map (succ a) id)) (split {Γ₁} {Γ₂} (in₁ {Γ₁} a x)))
≡⟨ copair-sum-map-merge {f₁ = succ a} (split {Γ₁} {Γ₂} (in₁ {Γ₁} a x)) ⟩
([ (f ● succ) a , g a ]′ (split {Γ₁} {Γ₂} (in₁ {Γ₁} a x)))
≡⟨ cong ([ (f ● succ) a , g a ]′) (split-lemma₁ Γ₁ Γ₂ x) ⟩
f a (succ a x)
∎
commute-in₁ : (Γ₁ : Ctx) → (Γ₂ : Ctx) → {Δ : Ctx} {f : Γ₁ ▹ Δ} {g : Γ₂ ▹ Δ}
→ ([ f , g ] ● in₁) ≡C f
commute-in₁ Γ₁ Γ₂ {Δ} {f} {g} =
record { ≡C-proof = λ {a} {x} → commute-in₁-arg {f = f} {g} a x }
commute-in₂-arg : ∀ {Γ₁ Γ₂ Δ : Ctx} {f : Γ₁ ▹ Δ} {g : Γ₂ ▹ Δ}
(a : Ty) (x : Var Γ₂ a) →
([ f , g ] ● in₂) a x ≡ g a x
commute-in₂-arg {[]} _ _ = refl
commute-in₂-arg {tt ∷ Γ₁} {Γ₂} {Δ} {f} {g} a x =
begin
([ f , g ] ● in₂ {{tt ∷ Γ₁}} ) a x
≡⟨ refl ⟩
[ f , g ] a ((succ ● in₂) a x)
≡⟨ refl ⟩
[ f a , g a ]′ (split {tt ∷ Γ₁} (succ a (in₂ a x)))
≡⟨ cong (λ u → [ f a , g a ]′ u) {x = split {tt ∷ Γ₁} (succ a (in₂ a x))} refl ⟩
[ f a , g a ]′ ((Sum.map (succ a) id) (split {Γ₁} (in₂ a x)))
≡⟨ cong (λ u → [ f a , g a ]′ (Sum.map (succ a) id u)) (split-lemma₂ Γ₁ Γ₂ x) ⟩
[ f a , g a ]′ ((Sum.map (succ a) id) (inj₂ x))
≡⟨ copair-sum-map-merge {f₁ = succ {Γ₁} a} {f₂ = id} {g₁ = f a} {g₂ = g a} (inj₂ x) ⟩
[ (f ● succ) a , (g ● ctx-id) a ]′ (inj₂ x)
≡⟨ copair-elimʳ {f = (f ● succ) a} {g = (g ● ctx-id) a} x ⟩
g a x
∎
commute-in₂ : (Γ₁ : Ctx) → (Γ₂ : Ctx) → {Δ : Ctx} {f : Γ₁ ▹ Δ} {g : Γ₂ ▹ Δ}
→ ([ f , g ] ● in₂) ≡C g
commute-in₂ Γ₁ Γ₂ {Δ} {f} {g} =
record { ≡C-proof = λ {a} {x} → commute-in₂-arg {f = f} {g} a x }
open import Categories.Object.Coproduct ctx-cat
ctx-coproduct : ∀{Γ₁ Γ₂ : Ctx} → Coproduct Γ₁ Γ₂
ctx-coproduct {Γ₁} {Γ₂} = record
{ A+B = Γ₁ ⊕ Γ₂
; i₁ = in₁
; i₂ = in₂
; [_,_] = [_,_]
; commute₁ = commute-in₁ Γ₁ Γ₂
; commute₂ = commute-in₂ Γ₁ Γ₂
; universal = ⊕-is-coprod₁
}
open import Categories.Object.BinaryCoproducts ctx-cat
ctx-bin-coproducts : BinaryCoproducts
ctx-bin-coproducts = record { coproduct = ctx-coproduct }
open BinaryCoproducts ctx-bin-coproducts
-}
|
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-- Andreas, 2017-01-24, issue #2429
-- ..-annotation in lambdas should be taken seriously
-- A ≤ ..A ≤ .A
-- (.A → B) ≤ (..A → B) ≤ A → B
should-fail : ∀{A B : Set} → (.A → B) → (.A → B)
should-fail f = λ ..a → f a
-- Expected error:
-- Found a non-strict lambda where a irrelevant lambda was expected
-- when checking that the expression λ ..a → f a has type ..A → .B
-- Note: Since A and B are not in scope, they are printed as .A and .B
-- This makes this error message super confusing.
|
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{- Name: Bowornmet (Ben) Hudson
--Preorders in Agda!--
-}
open import Preliminaries
module Preorder where
{- definition: a Preorder on a set A is a binary relation
that is reflexive and transitive.
-}
record Preorder-str (A : Set) : Set1 where
constructor preorder
field
≤ : A → A → Set
refl : ∀ x → ≤ x x
trans : ∀ x y z → ≤ x y → ≤ y z → ≤ x z
------------------------------------------
-- Task 1: Show that the Natural numbers with ≤ form a preorder
-- the ≤ relation on Natural numbers
≤nat : Nat → Nat → Set
≤nat Z Z = Unit
≤nat Z (S y) = Unit
≤nat (S x) Z = Void
≤nat (S x) (S y) = ≤nat x y
-- proof that Nat is reflexive under ≤
nat-refl : ∀ (x : Nat) → ≤nat x x
nat-refl Z = <>
nat-refl (S x) = nat-refl x
-- proof that Nat is transitive under ≤
nat-trans : ∀ (x y z : Nat) → ≤nat x y → ≤nat y z → ≤nat x z
nat-trans Z Z Z p q = <>
nat-trans Z Z (S z) p q = <>
nat-trans Z (S y) Z p q = abort q
nat-trans Z (S y) (S z) p q = <>
nat-trans (S x) Z Z () q
nat-trans (S x) Z (S z) () q
nat-trans (S x) (S y) Z p ()
nat-trans (S x) (S y) (S z) p q = nat-trans x y z p q
-- proof that Nat and ≤ (the ≤ relation defined on the natural numbers) form a preorder
nat-ispreorder : Preorder-str Nat
nat-ispreorder = record { ≤ = ≤nat; refl = nat-refl; trans = nat-trans }
------------------------------------------
-- Task 2: Show that the product of two preorders is a preorder
{- defining the relation: when is one cartesian product 'less than' another?
if A and B are preorders and we have cartesian products (a1,b1) and (a2,b2)
such that a1,a2 ∈ A and b1,b2 ∈ B,
then (a1,b1)≤(a2,b2) iff a1≤a2 and b1≤b2
-}
≤axb : ∀ {A B : Set} → Preorder-str A → Preorder-str B → (A × B) → (A × B) → Set
≤axb PA PB (a1 , b1) (a2 , b2) = Preorder-str.≤ PA a1 a2 × Preorder-str.≤ PB b1 b2
{- a cartesian product (a,b) is 'less than' itself
if each component of the product is reflexive, i.e.
just show that a is reflexive and b is reflexive
-}
axb-refl : ∀ {A B : Set} → (PA : Preorder-str A) → (PB : Preorder-str B) → (x : (A × B)) → ≤axb PA PB x x
axb-refl PA PB (a , b) = Preorder-str.refl PA a , Preorder-str.refl PB b
-- same idea for transitivity...
axb-trans : ∀ {A B : Set} → (PA : Preorder-str A) → (PB : Preorder-str B) → (x y z : (A × B)) → ≤axb PA PB x y → ≤axb PA PB y z → ≤axb PA PB x z
axb-trans PA PB (a1 , b1) (a2 , b2) (a3 , b3) (a1<a2 , b1<b2) (a2<a3 , b2<b3) =
Preorder-str.trans PA a1 a2 a3 a1<a2 a2<a3 , Preorder-str.trans PB b1 b2 b3 b1<b2 b2<b3
-- proof that AxB is a preorder
AxB-ispreorder : ∀ (A B : Set) → Preorder-str A → Preorder-str B → Preorder-str (A × B)
AxB-ispreorder A B pre-a pre-b = record { ≤ = ≤axb pre-a pre-b; refl = axb-refl pre-a pre-b; trans = axb-trans pre-a pre-b }
------------------------------------------
-- Task 3: Show that the sum of two preorders is a preorder
≤a+b : ∀ {A B : Set} → Preorder-str A → Preorder-str B → (Either A B) → (Either A B) → Set
≤a+b (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inl a2) = (a1 ≤a a2)
≤a+b (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inr b2) = Void
≤a+b (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inl a2) = Void
≤a+b (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inr b2) = (b1 ≤b b2)
a+b-refl : ∀ {A B : Set} → (a : Preorder-str A) → (b : Preorder-str B) → (x : (Either A B)) → ≤a+b a b x x
a+b-refl (preorder _≤a_ refla transa) pre-b (Inl a) = refla a
a+b-refl pre-a (preorder _≤b_ reflb transb) (Inr b) = reflb b
a+b-trans : ∀ {A B : Set} → (a : Preorder-str A) → (b : Preorder-str B) → (x y z : (Either A B)) → ≤a+b a b x y → ≤a+b a b y z → ≤a+b a b x z
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inl a2) (Inl a3) a1<a2 a2<a3 = transa a1 a2 a3 a1<a2 a2<a3
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inl a2) (Inr b3) p ()
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inr b2) (Inl a3) () q
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inl a1) (Inr b2) (Inr b3) () q
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inl a2) (Inl a3) () q
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inl a2) (Inr b3) () q
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inr b2) (Inl a3) p ()
a+b-trans (preorder _≤a_ refla transa) (preorder _≤b_ reflb transb) (Inr b1) (Inr b2) (Inr b3) b1<b2 b2<b3 = transb b1 b2 b3 b1<b2 b2<b3
-- proof that A+B is a preorder
A+B-ispreorder : ∀ (A B : Set) → Preorder-str A → Preorder-str B → Preorder-str (Either A B)
A+B-ispreorder A B pre-a pre-b = record { ≤ = ≤a+b pre-a pre-b; refl = a+b-refl pre-a pre-b; trans = a+b-trans pre-a pre-b }
------------------------------------------
-- Task 4: Show that given a Preorder A and Preorder B, Preorder (Monotone A B)
-- the type of monotone functions from A to B
-- i.e. functions which give you bigger outputs when you give them bigger inputs
record Monotone (A : Set) (B : Set) (PA : Preorder-str A) (PB : Preorder-str B) : Set where
constructor monotone
field
f : A → B
is-monotone : ∀ (x y : A) → Preorder-str.≤ PA x y → Preorder-str.≤ PB (f x) (f y)
-- the order on monotone functions is just the
-- pointwise order on the underlying functions
≤mono : ∀ {A B : Set} → (PA : Preorder-str A) → (PB : Preorder-str B) → (Monotone A B PA PB) → (Monotone A B PA PB) → Set
≤mono {A} PA PB f g = (x : A) → Preorder-str.≤ PB (Monotone.f f x) (Monotone.f g x)
mono-refl : ∀ {A B : Set} → (PA : Preorder-str A) → (PB : Preorder-str B) → (x : (Monotone A B PA PB)) → ≤mono PA PB x x
mono-refl PA PB f = λ x → Preorder-str.refl PB (Monotone.f f x)
mono-trans : ∀ {A B : Set} → (PA : Preorder-str A) → (PB : Preorder-str B) → (x y z : (Monotone A B PA PB)) → ≤mono PA PB x y → ≤mono PA PB y z → ≤mono PA PB x z
mono-trans PA PB f g h p q = λ x → Preorder-str.trans PB (Monotone.f f x) (Monotone.f g x) (Monotone.f h x) (p x) (q x)
monotone-ispreorder : ∀ (A B : Set) → (PA : Preorder-str A) → (PB : Preorder-str B) → Preorder-str (Monotone A B PA PB)
monotone-ispreorder A B PA PB = preorder (≤mono PA PB) (mono-refl PA PB) (mono-trans PA PB)
------------------------------------------
-- New stuff: Interpreting types as preorders
-- repackaging preorder as a type together with a Preorder structure on that type
PREORDER = Σ (λ (A : Set) → Preorder-str A)
MONOTONE : (PΓ PA : PREORDER) → Set
MONOTONE (Γ , PΓ) (A , PA) = Monotone Γ A PΓ PA
-- some operations
_×p_ : PREORDER → PREORDER → PREORDER
(A , PA) ×p (B , PB) = A × B , AxB-ispreorder A B PA PB
_+p_ : PREORDER → PREORDER → PREORDER
(A , PA) +p (B , PB) = Either A B , A+B-ispreorder A B PA PB
_->p_ : PREORDER → PREORDER → PREORDER
(A , PA) ->p (B , PB) = Monotone A B PA PB , monotone-ispreorder A B PA PB
-- Unit is a preorder
unit-p : PREORDER
unit-p = Unit , preorder (λ x y → Unit) (λ x → <>) (λ x y z _ _ → <>)
-- identity preserves monotonicity
id : ∀ {Γ} → MONOTONE Γ Γ
id = λ {Γ} → monotone (λ x → x) (λ x y x₁ → x₁)
-- composition preserves monotonicity
comp : ∀ {PA PB PC} → MONOTONE PA PB → MONOTONE PB PC → MONOTONE PA PC
comp (monotone f f-ismono) (monotone g g-ismono) =
monotone (λ x → g (f x)) (λ x y x₁ → g-ismono (f x) (f y) (f-ismono x y x₁))
-- proofs that types like pairs etc. with preorders are monotone
pair' : ∀ {PΓ PA PB} → MONOTONE PΓ PA → MONOTONE PΓ PB → MONOTONE PΓ (PA ×p PB)
pair' (monotone f f-ismono) (monotone g g-ismono) =
monotone (λ x → f x , g x) (λ x y z → f-ismono x y z , g-ismono x y z)
fst' : ∀ {PΓ PA PB} → MONOTONE PΓ (PA ×p PB) → MONOTONE PΓ PA
fst' (monotone f f-ismono) =
monotone (λ x → fst (f x)) (λ x y z → fst (f-ismono x y z))
snd' : ∀ {PΓ PA PB} → MONOTONE PΓ (PA ×p PB) → MONOTONE PΓ PB
snd' (monotone f f-ismono) =
monotone (λ x → snd (f x)) (λ x y z → snd (f-ismono x y z))
lam' : ∀ {PΓ PA PB} → MONOTONE (PΓ ×p PA) PB → MONOTONE PΓ (PA ->p PB)
lam' {Γ , preorder ≤Γ reflΓ transΓ} {a , preorder ≤a refla transa} {b , preorder ≤b reflb transb} (monotone f f-ismono) =
monotone (λ x → monotone (λ p → f (x , p)) (λ a b c → f-ismono (x , a) (x , b) (reflΓ x , c))) (λ x y z w → f-ismono (x , w) (y , w) (z , refla w))
app' : ∀ {PΓ PA PB} → MONOTONE PΓ (PA ->p PB) → MONOTONE PΓ PA → MONOTONE PΓ PB
app' {Γ , preorder ≤Γ reflΓ transΓ} {a , preorder ≤a refla transa} {b , preorder ≤b reflb transb} (monotone f f-ismono) (monotone g g-ismono) =
monotone (λ x → Monotone.f (f x) (g x)) (λ x y z → transb (Monotone.f (f x) (g x)) (Monotone.f (f y) (g x)) (Monotone.f (f y) (g y))
(f-ismono x y z (g x)) (Monotone.is-monotone (f y) (g x) (g y) (g-ismono x y z)))
inl' : ∀ {PΓ PA PB} → MONOTONE PΓ PA → MONOTONE PΓ (PA +p PB)
inl' (monotone f f-ismono) =
monotone (λ x → Inl (f x)) (λ x y z → f-ismono x y z)
inr' : ∀ {PΓ PA PB} → MONOTONE PΓ PB → MONOTONE PΓ (PA +p PB)
inr' (monotone f f-ismono) =
monotone (λ x → Inr (f x)) (λ x y z → f-ismono x y z)
case : ∀ {A B C : Set} → (A → C) → (B → C) → (Either A B → C)
case a b (Inl x) = a x
case a b (Inr x) = b x
el : PREORDER → Set
el = fst
PREORDER≤ : (PA : PREORDER) → el PA → el PA → Set
PREORDER≤ PA = Preorder-str.≤ (snd PA)
-- oh my god
lemma : ∀ {PA PB PC} {c1 c2 : el (PA +p PB)} {f1 f2 : el (PA ->p PC)} {g1 g2 : el (PB ->p PC)}
→ (PREORDER≤ (PA +p PB) c1 c2)
→ (PREORDER≤ (PA ->p PC) f1 f2)
→ (PREORDER≤ (PB ->p PC) g1 g2)
→ (PREORDER≤ PC (case (Monotone.f f1) (Monotone.f g1) c1) (case (Monotone.f f2) (Monotone.f g2) c2))
lemma {A , preorder ≤a refla transa} {B , preorder ≤b reflb transb} {C , preorder ≤c reflc transc}
{Inl a1} {Inl a2}
{monotone f1 f1-ismono} {monotone f2 f2-ismono}
a b c = transc (f1 a1) (f1 a2) (f2 a2) (f1-ismono a1 a2 a) (b a2)
lemma {PA} {PB} {PC} {Inl a1} {Inr b1} () b c
lemma {PA} {PB} {PC} {Inr b1} {Inl a1} () b c
lemma {A , preorder ≤a refla transa} {B , preorder ≤b reflb transb} {C , preorder ≤c reflc transc}
{Inr b1} {Inr b2}
{monotone f1 f1-ismono} {monotone f2 f2-ismono} {monotone g1 g1-ismono} {monotone g2 g2-ismono}
a b c = transc (g1 b1) (g1 b2) (g2 b2) (g1-ismono b1 b2 a) (c b2)
lemma2 : ∀ {PΓ PA PC} → MONOTONE (PΓ ×p PA) PC → el PΓ → MONOTONE PA PC
lemma2 {Γ , preorder ≤Γ reflΓ transΓ} (monotone f f-ismono) q = monotone (λ a → f (q , a)) (λ x y z → f-ismono (q , x) (q , y) (reflΓ q , z))
case' : ∀ {PΓ PA PB PC} → MONOTONE (PΓ ×p PA) PC -> MONOTONE (PΓ ×p PB) PC -> MONOTONE PΓ (PA +p PB) -> MONOTONE PΓ PC
case' {Γ , preorder ≤Γ reflΓ transΓ} {a , preorder ≤a refla transa} {b , preorder ≤b reflb transb} {c , preorder ≤c reflc transc}
(monotone f f-ismono) (monotone g g-ismono) (monotone h h-ismono) =
monotone (λ x → case (λ a → f (x , a)) (λ b → g (x , b)) (h x))
(λ x y z → lemma {a , preorder ≤a refla transa} {b , preorder ≤b reflb transb} {c , preorder ≤c reflc transc} {h x} {h y}
{lemma2 {Γ , preorder ≤Γ reflΓ transΓ}
{a , preorder ≤a refla transa} {c , preorder ≤c reflc transc}
(monotone f f-ismono) x}
{lemma2 {Γ , preorder ≤Γ reflΓ transΓ}
{a , preorder ≤a refla transa} {c , preorder ≤c reflc transc}
(monotone f f-ismono) y}
{lemma2 {Γ , preorder ≤Γ reflΓ transΓ}
{b , preorder ≤b reflb transb} {c , preorder ≤c reflc transc}
(monotone g g-ismono) x}
{lemma2 {Γ , preorder ≤Γ reflΓ transΓ}
{b , preorder ≤b reflb transb} {c , preorder ≤c reflc transc}
(monotone g g-ismono) y}
(h-ismono x y z) (λ a₁ → f-ismono (x , a₁) (y , a₁) (z , refla a₁)) (λ b₁ → g-ismono (x , b₁) (y , b₁) (z , reflb b₁)))
|
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{-# OPTIONS --cubical --safe #-}
module Cubical.HITs.Colimit.Examples where
open import Cubical.Core.Glue
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Data.SumFin
open import Cubical.Data.Graph
open import Cubical.HITs.Colimit.Base
open import Cubical.HITs.Pushout
-- Pushouts are colimits over the graph ⇐⇒
module _ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} where
PushoutDiag : (A → B) → (A → C) → Diag (ℓ-max ℓ (ℓ-max ℓ' ℓ'')) ⇐⇒
(PushoutDiag f g) $ fzero = Lift {j = ℓ-max ℓ ℓ'' } B
(PushoutDiag f g) $ fsuc fzero = Lift {j = ℓ-max ℓ' ℓ'' } A
(PushoutDiag f g) $ fsuc (fsuc fzero) = Lift {j = ℓ-max ℓ ℓ' } C
_<$>_ (PushoutDiag f g) {fsuc fzero} {fzero} tt (lift a) = lift (f a)
_<$>_ (PushoutDiag f g) {fsuc fzero} {fsuc (fsuc fzero)} tt (lift a) = lift (g a)
module _ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''} {f : A → B} {g : A → C} where
PushoutCocone : Cocone _ (PushoutDiag f g) (Pushout f g)
leg PushoutCocone fzero (lift b) = inl b
leg PushoutCocone (fsuc fzero) (lift a) = inr (g a)
leg PushoutCocone (fsuc (fsuc fzero)) (lift c) = inr c
com PushoutCocone {fsuc fzero} {fzero} tt i (lift a) = push a i
com PushoutCocone {fsuc fzero} {fsuc (fsuc fzero)} tt i (lift a) = inr (g a)
private
module _ ℓq (Y : Type ℓq) where
fwd : (Pushout f g → Y) → Cocone ℓq (PushoutDiag f g) Y
fwd = postcomp PushoutCocone
module _ (C : Cocone ℓq (PushoutDiag f g) Y) where
coml : ∀ a → leg C fzero (lift (f a)) ≡ leg C (fsuc fzero) (lift a)
comr : ∀ a → leg C (fsuc (fsuc fzero)) (lift (g a)) ≡ leg C (fsuc fzero) (lift a)
coml a i = com C {j = fsuc fzero} {k = fzero} tt i (lift a)
comr a i = com C {j = fsuc fzero} {k = fsuc (fsuc fzero)} tt i (lift a)
bwd : Cocone ℓq (PushoutDiag f g) Y → (Pushout f g → Y)
bwd C (inl b) = leg C fzero (lift b)
bwd C (inr c) = leg C (fsuc (fsuc fzero)) (lift c)
bwd C (push a i) = (coml C a ∙ sym (comr C a)) i
bwd-fwd : ∀ F → bwd (fwd F) ≡ F
bwd-fwd F i (inl b) = F (inl b)
bwd-fwd F i (inr c) = F (inr c)
bwd-fwd F i (push a j) = compPath-filler (coml (fwd F) a) (sym (comr (fwd F) a)) (~ i) j
fwd-bwd : ∀ C → fwd (bwd C) ≡ C
leg (fwd-bwd C i) fzero (lift b) = leg C fzero (lift b)
leg (fwd-bwd C i) (fsuc fzero) (lift a) = comr C a i
leg (fwd-bwd C i) (fsuc (fsuc fzero)) (lift c) = leg C (fsuc (fsuc fzero)) (lift c)
com (fwd-bwd C i) {fsuc fzero} {fzero} tt j (lift a) -- coml (fwd-bwd C i) = ...
= compPath-filler (coml C a) (sym (comr C a)) (~ i) j
com (fwd-bwd C i) {fsuc fzero} {fsuc (fsuc fzero)} tt j (lift a) -- comr (fwd-bwd C i) = ...
= comr C a (i ∧ j)
eqv : isEquiv {A = (Pushout f g → Y)} {B = Cocone ℓq (PushoutDiag f g) Y} (postcomp PushoutCocone)
eqv = isoToIsEquiv (iso fwd bwd fwd-bwd bwd-fwd)
isColimPushout : isColimit (PushoutDiag f g) (Pushout f g)
cone isColimPushout = PushoutCocone
univ isColimPushout = eqv
colim≃Pushout : colim (PushoutDiag f g) ≃ Pushout f g
colim≃Pushout = uniqColimit colimIsColimit isColimPushout
|
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module Luau.OpSem where
open import Agda.Builtin.Equality using (_≡_)
open import Agda.Builtin.Float using (Float; primFloatPlus; primFloatMinus; primFloatTimes; primFloatDiv)
open import FFI.Data.Maybe using (just)
open import Luau.Heap using (Heap; _≡_⊕_↦_; _[_]; function_is_end)
open import Luau.Substitution using (_[_/_]ᴮ)
open import Luau.Syntax using (Expr; Stat; Block; nil; addr; var; function_is_end; _$_; block_is_end; local_←_; _∙_; done; return; name; fun; arg; binexp; BinaryOperator; +; -; *; /; number)
open import Luau.Value using (addr; val; number)
evalBinOp : Float → BinaryOperator → Float → Float
evalBinOp x + y = primFloatPlus x y
evalBinOp x - y = primFloatMinus x y
evalBinOp x * y = primFloatTimes x y
evalBinOp x / y = primFloatDiv x y
data _⊢_⟶ᴮ_⊣_ {a} : Heap a → Block a → Block a → Heap a → Set
data _⊢_⟶ᴱ_⊣_ {a} : Heap a → Expr a → Expr a → Heap a → Set
data _⊢_⟶ᴱ_⊣_ where
nil : ∀ {H} →
-------------------
H ⊢ nil ⟶ᴱ nil ⊣ H
function : ∀ {H H′ a F B} →
H′ ≡ H ⊕ a ↦ (function F is B end) →
-------------------------------------------
H ⊢ (function F is B end) ⟶ᴱ (addr a) ⊣ H′
app₁ : ∀ {H H′ M M′ N} →
H ⊢ M ⟶ᴱ M′ ⊣ H′ →
-----------------------------
H ⊢ (M $ N) ⟶ᴱ (M′ $ N) ⊣ H′
app₂ : ∀ {H H′ V N N′} →
H ⊢ N ⟶ᴱ N′ ⊣ H′ →
-----------------------------
H ⊢ (val V $ N) ⟶ᴱ (val V $ N′) ⊣ H′
beta : ∀ {H a F B V} →
H [ a ] ≡ just(function F is B end) →
-----------------------------------------------------------------------------
H ⊢ (addr a $ val V) ⟶ᴱ (block (fun F) is (B [ V / name(arg F) ]ᴮ) end) ⊣ H
block : ∀ {H H′ B B′ b} →
H ⊢ B ⟶ᴮ B′ ⊣ H′ →
----------------------------------------------------
H ⊢ (block b is B end) ⟶ᴱ (block b is B′ end) ⊣ H′
return : ∀ {H V B b} →
--------------------------------------------------------
H ⊢ (block b is return (val V) ∙ B end) ⟶ᴱ (val V) ⊣ H
done : ∀ {H b} →
---------------------------------
H ⊢ (block b is done end) ⟶ᴱ nil ⊣ H
binOpEval :
∀ {H x op y} →
--------------------------------------------------------------------------
H ⊢ (binexp (number x) op (number y)) ⟶ᴱ (number (evalBinOp x op y)) ⊣ H
binOp₁ :
∀ {H H′ x x′ op y} →
H ⊢ x ⟶ᴱ x′ ⊣ H′ →
---------------------------------------------
H ⊢ (binexp x op y) ⟶ᴱ (binexp x′ op y) ⊣ H′
binOp₂ :
∀ {H H′ x op y y′} →
H ⊢ y ⟶ᴱ y′ ⊣ H′ →
---------------------------------------------
H ⊢ (binexp x op y) ⟶ᴱ (binexp x op y′) ⊣ H′
data _⊢_⟶ᴮ_⊣_ where
local : ∀ {H H′ x M M′ B} →
H ⊢ M ⟶ᴱ M′ ⊣ H′ →
-------------------------------------------------
H ⊢ (local x ← M ∙ B) ⟶ᴮ (local x ← M′ ∙ B) ⊣ H′
subst : ∀ {H x v B} →
------------------------------------------------------
H ⊢ (local x ← val v ∙ B) ⟶ᴮ (B [ v / name x ]ᴮ) ⊣ H
function : ∀ {H H′ a F B C} →
H′ ≡ H ⊕ a ↦ (function F is C end) →
--------------------------------------------------------------
H ⊢ (function F is C end ∙ B) ⟶ᴮ (B [ addr a / fun F ]ᴮ) ⊣ H′
return : ∀ {H H′ M M′ B} →
H ⊢ M ⟶ᴱ M′ ⊣ H′ →
--------------------------------------------
H ⊢ (return M ∙ B) ⟶ᴮ (return M′ ∙ B) ⊣ H′
data _⊢_⟶*_⊣_ {a} : Heap a → Block a → Block a → Heap a → Set where
refl : ∀ {H B} →
----------------
H ⊢ B ⟶* B ⊣ H
step : ∀ {H H′ H″ B B′ B″} →
H ⊢ B ⟶ᴮ B′ ⊣ H′ →
H′ ⊢ B′ ⟶* B″ ⊣ H″ →
------------------
H ⊢ B ⟶* B″ ⊣ H″
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Arguments used in the reflection machinery
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
module Reflection.Argument where
open import Data.List.Base as List using (List; []; _∷_)
open import Data.Product using (_×_; _,_; uncurry; <_,_>)
open import Data.Nat using (ℕ)
open import Reflection.Argument.Visibility
open import Reflection.Argument.Relevance
open import Reflection.Argument.Information as Information
open import Relation.Nullary
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality
open import Level
private
variable
a b : Level
A B : Set a
------------------------------------------------------------------------
-- Re-exporting the builtins publicly
open import Agda.Builtin.Reflection public using (Arg)
open Arg public
-- Pattern synonyms
pattern vArg ty = arg (arg-info visible relevant) ty
pattern hArg ty = arg (arg-info hidden relevant) ty
pattern iArg ty = arg (arg-info instance′ relevant) ty
------------------------------------------------------------------------
-- Lists of arguments
Args : {a : Level} (A : Set a) → Set a
Args A = List (Arg A)
infixr 5 _⟨∷⟩_ _⟅∷⟆_
pattern _⟨∷⟩_ x xs = vArg x ∷ xs
pattern _⟅∷⟆_ x xs = hArg x ∷ xs
------------------------------------------------------------------------
-- Operations
map : (A → B) → Arg A → Arg B
map f (arg i x) = arg i (f x)
map-Args : (A → B) → Args A → Args B
map-Args f xs = List.map (map f) xs
------------------------------------------------------------------------
-- Decidable equality
arg-injective₁ : ∀ {i i′} {a a′ : A} → arg i a ≡ arg i′ a′ → i ≡ i′
arg-injective₁ refl = refl
arg-injective₂ : ∀ {i i′} {a a′ : A} → arg i a ≡ arg i′ a′ → a ≡ a′
arg-injective₂ refl = refl
arg-injective : ∀ {i i′} {a a′ : A} → arg i a ≡ arg i′ a′ → i ≡ i′ × a ≡ a′
arg-injective = < arg-injective₁ , arg-injective₂ >
-- We often need decidability of equality for Arg A when implementing it
-- for A. Unfortunately ≡-dec makes the termination checker unhappy.
-- Instead, we can match on both Arg A and use unArg-dec for an obviously
-- decreasing recursive call.
unArg : Arg A → A
unArg (arg i a) = a
unArg-dec : {x y : Arg A} → Dec (unArg x ≡ unArg y) → Dec (x ≡ y)
unArg-dec {x = arg i a} {arg i′ a′} a≟a′ =
Dec.map′ (uncurry (cong₂ arg)) arg-injective ((i Information.≟ i′) ×-dec a≟a′)
≡-dec : DecidableEquality A → DecidableEquality (Arg A)
≡-dec _≟_ x y = unArg-dec (unArg x ≟ unArg y)
|
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2021, Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
open import LibraBFT.Base.Types
open import LibraBFT.Concrete.Records
open import LibraBFT.Impl.Consensus.Network
open import LibraBFT.Impl.Properties.Util
open import LibraBFT.ImplShared.Base.Types
open import LibraBFT.ImplShared.Consensus.Types
open import LibraBFT.ImplShared.NetworkMsg
open import Optics.All
open import Util.Prelude
module LibraBFT.Impl.Consensus.Network.Properties where
open Invariants
module processProposalSpec (proposal : ProposalMsg) (myEpoch : Epoch) (vv : ValidatorVerifier) where
postulate -- TODO-2: Refine contract
-- We also need to know that the the proposal message was successfully
-- checked by `ProposalMsg.verify`
contract
: case (processProposal proposal myEpoch vv) of λ where
(Left _) → Unit
(Right _) → proposal ^∙ pmProposal ∙ bEpoch ≡ myEpoch
× BlockId-correct (proposal ^∙ pmProposal)
|
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.Data.Fin where
open import Cubical.Data.Fin.Base public
open import Cubical.Data.Fin.Properties public
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed containers aka interaction structures aka polynomial
-- functors. The notation and presentation here is closest to that of
-- Hancock and Hyvernat in "Programming interfaces and basic topology"
-- (2006/9).
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe --guardedness #-}
module Data.Container.Indexed where
open import Level
open import Codata.Musical.M.Indexed
open import Data.Product as Prod hiding (map)
open import Data.W.Indexed
open import Function renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
open import Function.Equality using (_⟨$⟩_)
open import Function.Inverse using (_↔_; module Inverse)
open import Relation.Unary using (Pred; _⊆_)
import Relation.Binary as B
open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_; refl)
------------------------------------------------------------------------
-- The type and its semantics ("extension").
open import Data.Container.Indexed.Core public
open Container public
-- Abbreviation for the commonly used level one version of indexed
-- containers.
_▷_ : Set → Set → Set₁
I ▷ O = Container I O zero zero
-- The least and greatest fixpoint.
μ ν : ∀ {o c r} {O : Set o} → Container O O c r → Pred O _
μ = W
ν = M
-- An equivalence relation is defined in Data.Container.Indexed.WithK.
------------------------------------------------------------------------
-- Functoriality
-- Indexed containers are functors.
map : ∀ {i o c r ℓ₁ ℓ₂} {I : Set i} {O : Set o}
(C : Container I O c r) {X : Pred I ℓ₁} {Y : Pred I ℓ₂} →
X ⊆ Y → ⟦ C ⟧ X ⊆ ⟦ C ⟧ Y
map _ f = Prod.map ⟨id⟩ (λ g → f ⟨∘⟩ g)
-- Some properties are proved in Data.Container.Indexed.WithK.
------------------------------------------------------------------------
-- Container morphisms
module _ {i₁ i₂ o₁ o₂}
{I₁ : Set i₁} {I₂ : Set i₂} {O₁ : Set o₁} {O₂ : Set o₂} where
-- General container morphism.
record ContainerMorphism {c₁ c₂ r₁ r₂ ℓ₁ ℓ₂}
(C₁ : Container I₁ O₁ c₁ r₁) (C₂ : Container I₂ O₂ c₂ r₂)
(f : I₁ → I₂) (g : O₁ → O₂)
(_∼_ : B.Rel I₂ ℓ₁) (_≈_ : B.REL (Set r₂) (Set r₁) ℓ₂)
(_·_ : ∀ {A B} → A ≈ B → A → B) :
Set (i₁ ⊔ i₂ ⊔ o₁ ⊔ o₂ ⊔ c₁ ⊔ c₂ ⊔ r₁ ⊔ r₂ ⊔ ℓ₁ ⊔ ℓ₂) where
field
command : Command C₁ ⊆ Command C₂ ⟨∘⟩ g
response : ∀ {o} {c₁ : Command C₁ o} →
Response C₂ (command c₁) ≈ Response C₁ c₁
coherent : ∀ {o} {c₁ : Command C₁ o} {r₂ : Response C₂ (command c₁)} →
f (next C₁ c₁ (response · r₂)) ∼ next C₂ (command c₁) r₂
open ContainerMorphism public
-- Plain container morphism.
_⇒[_/_]_ : ∀ {c₁ c₂ r₁ r₂} →
Container I₁ O₁ c₁ r₁ → (I₁ → I₂) → (O₁ → O₂) →
Container I₂ O₂ c₂ r₂ → Set _
C₁ ⇒[ f / g ] C₂ = ContainerMorphism C₁ C₂ f g _≡_ (λ R₂ R₁ → R₂ → R₁) _$_
-- Linear container morphism.
_⊸[_/_]_ : ∀ {c₁ c₂ r₁ r₂} →
Container I₁ O₁ c₁ r₁ → (I₁ → I₂) → (O₁ → O₂) →
Container I₂ O₂ c₂ r₂ → Set _
C₁ ⊸[ f / g ] C₂ = ContainerMorphism C₁ C₂ f g _≡_ _↔_
(λ r₂↔r₁ r₂ → Inverse.to r₂↔r₁ ⟨$⟩ r₂)
-- Cartesian container morphism.
_⇒C[_/_]_ : ∀ {c₁ c₂ r} →
Container I₁ O₁ c₁ r → (I₁ → I₂) → (O₁ → O₂) →
Container I₂ O₂ c₂ r → Set _
C₁ ⇒C[ f / g ] C₂ = ContainerMorphism C₁ C₂ f g _≡_ (λ R₂ R₁ → R₂ ≡ R₁)
(λ r₂≡r₁ r₂ → P.subst ⟨id⟩ r₂≡r₁ r₂)
-- Degenerate cases where no reindexing is performed.
module _ {i o c r} {I : Set i} {O : Set o} where
_⇒_ : B.Rel (Container I O c r) _
C₁ ⇒ C₂ = C₁ ⇒[ ⟨id⟩ / ⟨id⟩ ] C₂
_⊸_ : B.Rel (Container I O c r) _
C₁ ⊸ C₂ = C₁ ⊸[ ⟨id⟩ / ⟨id⟩ ] C₂
_⇒C_ : B.Rel (Container I O c r) _
C₁ ⇒C C₂ = C₁ ⇒C[ ⟨id⟩ / ⟨id⟩ ] C₂
------------------------------------------------------------------------
-- Plain morphisms
-- Interpretation of _⇒_.
⟪_⟫ : ∀ {i o c r ℓ} {I : Set i} {O : Set o} {C₁ C₂ : Container I O c r} →
C₁ ⇒ C₂ → (X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X
⟪ m ⟫ X (c , k) = command m c , λ r₂ →
P.subst X (coherent m) (k (response m r₂))
module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
-- Identity.
id : (C : Container I O c r) → C ⇒ C
id _ = record
{ command = ⟨id⟩
; response = ⟨id⟩
; coherent = refl
}
-- Composition.
infixr 9 _∘_
_∘_ : {C₁ C₂ C₃ : Container I O c r} →
C₂ ⇒ C₃ → C₁ ⇒ C₂ → C₁ ⇒ C₃
f ∘ g = record
{ command = command f ⟨∘⟩ command g
; response = response g ⟨∘⟩ response f
; coherent = coherent g ⟨ P.trans ⟩ coherent f
}
-- Identity commutes with ⟪_⟫.
id-correct : ∀ {ℓ} {C : Container I O c r} → ∀ {X : Pred I ℓ} {o} →
⟪ id C ⟫ X {o} ≗ ⟨id⟩
id-correct _ = refl
-- More properties are proved in Data.Container.Indexed.WithK.
------------------------------------------------------------------------
-- Linear container morphisms
module LinearMorphism
{i o c r} {I : Set i} {O : Set o} {C₁ C₂ : Container I O c r}
(m : C₁ ⊸ C₂)
where
morphism : C₁ ⇒ C₂
morphism = record
{ command = command m
; response = _⟨$⟩_ (Inverse.to (response m))
; coherent = coherent m
}
⟪_⟫⊸ : ∀ {ℓ} (X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X
⟪_⟫⊸ = ⟪ morphism ⟫
open LinearMorphism public using (⟪_⟫⊸)
------------------------------------------------------------------------
-- Cartesian morphisms
module CartesianMorphism
{i o c r} {I : Set i} {O : Set o} {C₁ C₂ : Container I O c r}
(m : C₁ ⇒C C₂)
where
morphism : C₁ ⇒ C₂
morphism = record
{ command = command m
; response = P.subst ⟨id⟩ (response m)
; coherent = coherent m
}
⟪_⟫C : ∀ {ℓ} (X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X
⟪_⟫C = ⟪ morphism ⟫
open CartesianMorphism public using (⟪_⟫C)
------------------------------------------------------------------------
-- All and any
-- □ and ◇ are defined in the core module.
module _ {i o c r ℓ₁ ℓ₂} {I : Set i} {O : Set o} (C : Container I O c r)
{X : Pred I ℓ₁} {P Q : Pred (Σ I X) ℓ₂} where
-- All.
□-map : P ⊆ Q → □ C P ⊆ □ C Q
□-map P⊆Q = _⟨∘⟩_ P⊆Q
-- Any.
◇-map : P ⊆ Q → ◇ C P ⊆ ◇ C Q
◇-map P⊆Q = Prod.map ⟨id⟩ P⊆Q
-- Membership is defined in Data.Container.Indexed.WithK.
|
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module Monads.ExceptT where
open import Class.Monad
open import Class.Monad.Except
open import Class.Monad.State
open import Class.MonadTrans
open import Data.Sum
open import Function
open import Level
private
variable
a : Level
ExceptT : (M : Set a -> Set a) -> Set a -> Set a -> Set a
ExceptT M E A = M (E ⊎ A)
ExceptT-MonadTrans : {E : Set a} -> MonadTrans (λ (M : Set a -> Set a) -> ExceptT M E)
ExceptT-MonadTrans = record { embed = λ x -> x >>= (return ∘ inj₂) }
module _ {M : Set a -> Set a} {{_ : Monad M}} {E : Set a} where
ExceptT-Monad : Monad (ExceptT M E)
ExceptT-Monad = record { _>>=_ = helper ; return = λ x → (return $ inj₂ x) }
where
helper : ∀ {A B} -> ExceptT M E A -> (A -> ExceptT M E B) -> ExceptT M E B
helper x f = x >>= λ { (inj₁ y) -> return $ inj₁ y ; (inj₂ y) -> f y }
private
throwError' : ∀ {A : Set a} -> E -> ExceptT M E A
throwError' = return ∘ inj₁
catchError' : ∀ {A} -> ExceptT M E A -> (E -> ExceptT M E A) -> ExceptT M E A
catchError' x f = x >>= λ { (inj₁ x) → f x ; (inj₂ y) → return {{ExceptT-Monad}} y }
ExceptT-MonadExcept : MonadExcept (ExceptT M E) {{ExceptT-Monad}} E
ExceptT-MonadExcept = record { throwError = throwError' ; catchError = catchError' }
ExceptT-MonadState : ∀ {S} {{_ : MonadState M S}} -> MonadState (ExceptT M E) {{ExceptT-Monad}} S
ExceptT-MonadState = record
{ get = embed {{ExceptT-MonadTrans}} get
; put = embed {{ExceptT-MonadTrans}} ∘ put }
|
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|
-- Andreas, 2020-06-17, issue #4135
-- Constructor disambiguation on should only instantiate metas
-- in a unique way.
-- {-# OPTIONS -v tc.lhs:10 #-}
-- {-# OPTIONS -v tc.lhs.split:40 #-}
-- {-# OPTIONS -v tc.lhs.disamb:40 #-}
data Bool : Set where
true false : Bool
module Foo (b : Bool) where
data D : Set where
c : D
open module True = Foo true
open module False = Foo false
test : Foo.D ? → Set
test c = ?
-- C-c C-=
-- EXPECTED ERROR:
-- Ambiguous constructor c.
-- ...
-- when checking that the pattern c has type Foo.D ?0
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed binary relations
------------------------------------------------------------------------
-- This file contains some core definitions which are re-exported by
-- Relation.Binary.Indexed.Heterogeneous.
{-# OPTIONS --without-K --safe #-}
module Relation.Binary.Indexed.Heterogeneous.Core where
open import Level
import Relation.Binary.Core as B
import Relation.Binary.PropositionalEquality.Core as P
------------------------------------------------------------------------
-- Indexed binary relations
-- Heterogeneous types
IREL : ∀ {i₁ i₂ a₁ a₂} {I₁ : Set i₁} {I₂ : Set i₂} →
(I₁ → Set a₁) → (I₂ → Set a₂) → (ℓ : Level) → Set _
IREL A₁ A₂ ℓ = ∀ {i₁ i₂} → A₁ i₁ → A₂ i₂ → Set ℓ
-- Homogeneous types
IRel : ∀ {i a} {I : Set i} → (I → Set a) → (ℓ : Level) → Set _
IRel A ℓ = IREL A A ℓ
------------------------------------------------------------------------
-- Simple properties of indexed binary relations
Reflexive : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → IRel A ℓ → Set _
Reflexive _ _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})
Symmetric : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → IRel A ℓ → Set _
Symmetric _ _∼_ = ∀ {i j} → B.Sym (_∼_ {i} {j}) _∼_
Transitive : ∀ {i a ℓ} {I : Set i} (A : I → Set a) → IRel A ℓ → Set _
Transitive _ _∼_ = ∀ {i j k} → B.Trans _∼_ (_∼_ {j}) (_∼_ {i} {k})
-- Generalised implication.
infixr 4 _=[_]⇒_
_=[_]⇒_ : ∀ {a b ℓ₁ ℓ₂} {A : Set a} {B : A → Set b} →
B.Rel A ℓ₁ → ((x : A) → B x) → IRel B ℓ₂ → Set _
P =[ f ]⇒ Q = ∀ {i j} → P i j → Q (f i) (f j)
|
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module System.File where
open import System.FilePath
open import Prelude.IO
open import Prelude.String
open import Prelude.Unit
open import Prelude.Function
open import Prelude.Bytes
{-# FOREIGN GHC import qualified Data.Text as Text #-}
{-# FOREIGN GHC import qualified Data.Text.IO as Text #-}
{-# FOREIGN GHC import qualified Data.ByteString as B #-}
private
module Internal where
StrFilePath : Set
StrFilePath = String
postulate
readTextFile : StrFilePath → IO String
writeTextFile : StrFilePath → String → IO Unit
readBinaryFile : StrFilePath → IO Bytes
writeBinaryFile : StrFilePath → Bytes → IO Unit
{-# COMPILE GHC readTextFile = Text.readFile . Text.unpack #-}
{-# COMPILE GHC writeTextFile = Text.writeFile . Text.unpack #-}
{-# COMPILE GHC readBinaryFile = B.readFile . Text.unpack #-}
{-# COMPILE GHC writeBinaryFile = B.writeFile . Text.unpack #-}
{-# COMPILE UHC readTextFile = UHC.Agda.Builtins.primReadFile #-}
{-# COMPILE UHC writeTextFile = UHC.Agda.Builtins.primWriteFile #-}
readTextFile : ∀ {k} → Path k → IO String
readTextFile = Internal.readTextFile ∘ toString
writeTextFile : ∀ {k} → Path k → String → IO Unit
writeTextFile = Internal.writeTextFile ∘ toString
readBinaryFile : ∀ {k} → Path k → IO Bytes
readBinaryFile = Internal.readBinaryFile ∘ toString
writeBinaryFile : ∀ {k} → Path k → Bytes → IO Unit
writeBinaryFile = Internal.writeBinaryFile ∘ toString
|
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{-# OPTIONS --prop --rewriting #-}
module Examples.Gcd.Refine where
open import Calf.CostMonoid
import Calf.CostMonoids as CM
open import Calf CM.ℕ-CostMonoid
open import Calf.Types.Nat
open import Calf.Types.Bounded CM.ℕ-CostMonoid
open import Examples.Gcd.Euclid
open import Examples.Gcd.Clocked as Clocked
open import Data.Nat.DivMod
open import Data.Nat
open import Relation.Binary.PropositionalEquality as P
open import Function
open import Data.Nat.Properties
open import Data.Product
open import Data.Bool using (Bool; false; true)
open import Relation.Nullary
open import Relation.Nullary.Negation
open import Relation.Binary
fib : ℕ → ℕ
fib 0 = 0
fib 1 = 1
fib (suc (suc n)) = fib (suc n) + fib n
fib⁻¹/helper : ℕ → ℕ → ℕ
fib⁻¹/helper F zero = 0
fib⁻¹/helper F (suc i) with fib (suc i) ≤? F
... | (true because (ofʸ py)) = suc i
... | (false because (ofⁿ pn)) = fib⁻¹/helper F i
fib⁻¹ : ℕ → ℕ
fib⁻¹ F = fib⁻¹/helper F (1 + F)
fib⁻¹-unfold : ∀ F i → ¬ (fib (suc i) ≤ F) →
fib⁻¹/helper F (suc i) ≡ fib⁻¹/helper F i
fib⁻¹-unfold F i h with fib (suc i) ≤? F
... | (true because (ofʸ py)) = case (h py) of λ {()}
... | (false because (ofⁿ pn)) = refl
fib-base : ∀ n → 1 ≤ fib (1 + n)
fib-base zero = s≤s z≤n
fib-base (suc n') =
let g = fib-base n' in
≤-trans g (m≤m+n _ _)
fib-inc : ∀ n → n < fib (2 + n)
fib-inc zero = s≤s z≤n
fib-inc (suc n') =
let g = fib-inc n' in
let g1 = +-mono-≤ g (fib-base n') in
subst (λ k → k ≤ fib (2 + suc n')) (P.cong suc (P.trans (+-suc n' 0) (P.cong suc (+-identityʳ _)))) g1
fib-fib⁻¹/helper : ∀ F i → fib (suc i) > F → Σ (fib (fib⁻¹/helper F i) ≤ F) λ _ → F < fib (1 + fib⁻¹/helper F i)
fib-fib⁻¹/helper F zero h = z≤n , h
fib-fib⁻¹/helper F (suc i') h with fib (suc i') ≤? F
... | (true because (ofʸ py)) = py , h
... | (false because (ofⁿ pn)) = fib-fib⁻¹/helper F i' (≰⇒> pn)
fib-fib⁻¹ : ∀ F → Σ (fib (fib⁻¹ F) ≤ F) λ _ → F < fib (1 + fib⁻¹ F)
fib-fib⁻¹ F = fib-fib⁻¹/helper F (1 + F) (fib-inc F)
fib-mono-< : fib Preserves _<_ ⟶ _≤_
fib-mono-< {zero} {zero} h = case h of λ {()}
fib-mono-< {zero} {suc y} h = ≤-trans z≤n (fib-base y)
fib-mono-< {1} {1} (s≤s h) = case h of λ {()}
fib-mono-< {1} {suc (suc y)} h = fib-base (suc y)
fib-mono-< {suc (suc x)} {suc (suc y)} (s≤s (s≤s h)) =
let g = fib-mono-< h in
let g1 = fib-mono-< (s≤s h) in
+-mono-≤ g1 g
-- test : ℕ
-- test = gcd/depth (7 , 4 , s≤s (s≤s (s≤s (s≤s (s≤s z≤n)))))
gcd/fib : ∀ (n : ℕ) (i@(x , y , h) : m>n) →
gcd/depth i ≥ 1 + n →
Σ (x ≥ fib (2 + n)) λ _ → (y ≥ fib (1 + n))
gcd/fib zero (x , y , h) h1 with 1 ≤? y | 1 ≤? x
... | (true because (ofʸ py)) | (true because (ofʸ px)) = px , py
... | (true because _) | (false because (ofⁿ px)) =
let g = ≰⇒> px in
let g1 = n<1⇒n≡0 g in
let g2 = P.subst (λ x → y < x) g1 h in
case g2 of λ { () }
... | (false because (ofⁿ py)) | _ rewrite (n<1⇒n≡0 (≰⇒> py)) =
case h1 of λ { () }
gcd/fib (suc n) (x , y , h) h1 with y
... | zero = let g = n≤0⇒n≡0 h1 in case g of λ {()}
... | suc y' rewrite gcd/depth-unfold-suc {x} {y'} {h} =
let g : suc (gcd/depth (suc y' , x % suc y' , m%n<n x y')) ≥ 1 + (suc n)
g = h1 in
let g1 = +-cancelˡ-≤ 1 g in
let (r1 , r2) = gcd/fib n (suc y' , x % suc y' , m%n<n x y') g1 in
let r1' : fib n + fib (suc n) ≤ suc y'
r1' = P.subst (λ n → n ≤ suc y') (+-comm (fib (suc n)) (fib n)) r1 in
(let e1 = m≡m%n+[m/n]*n x y' in
let e2 = m/n*n≤m x (suc y') in
let e3 : 1 ≤ x / suc y'
e3 = m≥n⇒m/n>0 (≤-trans (n≤1+n (suc y')) h) in
let e4 : 1 * (suc y') ≤ x / suc y' * suc y'
e4 = *-monoˡ-≤ (suc y') e3 in
let e5 = P.subst (λ n → n ≤ x / suc y' * suc y') (*-identityˡ (suc y')) e4 in
P.subst (λ n → x ≥ n) (P.sym (+-assoc (fib (1 + n)) (fib n) (fib (1 + n)))) (
P.subst (λ x → x ≥ _) (P.sym e1)
(+-mono-≤ {x = fib (1 + n)} {y = x % (suc y')}
r2 (≤-trans r1' e5))
)), r1
gcd/depth/bound : ∀ (n : ℕ) (i@(x , y , h) : m>n) →
x < fib (2 + n) → y < (fib (1 + n)) →
gcd/depth i < 1 + n
gcd/depth/bound n i h1 h2 = ≰⇒> (contraposition (gcd/fib n i) (λ { (g1 , g2) → (<⇒≱ h1) g1}))
gcd/depth/closed : m>n → ℕ
gcd/depth/closed i@(x , y , h) = 1 + fib⁻¹ x
gcd/depth≤gcd/depth/closed : ∀ (i@(x , y , h) : m>n) → gcd/depth i ≤ gcd/depth/closed i
gcd/depth≤gcd/depth/closed i@(x , y , h) =
let g : x < fib (1 + fib⁻¹ x)
g = fib-fib⁻¹ x .proj₂ in
let g1 : fib (1 + fib⁻¹ x) ≤ fib (2 + fib⁻¹ x)
g1 = fib-mono-< {1 + fib⁻¹ x} {2 + fib⁻¹ x} (+-monoˡ-< (fib⁻¹ x) (s≤s (s≤s z≤n))) in
(<⇒≤ (gcd/depth/bound _ i (<-transˡ g g1) (<-trans h g)))
gcd≤gcd/depth/closed : ∀ i → IsBounded nat (gcd i) (gcd/depth/closed i)
gcd≤gcd/depth/closed i = bound/relax (λ _ → gcd/depth≤gcd/depth/closed i) (gcd≤gcd/depth i)
|
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-- Andreas, 2016-05-03 issue #1950
-- testcases from issue #679
-- {-# OPTIONS --show-implicit #-}
-- {-# OPTIONS -v reify.display:100 #-}
-- {-# OPTIONS -v tc.display.top:100 #-}
postulate anything : ∀{A : Set} → A
postulate Anything : ∀{A : Set1} → A
record ⊤ : Set where
data ty : Set where
♭ : ty
_`→_ : ty → ty → ty
⟦_⟧ : ty → Set
⟦ ♭ ⟧ = ⊤
⟦ A `→ B ⟧ = ⟦ A ⟧ → ⟦ B ⟧
eq : ∀ (σ : ty) (a b : ⟦ σ ⟧) → Set
eq ♭ a b = Anything
eq (A `→ B) f g = ∀ {a : ⟦ A ⟧} → eq B (f a) (g a)
eq-sym : ∀ σ {a b} (h : eq σ a b) → eq σ b a
eq-sym ♭ h = anything
eq-sym (A `→ B) h with B
... | B' = {!B'!}
-- splitting on B' should yield
-- eq-sym (A `→ B) h | ♭ = {!!}
-- eq-sym (A `→ B) h | B' `→ B'' = {!!}
data Unit : Set where
unit : Unit
T : Unit → Set
T unit = {u : Unit} → Unit
fails : (u : Unit) → T u
fails unit with unit
... | q = {!q!}
-- Splitting on q should yield
-- fails unit | unit = {!!}
|
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|
module sv20.compiler where
open import Data.Nat using (ℕ; zero; suc; _+_; _*_; _∸_)
--open import Data.Nat.DivMod using (_/_)
open import Data.List using (List; _++_; []; _∷_; head)
open import Data.Maybe as DM
open DM using (Maybe; just; nothing; maybe; _>>=_; from-just; From-just)
import Relation.Binary.PropositionalEquality as Eq
open Eq using (_≡_; refl; sym; trans; cong; subst)
open import Data.Product using (∃-syntax) renaming (_,_ to ⟨_,_⟩)
open import Function.Base using (_∘_; flip)
pattern [_] z = z ∷ []
pattern [_,_] y z = y ∷ z ∷ []
pattern [_,_,_] x y z = x ∷ y ∷ z ∷ []
pattern [_,_,_,_] w x y z = w ∷ x ∷ y ∷ z ∷ []
pattern [_,_,_,_,_] v w x y z = v ∷ w ∷ x ∷ y ∷ z ∷ []
pattern [_,_,_,_,_,_] u v w x y z = u ∷ v ∷ w ∷ x ∷ y ∷ z ∷ []
data ExOp : Set where
sum diff prod : ExOp
data Exp : Set where
const : ℕ → Exp
exop : ExOp → Exp → Exp → Exp
--exop sum (const 5) (const 2)
prim-ex : ExOp → ℕ → ℕ → ℕ
prim-ex sum = _+_
prim-ex diff = _∸_
prim-ex prod = _*_
interpreter : Exp → ℕ
interpreter (const ℕ) = ℕ
interpreter (exop op x y) = prim-ex op (interpreter x) (interpreter y)
--interpreter (exop sum (const 5) (const 2))
data BinOp : Set where
add sub mul : BinOp
data StackCmd : Set where
push : ℕ → StackCmd
binop : BinOp → StackCmd
Program = List StackCmd
Stack = List ℕ
prim-op : BinOp → ℕ → ℕ → ℕ
prim-op add = _+_
prim-op sub = _∸_
prim-op mul = _*_
run-vm : Program → Stack → Maybe Stack
run-vm [] st = just st
run-vm (push n ∷ ps) st = run-vm ps (n ∷ st)
run-vm (binop op ∷ ps) (n ∷ m ∷ st) = run-vm ps ((prim-op op n m) ∷ st)
run-vm _ _ = nothing
exop-binop : ExOp → BinOp
exop-binop sum = add
exop-binop diff = sub
exop-binop prod = mul
compile : Exp → Program
compile (const n) = [ push n ]
compile (exop op x y) = compile y ++ compile x ++ [ binop (exop-binop op) ]
-- (3 - 2) + 6 => + (- 3 2) 6 => "sum (diff 3 2) 6"
_ : compile (exop sum (exop diff (const 3) (const 2)) (const 6))
≡ [ push 6 , push 2 , push 3 , binop sub , binop add ]
_ = refl
_ : run-vm (compile (const 3)) [] ≡ just [ interpreter (const 3) ]
_ = refl
--run-vm [ push 2 , push 1 , binop add ] []
--run-vm [ push 1 , binop add ] [ 2 ]
--run-vm [ binop add ] [ 1 , 2 ]
_ : (run-vm [ push 1 ] [ 2 ] >>= run-vm [ binop add ])
≡ run-vm [ push 1 , binop add ] [ 2 ]
_ = refl
lemma₁ : ∀ (a b : Program) (s : Stack) → run-vm (a ++ b) s ≡ (run-vm a s >>= run-vm b)
lemma₁ [] _ _ = refl
lemma₁ (push n ∷ ps) b s rewrite lemma₁ ps b (n ∷ s) = refl
lemma₁ (binop _ ∷ ps) b [] = refl
lemma₁ (binop _ ∷ _) _ (_ ∷ []) = refl
lemma₁ (binop op ∷ ps) b (n ∷ m ∷ st) rewrite lemma₁ ps b (prim-op op n m ∷ st) = refl
lemma₂ : ∀ {op : ExOp} → prim-op (exop-binop op) ≡ prim-ex op
lemma₂ {sum} = refl
lemma₂ {diff} = refl
lemma₂ {prod} = refl
compiler-correctness₁ : ∀ {e : Exp} {s : Stack} → run-vm (compile e) s ≡ just (interpreter e ∷ s)
compiler-correctness₁ {const n} = refl
compiler-correctness₁ {exop op x y} {s}
rewrite
-- run-vm (compile (exop op x y)) s ≡ just (interpreter (exop op x y) ∷ s)
--
-- run-vm (compile y ++ compile x ++ [ binop (exop-binop op) ]) s ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
lemma₁ (compile y) (compile x ++ [ binop (exop-binop op) ]) s
-- (run-vm (compile y) s >>= run-vm (compile x ++ [ binop (exop-binop op) ])) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
| compiler-correctness₁ {y} {s}
-- (just (interpreter y ∷ s) >>= run-vm (compile x ++ [ binop (exop-binop op) ])) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
-- run-vm (compile x ++ [ binop (exop-binop op) ]) (interpreter y ∷ s) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
| lemma₁ (compile x) [ binop (exop-binop op) ] (interpreter y ∷ s)
-- (run-vm (compile x) (interpreter y ∷ s) >>= run-vm [ binop (exop-binop op) ]) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
| compiler-correctness₁ {x} {interpreter y ∷ s}
-- (just (interpreter x ∷ interpreter y ∷ s) >>= run-vm [ binop (exop-binop op) ]) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
-- just (prim-op (exop-binop op) (interpreter x) (interpreter y) ∷ s) ≡ just (prim-ex op (interpreter x) (interpreter y) ∷ s)
| lemma₂ {op}
= refl
compiler-correctness : ∀ (e : Exp) → run-vm (compile e) [] ≡ just [ interpreter e ]
compiler-correctness e = compiler-correctness₁ {e} {[]}
-- Which algorithm is it using when executed "interpreter" or "compile then
-- run-vm"?
-- Well, if I understand correctly. "from-just" will return the value
-- contained in "just" and "subst" will pass the values untouched. Which
-- would mean that there is an associated cost in proving stuff but the code
-- that is being used is "compile then run-vm"
run-vm∘compile : Exp → Maybe ℕ
run-vm∘compile e =
let
justln = from-just (run-vm (compile e) [])
list = subst From-just (compiler-correctness e) justln
in head list
_ : run-vm∘compile (const 5) ≡ just 5
_ = refl
reduce : {A : Set} → Maybe (List A) → Maybe A
reduce (just (x ∷ _)) = just x
reduce _ = nothing
_ : reduce (just [ 5 ]) ≡ just 5
_ = refl
-- Look, it is possible to execute compile and run-vm and get an unwrapped result
-- For an even prettier proof look at the end of the file
run-vm∘compile` : Exp → ℕ
run-vm∘compile` e =
let
computing = reduce (run-vm (compile e) [])
justn = from-just computing
redruncomp≡jinter = cong reduce (compiler-correctness e) -- reduce (run-vm (compile e) []) ≡ just (interpreter e)
in subst From-just redruncomp≡jinter justn
_ : run-vm∘compile` (const 5) ≡ 5
_ = refl
---- So. I wanted to generalise the approach used before, but I'm stuck. Agda
---- seems not able to infer some variable.
---- It's because it is never using 'g', I think
---- Ans: NO. IT WAS BECAUSE x WAS IMPLICIT IN eq
--
--unwrap : {A B : Set}
-- → {g : A → B}
-- → (f : A → Maybe (List B))
-- → ({x : A} → f x ≡ just [ g x ]) -- <- This was the problem! It should be explicit!
-- --------------------------------
-- → A → B -- Computed using f
--unwrap {A} {B} {g} f eq x =
-- let
-- computing = reduce (f x) -- type: Maybe B
-- justb = from-just computing -- type: From-just (f x)
-- in subst From-just new-eq justb -- type: B -- thanks to the magic of From-just
--
-- where
-- new-eq : {x : A} → reduce (f x) ≡ just (g x)
-- new-eq {x} rewrite eq {x} = refl
--
---- Why doesn't this work!? Ans: See above or below
----run-vm∘compile`` : Exp → ℕ
----run-vm∘compile`` = unwrap {Exp} {ℕ}
---- {interpreter}
---- (λ e → run-vm (compile e) [])
---- compiler-correctness
unwrap : {A B : Set} {g : A → B}
→ (f : A → Maybe (List B))
→ ((x : A) → f x ≡ just [ g x ])
--------------------------------
→ A → B -- Computed using f not g (g might still be run to prove the property correct)
unwrap f eq x =
let
-- Proof: f x ≡ just [ g x ]
fx≡mlgx = eq x
-- New proof: reduce (f x) ≡ just (g x)
redfx≡gx = cong reduce {x = f x} fx≡mlgx
compute = reduce (f x) -- type: Maybe B
justredfx = from-just compute -- type: From-just (reduce (f x)) -- lifting to the type level, so we can prove stuff
in subst From-just redfx≡gx justredfx -- type: From-just (just (g x)) which is the same as `B` -- thanks to the magic of From-just
--from-just (just 3) -- evaluates to: 3
-- -- with type: From-just (just 3)
-- -- which evaluates to: ℕ
-- ie,
-- from-just (just 3) : From-just (just 3)
-- from-just (just 3) : ℕ
-- 3 : ℕ
-- 3 : From-just (just 3)
_ : from-just (just 3) ≡ 3
_ = refl
_ : From-just (just 3) ≡ ℕ
_ = refl
_ : Set
_ = From-just (just 3)
_ : From-just (just 3) -- same as ℕ
_ = 3
_ = from-just (just 3)
_ = 2 -- This is an ℕ too
run-vm∘compile`` : Exp → ℕ
run-vm∘compile`` = unwrap ((flip run-vm []) ∘ compile) compiler-correctness
--run-vm∘compile`` = unwrap (λ e → run-vm (compile e) []) compiler-correctness
--LESSON LEARNT:
--Don't use implicit paramaters unless it makes for a clearer proof.
--Always use explicit parameters and make them implicit as you use the
--functions in more places (without breaking anything)
_ : run-vm∘compile`` (const 5) ≡ 5
_ = refl
I-am-a-number : ℕ
I-am-a-number = run-vm∘compile`` (exop sum (exop diff (const 3) (const 2)) (const 20))
|
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|
-- Andreas, 2016-05-06, issue 1967
postulate
D : {{A : Set}} → Set
test : (A : Set) → D A
-- Expected error:
-- Set should be a function type, but it isn't
-- when checking that A is a valid argument to a function of type
-- {{A = A₁ : Set}} → Set
|
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{-# OPTIONS --safe #-}
module Cubical.Algebra.DirectSum.DirectSumHIT.PseudoNormalForm where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Data.Nat renaming (_+_ to _+n_ ; _·_ to _·n_)
open import Cubical.Data.Sigma
open import Cubical.Data.List
open import Cubical.Data.Vec.DepVec
open import Cubical.HITs.PropositionalTruncation as PT
open import Cubical.Algebra.AbGroup
open import Cubical.Algebra.AbGroup.Instances.DirectSumHIT
open import Cubical.Algebra.DirectSum.DirectSumHIT.Base
private variable
ℓ : Level
open AbGroupStr
open AbGroupTheory
-----------------------------------------------------------------------------
-- Notation
module DefPNF
(G : (n : ℕ) → Type ℓ)
(Gstr : (n : ℕ) → AbGroupStr (G n))
where
open AbGroupStr (snd (⊕HIT-AbGr ℕ G Gstr)) using ()
renaming
( 0g to 0⊕HIT
; _+_ to _+⊕HIT_
; -_ to -⊕HIT_
; +Assoc to +⊕HIT-Assoc
; +IdR to +⊕HIT-IdR
; +IdL to +⊕HIT-IdL
; +InvR to +⊕HIT-InvR
; +InvL to +⊕HIT-InvL
; +Comm to +⊕HIT-Comm
; is-set to isSet⊕HIT)
-----------------------------------------------------------------------------
-- Lemma
-- def pseudo normal form
sumHIT : {n : ℕ} → depVec G n → ⊕HIT ℕ G Gstr
sumHIT {0} ⋆ = 0⊕HIT
sumHIT {suc n} (a □ dv) = (base n a) +⊕HIT (sumHIT dv)
-- 0 and sum
replicate0g : (n : ℕ) → depVec G n
replicate0g (zero) = ⋆
replicate0g (suc n) = (0g (Gstr n)) □ (replicate0g n)
sumHIT0g : (n : ℕ) → sumHIT (replicate0g n) ≡ 0⊕HIT
sumHIT0g (zero) = refl
sumHIT0g (suc n) = cong₂ _+⊕HIT_ (base-neutral n) (sumHIT0g n)
∙ +⊕HIT-IdL _
-- extension and sum
extendDVL : (k l : ℕ) → (dv : depVec G l) → depVec G (k +n l)
extendDVL zero l dv = dv
extendDVL (suc k) l dv = (0g (Gstr (k +n l))) □ (extendDVL k l dv)
extendDVLeq : (k l : ℕ) → (dv : depVec G l) → sumHIT (extendDVL k l dv) ≡ sumHIT dv
extendDVLeq (zero) l dv = refl
extendDVLeq (suc k) l dv = cong (λ X → X +⊕HIT sumHIT (extendDVL k l dv)) (base-neutral (k +n l))
∙ +⊕HIT-IdL _
∙ extendDVLeq k l dv
extendDVR : (k l : ℕ) → (dv : depVec G k) → depVec G (k +n l)
extendDVR k l dv = subst (λ X → depVec G X) (+-comm l k) (extendDVL l k dv)
extendDVReq : (k l : ℕ) → (dv : depVec G k) → sumHIT (extendDVR k l dv) ≡ sumHIT dv
extendDVReq k l dv = J (λ m p → sumHIT (subst (λ X → depVec G X) p (extendDVL l k dv)) ≡ sumHIT dv)
(sumHIT (subst (λ X → depVec G X) refl (extendDVL l k dv))
≡⟨ cong sumHIT (transportRefl (extendDVL l k dv)) ⟩
sumHIT (extendDVL l k dv)
≡⟨ extendDVLeq l k dv ⟩
sumHIT dv ∎)
(+-comm l k)
-- pointwise add
_pt+DV_ : {n : ℕ} → (dva dvb : depVec G n) → depVec G n
_pt+DV_ {0} ⋆ ⋆ = ⋆
_pt+DV_ {suc n} (a □ dva) (b □ dvb) = Gstr n ._+_ a b □ (dva pt+DV dvb)
sumHIT+ : {n : ℕ} → (dva dvb : depVec G n) → sumHIT (dva pt+DV dvb) ≡ sumHIT dva +⊕HIT sumHIT dvb
sumHIT+ {0} ⋆ ⋆ = sym (+⊕HIT-IdR _)
sumHIT+ {suc n} (a □ dva) (b □ dvb) = cong₂ _+⊕HIT_ (sym (base-add _ _ _)) (sumHIT+ dva dvb)
∙ comm-4 (⊕HIT-AbGr ℕ G Gstr) _ _ _ _
-----------------------------------------------------------------------------
-- Case Traduction
{- WARNING :
The pseudo normal form is not unique.
It is actually not unique enough so that it is not possible to raise one from ⊕HIT.
Hence we actually need to make it a prop to be able to eliminate.
-}
untruncatedPNF : (x : ⊕HIT ℕ G Gstr) → Type ℓ
untruncatedPNF x = Σ[ m ∈ ℕ ] Σ[ dv ∈ depVec G m ] x ≡ sumHIT dv
PNF : (x : ⊕HIT ℕ G Gstr) → Type ℓ
PNF x = ∥ untruncatedPNF x ∥₁
untruncatedPNF2 : (x y : ⊕HIT ℕ G Gstr) → Type ℓ
untruncatedPNF2 x y = Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b)
PNF2 : (x y : ⊕HIT ℕ G Gstr) → Type ℓ
PNF2 x y = ∥ untruncatedPNF2 x y ∥₁
-----------------------------------------------------------------------------
-- Translation
⊕HIT→PNF : (x : ⊕HIT ℕ G Gstr) → ∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] x ≡ sumHIT a ∥₁
⊕HIT→PNF = DS-Ind-Prop.f _ _ _ _
(λ _ → squash₁)
∣ (0 , (⋆ , refl)) ∣₁
base→PNF
add→PNF
where
base→PNF : (n : ℕ) → (a : G n) → PNF (base n a)
base→PNF n a = ∣ (suc n) , ((a □ replicate0g n) , sym (cong (λ X → base n a +⊕HIT X) (sumHIT0g n)
∙ +⊕HIT-IdR _)) ∣₁
add→PNF : {U V : ⊕HIT ℕ G Gstr} → (ind-U : PNF U) → (ind-V : PNF V) → PNF (U +⊕HIT V)
add→PNF {U} {V} = elim2 (λ _ _ → squash₁)
(λ { (k , dva , p) →
λ { (l , dvb , q) →
∣ ((k +n l)
, (((extendDVR k l dva) pt+DV (extendDVL k l dvb))
, cong₂ _+⊕HIT_ p q
∙ cong₂ _+⊕HIT_ (sym (extendDVReq k l dva)) (sym (extendDVLeq k l dvb))
∙ sym (sumHIT+ (extendDVR k l dva) (extendDVL k l dvb)) )) ∣₁}})
⊕HIT→PNF2 : (x y : ⊕HIT ℕ G Gstr) → ∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b) ∥₁
⊕HIT→PNF2 x y = helper (⊕HIT→PNF x) (⊕HIT→PNF y)
where
helper : PNF x → PNF y →
∥ Σ[ m ∈ ℕ ] Σ[ a ∈ depVec G m ] Σ[ b ∈ depVec G m ] (x ≡ sumHIT a) × (y ≡ sumHIT b) ∥₁
helper = elim2 (λ _ _ → squash₁)
(λ { (k , dva , p) →
λ { (l , dvb , q) →
∣ ((k +n l)
, ((extendDVR k l dva)
, (extendDVL k l dvb
, p ∙ sym (extendDVReq k l dva)
, q ∙ sym (extendDVLeq k l dvb)))) ∣₁}})
-----------------------------------------------------------------------------
-- Some idea
{-
This file should be generalizable to a general decidable index by adding a second vector
-}
{-
It maybe possible to give a normal for without need the prop truncation.
The issue with the current one is that we rely on a underline data type depVec
which forces us to give an explict length. That's what forces the ∥_∥₁.
Hence by getting rid of it and be rewrittinf the term it might be possible
to get a normal form without the PT.
Indeed this basically about pemuting and summing them by G n
∑ base (σ i) a (σ i) -> ∑[i ∈ ℕ] ∑[j ∈ I] base i (b i j) -> ∑ base i (c i)
where a b c are informal "sequences"
Then prove that if we extract the integer, we get an inceasing list
with no coefficient being present twice.
-}
|
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{-# OPTIONS --without-K #-}
{- This module defines our notion of connectedness not requiring HITs.
We define the generic n-connected version of a type, show that it is
n-connected, and that it behaves like the original type above level n.
We conclude by shoing this notion coincides with the usual notion.
of connectedness in the presence of HITs. -}
module Universe.Trunc.Connection where
open import lib.Basics
open import lib.NType2
open import lib.Equivalences2
open import lib.types.Unit
open import lib.types.Nat hiding (_+_)
open import lib.types.Pi
open import lib.types.Sigma
open import lib.types.Paths
open import lib.types.TLevel
open import Universe.Utility.General
open import Universe.Utility.Pointed
open import Universe.Utility.TruncUniverse
open import Universe.Hierarchy
open import Universe.Trunc.Universal
open import Universe.Trunc.Basics
open import Universe.Trunc.TypeConstructors
open trunc-ty
open trunc-props
-- *** Definition 6.13 ***
{- Our plain MLTT connectedness predicate.
Because of predicativity issues, it has to live in one universe higher
than its argument type C. Internally, we really want to quantify over
truncation operators not only at the level of C, but also at levels below C.
This would be the same in a theory with cumulative universes, but we are
working in Agda. Since we are only going to actually use the truncation
operator at one level lower than C, we restrict ourselves to this specific
assumption in this particular development for no particular reason other
than brevity of presentation. The reason for this difference of one level
will become apparent in the definition of 'connection'. -}
module _ {i} (n : ℕ₋₂) (C : Type (lsucc i)) where
is-connected⋆ : Type (lsucc (lsucc i))
is-connected⋆ = (Tr : (X : Type i) → trunc-ty n X (lsucc i))
(TrC : trunc-ty n C (lsucc i))
→ is-contr ⟦ type TrC ⟧
{- The (n+1)-connected version, or n-connection of some pointed type A.
Since we internally quantify over the type of truncations of a path space
from a before being able to require an element in the truncation,
predicativity issues force 'connection' to live in a universe one level
higher than A.
Take particular care in noting that the parameter n is in fact one-off,
for example, The type 'connection (-2)' denotes the (-1)-connection. -}
module _ {i} (A : Type• i) where
module con (n : ℕ₋₂) where
-- *** Definition 6.11 ***
connection• : Type• (lsucc i)
connection• =
Σ• (A
, ((λ b → (TrP : trunc-ty n (pt A == b) i) → ⟦ type TrP ⟧)
, (λ TrP → cons TrP idp)))
{- The base type of the (n+1)-connection:
Σ A (λ b → (TrP : trunc-ty n (a == b) i) → ⟦ type TrP ⟧. -}
connection : Type (lsucc i)
connection = base connection•
-- *** Lemma 6.14 ***
connection-is-connected : is-connected⋆ (S n) connection
connection-is-connected Tr TrC = equiv-preserves-level (e ⁻¹) h where
-- The supplied generic truncation operator is used only to truncate A.
TrA : trunc-ty (S n) (base A) (lsucc i)
TrA = Tr (base A)
TrP : (b : base A) → trunc-ty n (pt A == b) i
TrP = trunc-path.trunc {j = i} TrA (pt A)
{- This definition typechecks since
Σ A (λ b → ⟦ type (TrP b) ⟧
≡ Σ A (λ a → cons TrA (pt A) == cons TrA b)
by construction of trunc-path.trunc. -}
TrD : trunc-ty (S n) (Σ (base A) (λ b → ⟦ type (TrP b) ⟧)) (lsucc i)
TrD = trunc-Σ.trunc {j = lsucc i} TrA
(λ b → trunc-self.trunc (Path-≤ (type TrA) (cons TrA (pt A)) b))
u : connection ≃ Σ (base A) (λ b → ⟦ type (TrP b) ⟧)
u = equiv-Σ-snd {B = λ _ → Π _ _} -- No idea why Agda wants this.
(λ b → Π₁-contr (trunc-inhab-contr {j = i} (TrP b)))
e : ⟦ type TrC ⟧ ≃ ⟦ type TrD ⟧
e = trunc-functor.fmap-equiv {j = lsucc i} TrC TrD u
{- Note that
⟦ type TrD ⟧ ≡ Σ ⟦ type TrA ⟧ (λ tb → cons TrA (pt A) == tb)
by construction of trunc-Σ.trunc. -}
h : is-contr ⟦ type TrD ⟧
h = pathfrom-is-contr (cons TrA (pt A))
module con2 (n : ℕ) where
open con (n -2)
-- *** Lemma 6.12 ***
-- The (n+1)-connection of A coincides with A on dimension n+2 and above.
connection-higher-dim : (Ω ^ n) connection• ≃• (Ω ^ n) A
connection-higher-dim =
forget-Ω^-Σ•₂ _ n (λ _ → Π-level (λ Tr → snd (type Tr)))
open con public
open con2 public
-- For the first time in the dependency chain, we assume HITs.
module with-hits where
open import lib.types.Truncation
open import lib.NConnected
{- With the truncations of the HoTT community's library,
our truncation types are always inhabited, and hence contractible. -}
module _ {i j} where
trunc : (n : ℕ₋₂) (A : Type i) → trunc-ty n A (i ⊔ j)
trunc n A = record
{ type = (Trunc n A , Trunc-level)
; cons = [_]
; univ = λ U → is-eq
(λ f → f ∘ [_]) (Trunc-rec (snd U)) (λ f → idp)
(λ f → λ= (Trunc-elim (λ _ → =-preserves-level _ (snd U))
(λ a → idp)))}
trunc-contr : {n : ℕ₋₂} {A : Type i} → is-contr (trunc-ty n A (i ⊔ j))
trunc-contr = trunc-inhab-contr {j = j} (trunc _ _)
-- *** Lemma 6.15 ***
-- Our connectedness⋆ is equivalent to HIT connectedness.
module _ {i} {n : ℕ₋₂} {A : Type (lsucc i)} where
conn⋆-conn : is-connected⋆ n A ≃ is-connected n A
conn⋆-conn = Π₁-contr (trunc-contr {j = lsucc i})
∘e Π₁-contr (Π-level (λ _ → trunc-contr {j = lsucc i}))
-- *** Theorem 7.1 ***
module _ (n : ℕ) where
M• : Type• 「 n + 2 」
M• = connection• (_ , Loop n) (n -1)
assertion-0 : has-level ⟨ n + 1 ⟩ (base M•)
assertion-0 = snd (Σ-≤ (⟨ n ⟩ -Type-≤ 「 n 」)
(λ b → Π-≤ (trunc-ty _ (_ == b) _)
(λ tr → raise (raise (type tr)))))
assertion-1 : ¬ (has-level ⟨ n ⟩ (base M•))
assertion-1 =
main' n
∘ –> (equiv-is-contr• (connection-higher-dim _ (n + 1)))
∘ (λ z → z (pt M•))
∘ –> has-level-equiv-contr-loops
assertion-2 : is-connected⋆ ⟨ n ⟩ (base M•)
assertion-2 = connection-is-connected _ _
|
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{-# OPTIONS --rewriting #-}
open import Agda.Builtin.Equality
open import Agda.Builtin.Bool
open import Agda.Builtin.Nat
{-# BUILTIN REWRITE _≡_ #-}
not : Bool → Bool
not true = false
not false = true
postulate rew : Nat ≡ Bool
{-# REWRITE rew #-}
0' : Bool
0' = 0
test : not 0' ≡ true
test = refl
|
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module Data.SimpleMap where
open import Class.Equality
open import Class.Map
open import Data.Bool
open import Data.Maybe
open import Data.List hiding (lookup)
open import Data.Product
open import Relation.Nullary
open import Relation.Nullary.Negation
SimpleMap : Set -> Set -> Set
SimpleMap A B = List (A × B)
private
simpleRemove : ∀ {A B} {{_ : EqB A}} -> A -> SimpleMap A B -> SimpleMap A B
simpleRemove k m = boolFilter (λ {(k' , _) → not (k ≣ k')}) m
simpleInsert : ∀ {A B} {{_ : EqB A}} -> A -> B -> SimpleMap A B -> SimpleMap A B
simpleInsert k v m = (k , v) ∷ (simpleRemove k m)
simpleLookup : ∀ {A B} {{_ : EqB A}} -> A -> SimpleMap A B -> Maybe B
simpleLookup k [] = nothing
simpleLookup k ((fst , snd) ∷ m) with k ≣ fst
simpleLookup k ((fst , snd) ∷ m) | true = just snd
simpleLookup k ((fst , snd) ∷ m) | false = simpleLookup k m
simpleMapSnd : ∀ {A B C} -> (B -> C) -> SimpleMap A B -> SimpleMap A C
simpleMapSnd f [] = []
simpleMapSnd f ((fst , snd) ∷ m) = (fst , f snd) ∷ (simpleMapSnd f m)
instance
MapClass-Simple : {K : Set} {{_ : EqB K}} -> MapClass K (SimpleMap K)
MapClass-Simple = record
{ insert = simpleInsert
; remove = simpleRemove
; lookup = simpleLookup
; mapSnd = simpleMapSnd
; emptyMap = [] }
|
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{-# OPTIONS --without-K --safe #-}
-- Bundled version of a Cocartesian Category
module Categories.Category.Cocartesian.Bundle where
open import Level
open import Categories.Category.Core using (Category)
open import Categories.Category.Cocartesian using (Cocartesian)
record CocartesianCategory o ℓ e : Set (suc (o ⊔ ℓ ⊔ e)) where
field
U : Category o ℓ e -- U for underlying
cocartesian : Cocartesian U
open Category U public
open Cocartesian cocartesian public
|
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{- Byzantine Fault Tolerant Consensus Verification in Agda, version 0.9.
Copyright (c) 2020 Oracle and/or its affiliates.
Licensed under the Universal Permissive License v 1.0 as shown at https://opensource.oracle.com/licenses/upl
-}
{-# OPTIONS --allow-unsolved-metas #-}
open import LibraBFT.Prelude
open import LibraBFT.Lemmas
open import LibraBFT.Abstract.Types
-- For each desired property (VotesOnce and LockedRoundRule), we have a
-- module containing a Type that defines a property that an implementation
-- should prove, and a proof that it implies the corresponding rule used by
-- the abstract proofs. Then, we use those proofs to instantiate thmS5,
-- and the use thmS5 to prove a number of correctness conditions.
--
-- TODO-1: refactor this file to separate the definitions and proofs of
-- VotesOnce and LockedRoundRule from their use in proving the correctness
-- properties.
module LibraBFT.Abstract.Properties
(𝓔 : EpochConfig)
(UID : Set)
(_≟UID_ : (u₀ u₁ : UID) → Dec (u₀ ≡ u₁))
(𝓥 : VoteEvidence 𝓔 UID)
where
open import LibraBFT.Abstract.Records 𝓔 UID _≟UID_ 𝓥
open import LibraBFT.Abstract.Records.Extends 𝓔 UID _≟UID_ 𝓥
open import LibraBFT.Abstract.RecordChain 𝓔 UID _≟UID_ 𝓥
import LibraBFT.Abstract.RecordChain.Assumptions 𝓔 UID _≟UID_ 𝓥
as StaticAssumptions
open import LibraBFT.Abstract.System 𝓔 UID _≟UID_ 𝓥
open EpochConfig 𝓔
open import LibraBFT.Abstract.Obligations.VotesOnce 𝓔 UID _≟UID_ 𝓥 as VO
open import LibraBFT.Abstract.Obligations.LockedRound 𝓔 UID _≟UID_ 𝓥 as LR
--------------------------------------------------------------------------------------------
-- * A /ValidSysState/ is one in which both peer obligations are obeyed by honest peers * --
--------------------------------------------------------------------------------------------
record ValidSysState {ℓ}(𝓢 : AbsSystemState ℓ) : Set (ℓ+1 ℓ0 ℓ⊔ ℓ) where
field
vss-votes-once : VO.Type 𝓢
vss-locked-round : LR.Type 𝓢
open ValidSysState public
-- And a valid system state offers the desired /CommitsDoNotConflict/ property
-- and variants.
module _ {ℓ}(𝓢 : AbsSystemState ℓ) (st-valid : ValidSysState 𝓢) where
open AbsSystemState 𝓢
open All-InSys-props InSys
import LibraBFT.Abstract.RecordChain.Properties 𝓔 UID _≟UID_ 𝓥 as Props
CommitsDoNotConflict : ∀{q q'}
→ {rc : RecordChain (Q q)} → All-InSys rc
→ {rc' : RecordChain (Q q')} → All-InSys rc'
→ {b b' : Block}
→ CommitRule rc b
→ CommitRule rc' b'
→ NonInjective-≡ bId ⊎ ((B b) ∈RC rc' ⊎ (B b') ∈RC rc)
CommitsDoNotConflict = Props.WithInvariants.thmS5 InSys
(VO.proof 𝓢 (vss-votes-once st-valid))
(LR.proof 𝓢 (vss-locked-round st-valid))
-- When we are dealing with a /Complete/ AbsSystem, we can go a few steps
-- further and prove that commits do not conflict even if we have only partial
-- knowledge about Records represented in the system.
module _ (∈QC⇒AllSent : Complete 𝓢) where
-- For a /complete/ system we can go even further; if we have evidence that
-- only the tip of the record chains is in the system, we can infer
-- the rest of it is also in the system (or blockIDs are not injective).
CommitsDoNotConflict'
: ∀{q q'}{rc : RecordChain (Q q)}{rc' : RecordChain (Q q')}{b b' : Block}
→ InSys (Q q) → InSys (Q q')
→ CommitRule rc b
→ CommitRule rc' b'
→ NonInjective-≡ bId ⊎ ((B b) ∈RC rc' ⊎ (B b') ∈RC rc)
CommitsDoNotConflict' {q} {q'} {step {r = B bb} rc b←q} {step {r = B bb'} rc' b←q'} {b} {b'} q∈sys q'∈sys cr cr'
with bft-assumption (qVotes-C2 q) (qVotes-C2 q')
...| α , α∈qmem , α∈q'mem , hα
with Any-sym (Any-map⁻ α∈qmem) | Any-sym (Any-map⁻ α∈q'mem)
...| α∈q | α∈q'
with ∈QC⇒AllSent {q = q} hα α∈q q∈sys | ∈QC⇒AllSent {q = q'} hα α∈q' q'∈sys
...| ab , ab←q , arc , ais | ab' , ab←q' , arc' , ais'
with RecordChain-irrelevant (step arc ab←q) (step rc b←q) |
RecordChain-irrelevant (step arc' ab←q') (step rc' b←q')
...| inj₁ hb | _ = inj₁ hb
...| inj₂ _ | inj₁ hb = inj₁ hb
...| inj₂ arc≈rc | inj₂ arc'≈rc'
with CommitsDoNotConflict
(All-InSys-step ais ab←q q∈sys )
(All-InSys-step ais' ab←q' q'∈sys)
(transp-CR (≈RC-sym arc≈rc ) cr )
(transp-CR (≈RC-sym arc'≈rc') cr')
...| inj₁ hb = inj₁ hb
...| inj₂ (inj₁ b∈arc') = inj₂ (inj₁ (transp-B∈RC arc'≈rc' b∈arc'))
...| inj₂ (inj₂ b'∈arc) = inj₂ (inj₂ (transp-B∈RC arc≈rc b'∈arc))
-- The final property is even stronger; it states that even if an observer
-- has access only to suffixes of record chains that match the commit rule,
-- we can still guarantee that b and b' are non-conflicting blocks. This
-- will be important for showing that observers can have confidence in commit
-- messages without participating in the protocol and without having access to
-- all previously sent records.
CommitsDoNotConflict''
: ∀{o o' q q'}
→ {rcf : RecordChainFrom o (Q q)}
→ {rcf' : RecordChainFrom o' (Q q')}
→ {b b' : Block}
→ InSys (Q q)
→ InSys (Q q')
→ CommitRuleFrom rcf b
→ CommitRuleFrom rcf' b'
→ NonInjective-≡ bId ⊎ Σ (RecordChain (Q q')) ((B b) ∈RC_)
⊎ Σ (RecordChain (Q q)) ((B b') ∈RC_)
CommitsDoNotConflict'' {cb} {q = q} {q'} {rcf} {rcf'} q∈sys q'∈sys crf crf'
with bft-assumption (qVotes-C2 q) (qVotes-C2 q')
...| α , α∈qmem , α∈q'mem , hα
with Any-sym (Any-map⁻ α∈qmem) | Any-sym (Any-map⁻ α∈q'mem)
...| α∈q | α∈q'
with ∈QC⇒AllSent {q = q} hα α∈q q∈sys | ∈QC⇒AllSent {q = q'} hα α∈q' q'∈sys
...| ab , ab←q , arc , ais | ab' , ab←q' , arc' , ais'
with step arc ab←q | step arc' ab←q'
...| rcq | rcq'
with crf⇒cr rcf rcq crf | crf⇒cr rcf' rcq' crf'
...| inj₁ hb | _ = inj₁ hb
...| inj₂ _ | inj₁ hb = inj₁ hb
...| inj₂ cr | inj₂ cr'
with CommitsDoNotConflict' q∈sys q'∈sys cr cr'
...| inj₁ hb = inj₁ hb
...| inj₂ (inj₁ b∈arc') = inj₂ (inj₁ (rcq' , b∈arc'))
...| inj₂ (inj₂ b'∈arc) = inj₂ (inj₂ (rcq , b'∈arc))
|
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interleaved mutual
data Foo : Set → Set
data Foo_Bar : Set
constructor
foobar : Foo Bar
|
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{-# OPTIONS --rewriting #-}
module MessageClosureProperties where
open import Data.Nat using (ℕ; zero ; suc)
open import Data.Fin using (Fin; zero; suc)
open import Function using (_∘_)
open import Relation.Binary.PropositionalEquality using (_≡_; cong; cong₂; sym; refl)
open import Auxiliary.Extensionality
open import Auxiliary.RewriteLemmas
import Types.COI as COI
import Types.IND1 as IND
import Types.Tail1 as Tail
import DualTail1 as DT
import MessageClosure as MC
open COI using (_≈_; _≈'_; _≈ᵗ_)
open DT using (Stack; ε; ⟪_,_⟫)
private
variable
n : ℕ
σ σ′ : Stack n
G : IND.GType n
----------------------------------------------------------------------
var=shift-var : (i : Fin (suc n)) → IND.var i ≡ MC.shift{m = n}{n = 0} IND.var i
var=shift-var zero = refl
var=shift-var (suc i) = refl
apply-id-S : (S : IND.SType n) → MC.applyS{n = 0} IND.var S ≡ S
apply-id-G : (G : IND.GType n) → MC.applyG{n = 0} IND.var G ≡ G
apply-id-T : (T : IND.TType n) → MC.applyT{n = 0} IND.var T ≡ T
apply-id-S (IND.gdd G) = cong IND.gdd (apply-id-G G)
apply-id-S{n} (IND.rec G) rewrite sym (ext (var=shift-var{n})) = cong IND.rec (apply-id-G G)
apply-id-S (IND.var x) = refl
apply-id-G (IND.transmit d T S) = cong₂ (IND.transmit d) (apply-id-T T) (apply-id-S S)
apply-id-G (IND.choice d m alt) = cong (IND.choice d m) (ext (apply-id-S ∘ alt))
apply-id-G IND.end = refl
apply-id-T IND.TUnit = refl
apply-id-T IND.TInt = refl
apply-id-T (IND.TPair T T₁) = cong₂ IND.TPair (apply-id-T T) (apply-id-T T₁)
apply-id-T (IND.TChan S) = cong IND.TChan (apply-id-S S)
mc-equiv-S : (s : IND.SType 0)
→ DT.ind2coiS ε s ≈ DT.tail2coiS ε (MC.mclosureS s)
mc-equiv-G : (g : IND.GType 0)
→ DT.ind2coiG ε g ≈' DT.tail2coiG ε (MC.mclosureG g)
mc-equiv-T : (t : IND.TType 0)
→ DT.ind2coiT ε t ≈ᵗ DT.tail2coiT (MC.injectT (MC.applyT IND.var t))
COI.Equiv.force (mc-equiv-S (IND.gdd g)) = mc-equiv-G g
COI.Equiv.force (mc-equiv-S (IND.rec G)) = {!!}
-- mc-equiv-G (IND.st-substG G zero (IND.rec G))
mc-equiv-G (IND.transmit d t s) =
COI.eq-transmit d (mc-equiv-T t) (mc-equiv-S s)
mc-equiv-G (IND.choice d m alt) =
COI.eq-choice d (mc-equiv-S ∘ alt)
mc-equiv-G IND.end =
COI.eq-end
mc-equiv-T IND.TUnit = COI.eq-unit
mc-equiv-T IND.TInt = COI.eq-int
mc-equiv-T (IND.TPair t t₁) = COI.eq-pair (mc-equiv-T t) (mc-equiv-T t₁)
mc-equiv-T (IND.TChan S) rewrite apply-id-S S = COI.eq-chan COI.≈-refl
-- relation between two stacks (to fill above hole in mc-equiv-S)
data Related : DT.Stack {IND.GType} n → Stack {Tail.GType} n → Set where
base : Related {0} ε ε
step : Related {n} σ σ′
→ Related {suc n} ⟪ σ , G ⟫ ⟪ σ′ , MC.mcloG (MC.ext {!!} (IND.rec G)) G ⟫
|
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module Logic.Structure.Monoid where
import Logic.Equivalence
import Logic.Operations as Operations
open Logic.Equivalence using (Equivalence; module Equivalence)
open Operations.Param
data Monoid (A : Set)(Eq : Equivalence A) : Set where
monoid :
(z : A)
(_+_ : A -> A -> A)
(leftId : LeftIdentity Eq z _+_)
(rightId : RightIdentity Eq z _+_)
(assoc : Associative Eq _+_) ->
Monoid A Eq
-- There should be a simpler way of doing this. Local definitions to data declarations?
module Projections where
zero : {A : Set}{Eq : Equivalence A} -> Monoid A Eq -> A
zero (monoid z _ _ _ _) = z
plus : {A : Set}{Eq : Equivalence A} -> Monoid A Eq -> A -> A -> A
plus (monoid _ p _ _ _) = p
leftId : {A : Set}{Eq : Equivalence A}(Mon : Monoid A Eq) -> LeftIdentity Eq (zero Mon) (plus Mon)
leftId (monoid _ _ li _ _) = li
rightId : {A : Set}{Eq : Equivalence A}(Mon : Monoid A Eq) -> RightIdentity Eq (zero Mon) (plus Mon)
rightId (monoid _ _ _ ri _) = ri
assoc : {A : Set}{Eq : Equivalence A}(Mon : Monoid A Eq) -> Associative Eq (plus Mon)
assoc (monoid _ _ _ _ a) = a
module Monoid {A : Set}{Eq : Equivalence A}(Mon : Monoid A Eq) where
zero = Projections.zero Mon
_+_ = Projections.plus Mon
leftId = Projections.leftId Mon
rightId = Projections.rightId Mon
assoc = Projections.assoc Mon
|
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open import Relation.Binary.Core
module InsertSort.Impl2.Correctness.Permutation.Alternative {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_) where
open import Bound.Lower A
open import Bound.Lower.Order _≤_
open import Data.List
open import Data.Sum
open import Function
open import InsertSort.Impl2 _≤_ tot≤
open import List.Permutation.Alternative A renaming (_∼_ to _∼′_)
open import List.Permutation.Alternative.Correctness A
open import List.Permutation.Base A
open import OList _≤_
lemma-insert∼′ : {b : Bound}{x : A}(b≤x : LeB b (val x))(xs : OList b) → (x ∷ forget xs) ∼′ forget (insert b≤x xs)
lemma-insert∼′ b≤x onil = ∼refl
lemma-insert∼′ {x = x} b≤x (:< {x = y} b≤y ys)
with tot≤ x y
... | inj₁ x≤y = ∼refl
... | inj₂ y≤x = ∼trans (∼swap ∼refl) (∼head y (lemma-insert∼′ (lexy y≤x) ys))
lemma-insertSort∼′ : (xs : List A) → xs ∼′ forget (insertSort xs)
lemma-insertSort∼′ [] = ∼refl
lemma-insertSort∼′ (x ∷ xs) = ∼trans (∼head x (lemma-insertSort∼′ xs)) (lemma-insert∼′ lebx (insertSort xs))
theorem-insertSort∼ : (xs : List A) → xs ∼ forget (insertSort xs)
theorem-insertSort∼ = lemma-∼′-∼ ∘ lemma-insertSort∼′
|
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|
-- Currently open declarations are not allowed in mutual blocks.
-- This might change.
module OpenInMutual where
module A where
mutual
open A
T : Set -> Set
T A = A
|
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{-# OPTIONS --allow-unsolved-metas #-}
module TemporalOps.Linear where
open import CategoryTheory.Instances.Reactive
open import CategoryTheory.Functor
open import CategoryTheory.CartesianStrength
open import CategoryTheory.NatTrans
open import CategoryTheory.Monad
open import CategoryTheory.Linear
open import TemporalOps.Next
open import TemporalOps.Delay
open import TemporalOps.Diamond
open import TemporalOps.OtherOps
open import TemporalOps.Common.Other
open import TemporalOps.Common.Compare
open import TemporalOps.Common.Rewriting
open import Data.Product
open import Data.Product.Properties
open import Data.Sum
open import Data.Nat hiding (_*_)
open import Data.Nat.Properties
using (+-identityʳ ; +-comm ; +-suc ; +-assoc)
open import Relation.Binary.PropositionalEquality hiding (inspect)
open import Holes.Term using (⌞_⌟)
open import Holes.Cong.Propositional
open ≡-Reasoning
open L ℝeactive-BCCC M-◇ public
ℝeactive-linear : Linear ℝeactive-BCCC M-◇
ℝeactive-linear = record
{ linprod = λ A B →
record { ⟪_,_⟫ = prod-◇
; *π₁-comm = λ {C} {l₁} {l₂} {n} {a} →
*π₁-comm-◇ {A}{B}{C} {l₁} {l₂} {n} {a}
; *π₂-comm = {! !}
; ⊛-unique = {! !}
}
}
where
open Functor F-▹
open Functor F-◇ renaming (fmap to ◇-f)
private module ▹ᵏ-F k = Functor (F-delay k)
private module ▹ᵏ-◇ k = _⟹_ (▹ᵏ-to-◇ k)
private module ▹ᵏ-C k = CartesianFunctor (F-cart-delay k)
open Monad M-◇
open CartesianFunctor F-cart-▹ renaming (m to m-▹)
pair-delay-◇₁ : ∀{A B : τ} -> (k l : ℕ) -> (delay A by suc (k + l) ⊗ delay B by k)
⇴ delay (◇ A ⊗ B) by k
pair-delay-◇₁ zero l n (dA , dB) = (suc l , dA) , dB
pair-delay-◇₁ {A}{B} (suc k) l n p = fmap (pair-delay-◇₁ k l) n
(m-▹ (delay A by suc (k + l)) (delay B by k) n p)
pair-delay-◇₂ : ∀{A B : τ} -> (k l : ℕ) -> (delay A by k ⊗ delay B by suc (k + l))
⇴ delay (A ⊗ ◇ B) by k
pair-delay-◇₂ zero l n (dA , dB) = dA , (suc l , dB)
pair-delay-◇₂ {A}{B} (suc k) l n p = fmap (pair-delay-◇₂ k l) n
(m-▹ (delay A by k) (delay B by suc (k + l)) n p)
prod-◇-compare : ∀{A B : τ} -> (k₁ k₂ n : ℕ)
-> (a₁ : delay A by k₁ at n)(a₂ : delay B by k₂ at n)
-> Ordering k₁ k₂ -> ◇ (A ⊛ B) at n
prod-◇-compare {A} {B} k₁ .(suc (k₁ + l)) n a₁ a₂ (less .k₁ l) =
k₁ , ▹ᵏ-F.fmap k₁ ι₁ n (▹ᵏ-F.fmap k₁ ι₁ n (pair-delay-◇₂ k₁ l n (a₁ , a₂)))
prod-◇-compare {A} {B} .(suc (k₂ + l)) k₂ n a₁ a₂ (greater .k₂ l) =
k₂ , ▹ᵏ-F.fmap k₂ ι₁ n (▹ᵏ-F.fmap k₂ ι₂ n (pair-delay-◇₁ k₂ l n (a₁ , a₂)))
prod-◇-compare {A} {B} k₁ .k₁ n a₁ a₂ (equal .k₁) =
k₁ , ▹ᵏ-F.fmap k₁ ι₂ n (▹ᵏ-C.m k₁ A B n (a₁ , a₂))
◇-select : ∀{A B : τ} -> (◇ A ⊗ ◇ B) ⇴ ◇ (A ⊛ B)
◇-select {A}{B} n ((k₁ , a₁) , (k₂ , a₂)) =
prod-◇-compare {A}{B} k₁ k₂ n a₁ a₂ (compare k₁ k₂)
prod-◇ : ∀{A B L : τ} -> (L ⇴ ◇ A) -> (L ⇴ ◇ B) -> (L ⇴ ◇ (A ⊛ B))
prod-◇ fa fb n lp = ◇-select n (fa n lp , fb n lp)
*π₁-comm-◇ : ∀{A B L} -> {l₁ : L ⇴ ◇ A} {l₂ : L ⇴ ◇ B}
-> (μ.at A ∘ ◇-f *π₁) ∘ (prod-◇ l₁ l₂) ≈ l₁
*π₁-comm-◇ {A}{B}{L}{l₁}{l₂} {n} {a} with inspect (l₁ n a , l₂ n a)
*π₁-comm-◇ | ((k₁ , a₁) , (k₂ , a₂)) with≡ pf with inspect (compare k₁ k₂)
*π₁-comm-◇ {A}{B}{L}{l₁}{l₂}{n}{a} | ((k₁ , a₁) , .(suc (k₁ + l)) , a₂) with≡ pf
| less .k₁ l with≡ cpf =
begin
(μ.at A n (◇-f *π₁ n (◇-select n ⌞ (l₁ n a , l₂ n a) ⌟)))
≡⟨ cong! pf ⟩
μ.at A n (◇-f *π₁ n
(prod-◇-compare k₁ (suc (k₁ + l)) n a₁ a₂ (⌞ compare k₁ (suc (k₁ + l)) ⌟)))
≡⟨ cong! cpf ⟩
μ.at A n (◇-f *π₁ n
(prod-◇-compare k₁ (suc (k₁ + l)) n a₁ a₂ (less k₁ l)))
≡⟨⟩
μ.at A n (k₁ ,
▹ᵏ-F.fmap k₁ *π₁ n ⌞ (▹ᵏ-F.fmap k₁ ι₁ n (▹ᵏ-F.fmap k₁ ι₁ n
(pair-delay-◇₂ k₁ l n (a₁ , a₂)))) ⌟)
≡⟨ cong! (▹ᵏ-F.fmap-∘ k₁ {g = ι₁ {A ⊗ ◇ B ⊕ ◇ A ⊗ B} {A ⊗ B}}
{ι₁ {A ⊗ ◇ B} {◇ A ⊗ B}}{n} {pair-delay-◇₂ k₁ l n (a₁ , a₂)}) ⟩
μ.at A n (k₁ ,
⌞ ▹ᵏ-F.fmap k₁ *π₁ n (▹ᵏ-F.fmap k₁ (ι₁ ∘ ι₁) n
(pair-delay-◇₂ k₁ l n (a₁ , a₂))) ⌟)
≡⟨ cong! (▹ᵏ-F.fmap-∘ k₁ {g = *π₁} {ι₁ ∘ ι₁} {n} {pair-delay-◇₂ k₁ l n (a₁ , a₂)}) ⟩
μ.at A n (k₁ ,
▹ᵏ-F.fmap k₁ ([ η.at A ∘ π₁ ⁏ π₁ {B = L} ⁏ η.at A ∘ π₁ {B = L} ] ∘ ι₁ ∘ ι₁) n
(pair-delay-◇₂ k₁ l n (a₁ , a₂)))
≡⟨⟩
μ.at A n (k₁ ,
⌞ ▹ᵏ-F.fmap k₁ (η.at A ∘ π₁) n
(pair-delay-◇₂ k₁ l n (a₁ , a₂)) ⌟)
≡⟨ cong! (▹ᵏ-F.fmap-∘ k₁) ⟩
μ.at A n (k₁ , ▹ᵏ-F.fmap k₁ (η.at A) n
⌞ ▹ᵏ-F.fmap k₁ π₁ n (pair-delay-◇₂ k₁ l n (a₁ , a₂)) ⌟)
≡⟨ cong! (lemma k₁ l {n} {a₁ , a₂}) ⟩
μ.at A n (k₁ , ▹ᵏ-F.fmap k₁ (η.at A) n a₁)
≡⟨ η-unit2 ⟩
k₁ , a₁
≡⟨ sym (,-injectiveˡ pf) ⟩
l₁ n a
∎
where
lemma : ∀ (k l : ℕ) -> ▹ᵏ-F.fmap k π₁ ∘ pair-delay-◇₂ k l ≈ π₁
lemma zero l = refl
lemma (suc k) l {zero} = refl
lemma (suc k) l {suc n} = lemma k l {n}
*π₁-comm-◇ {A}{B}{L}{l₁}{l₂}{n}{a} | ((.(suc (k₂ + l)) , a₁) , k₂ , a₂) with≡ pf
| greater .k₂ l with≡ cpf =
begin
(μ.at A n (◇-f *π₁ n (◇-select n ⌞ (l₁ n a , l₂ n a) ⌟)))
≡⟨ cong! pf ⟩
μ.at A n (◇-f *π₁ n
(prod-◇-compare (suc (k₂ + l)) k₂ n a₁ a₂ (⌞ compare (suc (k₂ + l)) k₂ ⌟)))
≡⟨ cong! cpf ⟩
μ.at A n (◇-f *π₁ n
(prod-◇-compare (suc (k₂ + l)) k₂ n a₁ a₂ (greater k₂ l)))
≡⟨⟩
μ.at A n (k₂ ,
▹ᵏ-F.fmap k₂ *π₁ n ⌞ (▹ᵏ-F.fmap k₂ ι₁ n (▹ᵏ-F.fmap k₂ ι₂ n
(pair-delay-◇₁ k₂ l n (a₁ , a₂)))) ⌟)
≡⟨ cong! (▹ᵏ-F.fmap-∘ k₂ {g = ι₁ {A ⊗ ◇ B ⊕ ◇ A ⊗ B} {A ⊗ B}}
{ι₂ {A ⊗ ◇ B} {◇ A ⊗ B}}{n} {pair-delay-◇₁ k₂ l n (a₁ , a₂)}) ⟩
μ.at A n (k₂ ,
⌞ ▹ᵏ-F.fmap k₂ *π₁ n (▹ᵏ-F.fmap k₂ (ι₁ ∘ ι₂) n
(pair-delay-◇₁ k₂ l n (a₁ , a₂))) ⌟)
≡⟨ cong! (▹ᵏ-F.fmap-∘ k₂ {g = *π₁} {ι₁ ∘ ι₂} {n} {pair-delay-◇₁ k₂ l n (a₁ , a₂)}) ⟩
μ.at A n (k₂ ,
▹ᵏ-F.fmap k₂ ([ η.at A ∘ π₁ {B = L} ⁏ π₁ {B = B} ⁏ η.at A ∘ π₁ {B = L} ] ∘ ι₁ ∘ ι₂) n
(pair-delay-◇₁ k₂ l n (a₁ , a₂)))
≡⟨⟩
μ.at A n (k₂ , ⌞ ▹ᵏ-F.fmap k₂ π₁ n (pair-delay-◇₁ k₂ l n (a₁ , a₂)) ⌟)
≡⟨ refl ⟩
μ.at A n (◇-f π₁ n (k₂ , pair-delay-◇₁ k₂ l n (a₁ , a₂)))
≡⟨ {! !} ⟩
-- ≡⟨ cong! (lemma k₂ l {n} {a₁ , a₂}) ⟩
-- μ.at A n (k₂ , ▹ᵏ-F.fmap k₂ (▹ᵏ-◇.at (suc l) A) n (split-▹ᵏ k₂ l n a₁))
suc (k₂ + l) , a₁
≡⟨ sym (,-injectiveˡ pf) ⟩
l₁ n a
∎
where
split-▹ᵏ : ∀ {A} k l -> delay A by suc (k + l) ⇴ delay (delay A by suc l) by k
split-▹ᵏ zero l n a = a
split-▹ᵏ (suc k) l zero a = top.tt
split-▹ᵏ (suc k) l (suc n) a = split-▹ᵏ k l n a
lemma : ∀ {A B} (k l : ℕ)
-> ▹ᵏ-F.fmap k (π₁ {B = B}) ∘ pair-delay-◇₁ {A} k l
≈ ▹ᵏ-F.fmap k (▹ᵏ-◇.at (suc l) A) ∘ split-▹ᵏ k l ∘ π₁
lemma zero l {n} = refl
lemma (suc k) l {zero} = refl
lemma (suc k) l {suc n} = lemma k l
lemma2 : ∀ {A B} k l n (a₁ : delay A by suc (k + l) at n)
(a₂ : delay B by k at n)
-> μ.at A n (◇-f π₁ n (k , pair-delay-◇₁ k l n (a₁ , a₂)))
≡ (suc (k + l) , a₁)
lemma2 zero l n a₁ a₂ = refl
lemma2 (suc k) l zero a₁ a₂ = {! !}
lemma2 (suc k) l (suc n) a₁ a₂ = {! !}
*π₁-comm-◇ {l₁ = l₁} {l₂} {n} {a} | ((k₁ , a₁) , .k₁ , a₂) with≡ pf
| equal .k₁ with≡ cpf = {! !}
open Linear ℝeactive-linear public
-- Handle a linear product with three continuations
handle : ∀ {A B C D : τ}
-> (A ⊗ B ⊗ ◇ C ⇴ ◇ D)
-> (A ⊗ ◇ B ⊗ C ⇴ ◇ D)
-> (A ⊗ B ⊗ C ⇴ ◇ D)
-> A ⊗ (B ⊛ C) ⇴ ◇ D
handle a b c = [ a ∘ assoc-left ⁏ b ∘ assoc-left ⁏ c ∘ assoc-left ] ∘ dist2
|
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------------------------------------------------------------------------
-- Do the parser combinators form a Kleene algebra?
------------------------------------------------------------------------
module TotalParserCombinators.Laws.KleeneAlgebra where
open import Algebra
open import Data.List
open import Data.List.Properties
open import Data.Nat using (ℕ)
open import Data.Product using (_,_; proj₂)
open import Function using (_$_)
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence
using (_⇔_; equivalence; module Equivalence)
import Relation.Binary.PropositionalEquality as P
open import Relation.Nullary
open import TotalParserCombinators.Lib
open import TotalParserCombinators.Parser
open import TotalParserCombinators.Semantics
hiding (_>>=_) renaming (return to return′; _⊛_ to _⊛′_)
------------------------------------------------------------------------
-- A variant of _≲_
infix 4 _≲′_
-- The AdditiveMonoid module shows that _∣_ can be viewed as the join
-- operation of a join-semilattice (if language equivalence is used).
-- This means that the following definition of order is natural.
_≲′_ : ∀ {Tok R xs₁ xs₂} → Parser Tok R xs₁ → Parser Tok R xs₂ → Set₁
p₁ ≲′ p₂ = p₁ ∣ p₂ ≈ p₂
-- This order coincides with _≲_.
≲⇔≲′ : ∀ {Tok R xs₁ xs₂}
(p₁ : Parser Tok R xs₁) (p₂ : Parser Tok R xs₂) →
p₁ ≲ p₂ ⇔ p₁ ≲′ p₂
≲⇔≲′ {xs₁ = xs₁} p₁ p₂ = equivalence
(λ (p₁≲p₂ : p₁ ≲ p₂) {x s} → equivalence (helper p₁≲p₂) (∣-right xs₁))
(λ (p₁≲′p₂ : p₁ ≲′ p₂) s∈p₁ → Equivalence.to p₁≲′p₂ ⟨$⟩ ∣-left s∈p₁)
where
helper : p₁ ≲ p₂ → p₁ ∣ p₂ ≲ p₂
helper p₁≲p₂ (∣-left s∈p₁) = p₁≲p₂ s∈p₁
helper p₁≲p₂ (∣-right .xs₁ s∈p₂) = s∈p₂
------------------------------------------------------------------------
-- A limited notion of *-continuity
-- Least upper bounds.
record _LeastUpperBoundOf_
{Tok R xs} {f : ℕ → List R}
(lub : Parser Tok R xs)
(p : (n : ℕ) → Parser Tok R (f n)) : Set₁ where
field
upper-bound : ∀ n → p n ≲ lub
least : ∀ {ys} {ub : Parser Tok R ys} →
(∀ n → p n ≲ ub) → lub ≲ ub
-- For argument parsers which are not nullable we can prove that the
-- Kleene star operator is *-continuous.
*-continuous :
∀ {Tok R₁ R₂ R₃ fs xs}
(p₁ : Parser Tok (List R₁ → R₂ → R₃) fs)
(p₂ : Parser Tok R₁ [])
(p₃ : Parser Tok R₂ xs) →
(p₁ ⊛ p₂ ⋆ ⊛ p₃) LeastUpperBoundOf (λ n → p₁ ⊛ p₂ ↑ n ⊛ p₃)
*-continuous {Tok} {R₁ = R₁} {R₃ = R₃} {fs} {xs} p₁ p₂ p₃ =
record { upper-bound = upper-bound; least = least }
where
upper-bound : ∀ n → p₁ ⊛ p₂ ↑ n ⊛ p₃ ≲ p₁ ⊛ p₂ ⋆ ⊛ p₃
upper-bound n (∈p₁ ⊛′ ∈p₂ⁿ ⊛′ ∈p₃) =
[ ○ - ○ ] [ ○ - ○ ] ∈p₁ ⊛ Exactly.↑≲⋆ n ∈p₂ⁿ ⊛ ∈p₃
least : ∀ {ys} {p : Parser Tok R₃ ys} →
(∀ i → p₁ ⊛ p₂ ↑ i ⊛ p₃ ≲ p) → p₁ ⊛ p₂ ⋆ ⊛ p₃ ≲ p
least ub (∈p₁ ⊛′ ∈p₂⋆ ⊛′ ∈p₃) with Exactly.⋆≲∃↑ ∈p₂⋆
... | (n , ∈p₂ⁿ) = ub n ([ ○ - ○ ] [ ○ - ○ ] ∈p₁ ⊛ ∈p₂ⁿ ⊛ ∈p₃)
------------------------------------------------------------------------
-- The parser combinators do not form a Kleene algebra
-- If we allow arbitrary argument parsers, then we cannot prove the
-- following (variant of a) Kleene algebra axiom.
not-Kleene-algebra :
∀ {Tok} →
Tok →
(f : ∀ {R xs} → Parser Tok R xs → List (List R)) →
(_⋆′ : ∀ {R xs} (p : Parser Tok R xs) →
Parser Tok (List R) (f p)) →
¬ (∀ {R xs} {p : Parser Tok R xs} →
return [] ∣ (p >>= λ x → (p ⋆′) >>= λ xs → return (x ∷ xs))
≲ (p ⋆′))
not-Kleene-algebra {Tok} t f _⋆′ fold =
KleeneStar.unrestricted-incomplete t f _⋆′ ⋆′-complete
where
⋆′-complete : ∀ {xs ys s} {p : Parser Tok Tok ys} →
xs ∈[ p ]⋆· s → xs ∈ p ⋆′ · s
⋆′-complete [] = fold (∣-left return′)
⋆′-complete (∈p ∷ ∈p⋆) =
fold (∣-right [ [] ]
([ ○ - ○ ] ∈p >>=
fix ([ ○ - ○ ] ⋆′-complete ∈p⋆ >>= return′)))
where
fix = cast∈ P.refl P.refl $ ++-identityʳ _
-- This shows that the parser combinators do not form a Kleene
-- algebra (interpreted liberally) using _⊛_ for composition, return
-- for unit, etc. However, it should be straightforward to build a
-- recogniser library, based on the parser combinators, which does
-- satisfy the Kleene algebra axioms (see
-- TotalRecognisers.LeftRecursion.KleeneAlgebra).
|
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module Issue282 where
module Works where
record R : Set where
constructor c
foo = R.c
module Doesn't_work where
private
record R : Set where
constructor c
foo = R.c
-- Bug.agda:17,9-12
-- Not in scope:
-- R.c at Bug.agda:17,9-12
-- when scope checking R.c
module Doesn't_work_either where
private
data D : Set where
c : D
foo = D.c
|
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module AllStdLib where
-- Ensure that the entire standard library is compiled.
import README
open import Data.Unit.Polymorphic using (⊤)
open import Data.String
open import IO hiding (_>>_)
import IO.Primitive as Prim
import DivMod
import HelloWorld
import HelloWorldPrim
import ShowNat
import TrustMe
import Vec
import dimensions
infixr 1 _>>_
_>>_ : ∀ {A B : Set} → Prim.IO A → Prim.IO B → Prim.IO B
m >> m₁ = m Prim.>>= λ _ → m₁
main : Prim.IO ⊤
main = run (putStrLn "Hello World!") >>
DivMod.main >>
HelloWorld.main >>
HelloWorldPrim.main >>
ShowNat.main >>
TrustMe.main >>
Vec.main >>
dimensions.main
|
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Abelian.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Modules.Definition
module Modules.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+R_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+R_ _*_} {m n : _} {M : Set m} {T : Setoid {m} {n} M} {_+_ : M → M → M} {G' : Group T _+_} {G : AbelianGroup G'} {_·_ : A → M → M} (mod : Module R G _·_) where
open Group G'
open Ring R
open Setoid T
open Equivalence eq
open Module mod
moduleTimesZero : {x : M} → (0R · x) ∼ 0G
moduleTimesZero {x} = equalsDoubleImpliesZero G' (symmetric x=2x)
where
x=2x : (0R · x) ∼ (0R · x) + (0R · x)
x=2x = transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.identLeft additiveGroup)) reflexive) dotDistributesRight
moduleTimes-1 : {x : M} → ((Group.inverse additiveGroup 1R) · x) ∼ inverse x
moduleTimes-1 {x} = transitive (transferToRight' G' j) (inverseWellDefined G' dotIdentity)
where
i : ((1R · x) + ((Group.inverse additiveGroup 1R) · x)) ∼ 0G
i = transitive (symmetric (transitive (dotWellDefined (Equivalence.symmetric (Setoid.eq S) (Group.invRight additiveGroup {1R})) reflexive) dotDistributesRight)) (moduleTimesZero)
j : (((Group.inverse additiveGroup 1R) · x) + (1R · x)) ∼ 0G
j = transitive (AbelianGroup.commutative G) i
|
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module Extensions.Nat where
open import Prelude
open import Data.Nat.Base
open import Data.Nat.Properties.Simple
open import Data.Nat.Properties
m+1+n≡1+m+n : ∀ m n → m + suc n ≡ suc (m + n)
m+1+n≡1+m+n zero n = refl
m+1+n≡1+m+n (suc m) n = cong suc (m+1+n≡1+m+n m n)
<-+ : ∀ {m n m' n'} → m ≤ m' → n ≤ n' → m + n ≤ m' + n'
<-+ {zero} {zero} z≤n z≤n = z≤n
<-+ {suc m} (s≤s m<m') x = s≤s (<-+ m<m' x)
<-+ {zero} {suc n} {zero} z≤n (s≤s n<n') = s≤s n<n'
<-+ {zero} {suc n} {suc m'} z≤n (s≤s n<n') = s≤s (<-+ {m' = m'} z≤n (≤-step n<n'))
<-unique : ∀ {i u} (p q : i < u) → p ≡ q
<-unique (s≤s z≤n) (s≤s z≤n) = refl
<-unique (s≤s (s≤s p)) (s≤s (s≤s q)) = sym (cong s≤s (<-unique (s≤s q) (s≤s p)))
|
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------------------------------------------------------------------------
-- The Agda standard library
--
-- Consequences of a monomorphism between binary relations
------------------------------------------------------------------------
-- See Data.Nat.Binary.Properties for examples of how this and similar
-- modules can be used to easily translate properties between types.
{-# OPTIONS --without-K --safe #-}
open import Function
open import Relation.Binary
open import Relation.Binary.Morphism
module Relation.Binary.Morphism.RelMonomorphism
{a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
{_∼₁_ : Rel A ℓ₁} {_∼₂_ : Rel B ℓ₂}
{⟦_⟧ : A → B} (isMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧)
where
open import Data.Sum.Base as Sum
open import Relation.Nullary using (yes; no)
open import Relation.Nullary.Decidable
open IsRelMonomorphism isMonomorphism
------------------------------------------------------------------------
-- Properties
refl : Reflexive _∼₂_ → Reflexive _∼₁_
refl refl = injective refl
sym : Symmetric _∼₂_ → Symmetric _∼₁_
sym sym x∼y = injective (sym (cong x∼y))
trans : Transitive _∼₂_ → Transitive _∼₁_
trans trans x∼y y∼z = injective (trans (cong x∼y) (cong y∼z))
total : Total _∼₂_ → Total _∼₁_
total total x y = Sum.map injective injective (total ⟦ x ⟧ ⟦ y ⟧)
asym : Asymmetric _∼₂_ → Asymmetric _∼₁_
asym asym x∼y y∼x = asym (cong x∼y) (cong y∼x)
dec : Decidable _∼₂_ → Decidable _∼₁_
dec _∼?_ x y = map′ injective cong (⟦ x ⟧ ∼? ⟦ y ⟧)
------------------------------------------------------------------------
-- Structures
isEquivalence : IsEquivalence _∼₂_ → IsEquivalence _∼₁_
isEquivalence isEq = record
{ refl = refl E.refl
; sym = sym E.sym
; trans = trans E.trans
} where module E = IsEquivalence isEq
isDecEquivalence : IsDecEquivalence _∼₂_ → IsDecEquivalence _∼₁_
isDecEquivalence isDecEq = record
{ isEquivalence = isEquivalence E.isEquivalence
; _≟_ = dec E._≟_
} where module E = IsDecEquivalence isDecEq
|
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open import HoTT
module cohomology.Exactness where
module _ {i j k} {X : Ptd i} {Y : Ptd j} {Z : Ptd k}
(F : fst (X ⊙→ Y)) (G : fst (Y ⊙→ Z)) where
private
f = fst F
g = fst G
{- in image of F ⇒ in kernel of G -}
is-exact-itok : Type (lmax k (lmax j i))
is-exact-itok = (y : fst Y) → Trunc ⟨-1⟩ (Σ (fst X) (λ x → f x == y))
→ g y == snd Z
{- in kernel of G ⇒ in image of F -}
is-exact-ktoi : Type (lmax k (lmax j i))
is-exact-ktoi = (y : fst Y) → g y == snd Z
→ Trunc ⟨-1⟩ (Σ (fst X) (λ x → f x == y))
record is-exact : Type (lmax k (lmax j i)) where
field
itok : is-exact-itok
ktoi : is-exact-ktoi
open is-exact public
{- an equivalent version of is-exact-ktoi if Z is a set -}
itok-alt-in : has-level ⟨0⟩ (fst Z)
→ ((x : fst X) → g (f x) == snd Z) → is-exact-itok
itok-alt-in pZ h y = Trunc-rec (pZ _ _)
(λ {(x , p) → ap g (! p) ∙ h x})
itok-alt-out : is-exact-itok → ((x : fst X) → g (f x) == snd Z)
itok-alt-out h x = h (f x) [ x , idp ]
{- Convenient notation for exact sequences. At the moment this is only set
up for exact sequences of groups. Do we care about the general case? -}
infix 2 _⊣|
infixr 2 _⟨_⟩→_
data ExactDiag {i} : Group i → Group i → Type (lsucc i) where
_⊣| : (G : Group i) → ExactDiag G G
_⟨_⟩→_ : (G : Group i) {H K : Group i} (φ : G →ᴳ H)
→ ExactDiag H K → ExactDiag G K
data ExactSeq {i} : {G H : Group i} → ExactDiag G H → Type (lsucc i) where
exact-seq-zero : {G : Group i} → ExactSeq (G ⊣|)
exact-seq-one : {G H : Group i} {φ : G →ᴳ H} → ExactSeq (G ⟨ φ ⟩→ H ⊣|)
exact-seq-two : {G H K J : Group i} {φ : G →ᴳ H} {ψ : H →ᴳ K}
{diag : ExactDiag K J} → is-exact (GroupHom.⊙f φ) (GroupHom.⊙f ψ)
→ ExactSeq (H ⟨ ψ ⟩→ diag) → ExactSeq (G ⟨ φ ⟩→ H ⟨ ψ ⟩→ diag)
private
exact-get-type : ∀ {i} {G H : Group i} → ExactDiag G H → ℕ → Type i
exact-get-type (G ⊣|) _ = Lift Unit
exact-get-type (G ⟨ φ ⟩→ H ⊣|) _ = Lift Unit
exact-get-type (G ⟨ φ ⟩→ (H ⟨ ψ ⟩→ s)) O =
is-exact (GroupHom.⊙f φ) (GroupHom.⊙f ψ)
exact-get-type (_ ⟨ _ ⟩→ s) (S n) = exact-get-type s n
exact-get : ∀ {i} {G H : Group i} {diag : ExactDiag G H}
→ ExactSeq diag → (n : ℕ) → exact-get-type diag n
exact-get exact-seq-zero _ = lift unit
exact-get exact-seq-one _ = lift unit
exact-get (exact-seq-two ex s) O = ex
exact-get (exact-seq-two ex s) (S n) = exact-get s n
private
exact-build-arg-type : ∀ {i} {G H : Group i} → ExactDiag G H → List (Type i)
exact-build-arg-type (G ⊣|) = nil
exact-build-arg-type (G ⟨ φ ⟩→ H ⊣|) = nil
exact-build-arg-type (G ⟨ φ ⟩→ H ⟨ ψ ⟩→ s) =
is-exact (GroupHom.⊙f φ) (GroupHom.⊙f ψ)
:: exact-build-arg-type (H ⟨ ψ ⟩→ s)
exact-build-helper : ∀ {i} {G H : Group i} (diag : ExactDiag G H)
→ HList (exact-build-arg-type diag) → ExactSeq diag
exact-build-helper (G ⊣|) nil = exact-seq-zero
exact-build-helper (G ⟨ φ ⟩→ H ⊣|) nil = exact-seq-one
exact-build-helper (G ⟨ φ ⟩→ H ⟨ ψ ⟩→ s) (ie :: ies) =
exact-seq-two ie (exact-build-helper (H ⟨ ψ ⟩→ s) ies)
exact-build : ∀ {i} {G H : Group i} (diag : ExactDiag G H)
→ hlist-curry-type (exact-build-arg-type diag) (λ _ → ExactSeq diag)
exact-build diag = hlist-curry (exact-build-helper diag)
private
exact-snoc-diag : ∀ {i} {G H K : Group i}
→ ExactDiag G H → (H →ᴳ K) → ExactDiag G K
exact-snoc-diag (G ⊣|) ψ = G ⟨ ψ ⟩→ _ ⊣|
exact-snoc-diag (G ⟨ φ ⟩→ s) ψ = G ⟨ φ ⟩→ exact-snoc-diag s ψ
exact-concat-diag : ∀ {i} {G H K : Group i}
→ ExactDiag G H → ExactDiag H K → ExactDiag G K
exact-concat-diag (G ⊣|) s₂ = s₂
exact-concat-diag (G ⟨ φ ⟩→ s₁) s₂ = G ⟨ φ ⟩→ (exact-concat-diag s₁ s₂)
abstract
exact-concat : ∀ {i} {G H K L : Group i}
{diag₁ : ExactDiag G H} {φ : H →ᴳ K} {diag₂ : ExactDiag K L}
→ ExactSeq (exact-snoc-diag diag₁ φ) → ExactSeq (H ⟨ φ ⟩→ diag₂)
→ ExactSeq (exact-concat-diag diag₁ (H ⟨ φ ⟩→ diag₂))
exact-concat {diag₁ = G ⊣|} exact-seq-one es₂ = es₂
exact-concat {diag₁ = G ⟨ ψ ⟩→ H ⊣|} (exact-seq-two ex _) es₂ =
exact-seq-two ex es₂
exact-concat {diag₁ = G ⟨ ψ ⟩→ H ⟨ χ ⟩→ s} (exact-seq-two ex es₁) es₂ =
exact-seq-two ex (exact-concat {diag₁ = H ⟨ χ ⟩→ s} es₁ es₂)
|
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|
module Sigma.Renaming.Properties where
open import Data.Nat using (ℕ; _+_; zero; suc)
open import Data.Fin using (Fin; zero; suc)
open import Sigma.Subst.Core
open import Sigma.Renaming.Base
open import Sigma.Subst.Properties using (extensionality)
open import Function using (_∘_)
open import Relation.Binary.PropositionalEquality as Eq
using (_≡_; refl; cong; cong₂)
open Eq.≡-Reasoning
-- ------------------------------------------------------------------------
⇑-cong : ∀ { m n } { ρ₁ ρ₂ : Ren m n }
→ ρ₁ ≡ ρ₂
-- -----------
→ ρ₁ ⇑ ≡ ρ₂ ⇑
⇑-cong ρ₁≡ρ₂ = cong (_⇑) ρ₁≡ρ₂
⇑✶-cong : ∀ { m n } k { ρ₁ ρ₂ : Ren m n }
→ ρ₁ ≡ ρ₂
-- -----------
→ ρ₁ ⇑✶ k ≡ ρ₂ ⇑✶ k
⇑✶-cong k ρ₁≡ρ₂ = cong (_⇑✶ k) ρ₁≡ρ₂
-- ------------------------------------------------------------------------
∘-⇑-distrib : ∀ { m n k } ( ρ₁ : Ren m n ) ( ρ₂ : Ren n k )
→ (ρ₂ ⇑ ∘ ρ₁ ⇑) ≡ (ρ₂ ∘ ρ₁) ⇑
∘-⇑-distrib ρ₁ ρ₂ = extensionality lemma
where
lemma : ∀ x → (ρ₂ ⇑ ∘ ρ₁ ⇑) x ≡ ((ρ₂ ∘ ρ₁) ⇑) x
lemma zero = refl
lemma (suc x) = refl
-- TODO:
-- generalized for ⇑✶
-- ------------------------------------------------------------------------
⇑-id : ∀ { n } → (id { n }) ⇑ ≡ id
⇑-id = extensionality lemma
where
lemma : ∀ { n } x → (id { n } ⇑) x ≡ x
lemma zero = refl
lemma (suc x) = refl
|
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|
-- Minimal implicational logic, PHOAS approach, initial encoding
module STLC where
-- Types
infixr 0 _=>_
data Ty : Set where
UNIT : Ty
_=>_ : Ty -> Ty -> Ty
-- Context
Cx : Set1
Cx = Ty -> Set
-- Terms
infixl 1 _$_
data Tm (tc : Cx) : Ty -> Set
where
var : forall {a} -> tc a
----------
-> Tm tc a
lam : forall {a b} -> (tc a -> Tm tc b)
---------------------
-> Tm tc (a => b)
_$_ : forall {a b} -> Tm tc (a => b) -> Tm tc a
----------------------------
-> Tm tc b
T : Ty -> Set1
T a = forall {tc} -> Tm tc a
-- Example theorems
I : forall {a} -> T (a => a)
I = lam \x ->
var x
K : forall {a b} -> T (a => b => a)
K = lam \x ->
lam \_ ->
var x
S : forall {a b c} -> T ((a => b => c) => (a => b) => a => c)
S = lam \f ->
lam \g ->
lam \x ->
(var f $ var x) $ (var g $ var x)
SKK : forall {a} -> T (a => a)
SKK {a = a} = S {b = a => a} $ K $ K
|
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{-# OPTIONS --cubical --rewriting #-}
open import Agda.Builtin.Cubical.Path
open import Agda.Primitive.Cubical
data A : Set where
a : A
eq : a ≡ a
bad : ∀ i → eq i ≡ a
bad i j = eq (primIMin i (primINeg j))
{-# BUILTIN REWRITE _≡_ #-}
{-# REWRITE bad #-}
|
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|
-- Allow holes in modules to import, by introducing a single general postulate.
module UNDEFINED where
postulate
UNDEFINED : ∀ {ℓ} → {T : Set ℓ} → T
|
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|
-- Andreas, 2017-01-18, issue #819, reported by stevana
-- Underscores should be preserved when case-splitting
data List (A : Set) : Set where
_∷_ : (x : A)(xs : List A) → List A
data ⊥ : Set where
-- Case-splitting on x:
test : List ⊥ → ⊥
test (x ∷ _) = {!x!} -- split on x
-- I think the underscore should be kept.
expected : List ⊥ → ⊥
expected (() ∷ _)
-- Likewise here:
test′ : List ⊥ → List ⊥ → List ⊥ → ⊥
test′ (x ∷ _) ys _ = {!ys x!} -- split on ys and x
expected′ : List ⊥ → List ⊥ → List ⊥ → ⊥
expected′ (() ∷ _) (x ∷ ys) _
|
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|
module Class.Map where
open import Class.Equality
open import Data.Maybe using (Maybe; just; nothing)
open import Data.List using (List; []; _∷_; [_])
record MapClass (K : Set) {{_ : EqB K}} (M : Set -> Set) : Set₁ where
field
insert : ∀ {V} -> K -> V -> M V -> M V
remove : ∀ {V} -> K -> M V -> M V
lookup : ∀ {V} -> K -> M V -> Maybe V
mapSnd : ∀ {V C} -> (V -> C) -> M V -> M C
emptyMap : ∀ {V} -> M V
open MapClass {{...}} public
mapFromList : ∀ {K V M} {{_ : EqB K}} {{_ : MapClass K M}} -> (V -> K) -> List V -> M (List V)
mapFromList f [] = emptyMap
mapFromList f (x ∷ l) with mapFromList f l
... | m with lookup (f x) m
... | just x₁ = insert (f x) (x ∷ x₁) m
... | nothing = insert (f x) [ x ] m
|
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{-# OPTIONS --without-K #-}
open import M-types.Base
module M-types.Coalg.M {ℓ : Level} (A : Ty ℓ) (B : A → Ty ℓ) where
open import M-types.Coalg.Core A B
open import M-types.Coalg.Bisim A B
IsFinM : ∏[ M ∈ Coalg ] ∏[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
Ty (ℓ-suc ℓ)
IsFinM M coiter = ∏[ X ∈ Coalg ] ∏[ f ∈ CoalgMor X M ] f ≡ coiter X
FinM : Ty (ℓ-suc ℓ)
FinM = ∑[ M ∈ Coalg ] ∑[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
IsFinM M coiter
IsCohM : ∏[ M ∈ Coalg ] ∏[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
Ty (ℓ-suc ℓ)
IsCohM M coiter =
∏[ X ∈ Coalg ] ∏[ f₀ ∈ CoalgMor X M ] ∏[ f₁ ∈ CoalgMor X M ]
∑[ p ∈ fun f₀ ≡ fun f₁ ] ∏[ x ∈ ty X ]
ap (λ f → obs M (f x)) p · ≡-apply (com f₁) x ≡
≡-apply (com f₀) x · ap (λ f → P-Fun f (obs X x)) p
CohM : Ty (ℓ-suc ℓ)
CohM = ∑[ M ∈ Coalg ] ∑[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
IsCohM M coiter
IsBareM : ∏[ M ∈ Coalg ] ∏[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
Ty (ℓ-suc ℓ)
IsBareM M coiter =
∏[ X ∈ Coalg ] ∏[ f₀ ∈ CoalgMor X M ] ∏[ f₁ ∈ CoalgMor X M ]
(fun f₀ ≡ fun f₁)
BareM : Ty (ℓ-suc ℓ)
BareM = ∑[ M ∈ Coalg ] ∑[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
IsBareM M coiter
IsTyBisimM : ∏[ M ∈ Coalg ] ∏[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
Ty (ℓ-suc ℓ)
IsTyBisimM M coiter =
∏[ ∼ ∈ TyBisim M ] SpanRelMor (spanRel {M} ∼) ≡-spanRel
TyBisimM : Ty (ℓ-suc ℓ)
TyBisimM = ∑[ M ∈ Coalg ] ∑[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
IsTyBisimM M coiter
IsFunBisimM : ∏[ M ∈ Coalg ] ∏[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
Ty (ℓ-suc ℓ)
IsFunBisimM M coiter =
∏[ ∼ ∈ FunBisim M ] DepRelMor (depRel {M} ∼) ≡-depRel
FunBisimM : Ty (ℓ-suc ℓ)
FunBisimM = ∑[ M ∈ Coalg ] ∑[ coiter ∈ (∏[ X ∈ Coalg ] CoalgMor X M) ]
IsFunBisimM M coiter
FinM→CohM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsFinM M coiter → IsCohM M coiter
FinM→CohM {M} {coiter} isFin = λ X → λ f₀ → λ f₁ →
coh (isFin X f₀ · isFin X f₁ ⁻¹)
where
coh : {X : Coalg} {f₀ f₁ : CoalgMor X M} →
(f₀ ≡ f₁) → (
∑[ p ∈ fun f₀ ≡ fun f₁ ] ∏[ x ∈ ty X ]
ap (λ f → obs M (f x)) p · ≡-apply (com f₁) x ≡
≡-apply (com f₀) x · ap (λ f → P-Fun f (obs X x)) p
)
coh {X} {f} {f} refl =
(
refl ,
λ x →
·-neutr₀ ( ≡-apply (com f) x) ·
·-neutr₁ ( ≡-apply (com f) x) ⁻¹
)
CohM→FinM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsCohM M coiter → IsFinM M coiter
CohM→FinM {M} {coiter} isCoh = λ X → λ f →
fin (isCoh X f (coiter X))
where
fin : {X : Coalg} {f : CoalgMor X M} →
(
∑[ p ∈ fun f ≡ fun (coiter X) ] ∏[ x ∈ ty X ]
ap (λ f → obs M (f x)) p · ≡-apply (com (coiter X)) x ≡
≡-apply (com f) x · ap (λ f → P-Fun f (obs X x)) p
) → (f ≡ coiter X)
fin {X} {f} (refl , coh) =
≡-pair (
refl ,
(hom₀ (≡-apply , funext-axiom) (com f)) ⁻¹ ·
ap funext (funext (λ x →
(·-neutr₁ (≡-apply (com f) x)) ⁻¹ ·
(coh x) ⁻¹ ·
(·-neutr₀ (≡-apply (com (coiter X)) x))
)) ·
(hom₀ (≡-apply , funext-axiom) (com (coiter X)))
)
CohM→BareM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsCohM M coiter → IsBareM M coiter
CohM→BareM {M} {coiter} isCoh = λ X → λ f₀ → λ f₁ → pr₀ (isCoh X f₀ f₁)
BareM→TyBisimM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsBareM M coiter → IsTyBisimM M coiter
BareM→TyBisimM {M} {coiter} isBare = λ ∼ →
(
fun (ρ₀ ∼) ,
refl ,
isBare (ty ∼) (ρ₀ ∼) (ρ₁ ∼)
)
TyBisimM→BareM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsTyBisimM M coiter → IsBareM M coiter
TyBisimM→BareM {M} {coiter} isTyBisim = λ X → λ f₀ → λ f₁ →
funext (λ x →
(≡-apply (com₀ (isTyBisim (X , f₀ , f₁))) x) ⁻¹ ·
(≡-apply (com₁ (isTyBisim (X , f₀ , f₁))) x)
)
TyBisimM→FunBisimM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsTyBisimM M coiter → IsFunBisimM M coiter
TyBisimM→FunBisimM {M} {coiter} isTyBisim = λ ∼ → let
f : DepRelMor (depRel {M} ∼) (SpanRel→DepRel (DepRel→SpanRel (depRel {M} ∼)))
f = DepRel→DepRel-mor (depRel {M} ∼)
g : DepRelMor (SpanRel→DepRel (DepRel→SpanRel (depRel {M} ∼))) (SpanRel→DepRel ≡-spanRel)
g = SpanRelMor→DepRelMor (isTyBisim (FunBisim→TyBisim {M} ∼))
h : DepRelMor (SpanRel→DepRel ≡-spanRel) ≡-depRel
h = ≡-SpanRel→DepRel-mor
in
h ∘-depRel (g ∘-depRel f)
FunBisimM→TyBisimM : {M : Coalg} {coiter : ∏[ X ∈ Coalg ] CoalgMor X M} →
IsFunBisimM M coiter → IsTyBisimM M coiter
FunBisimM→TyBisimM {M} {coiter} isFunBisim = λ ∼ → let
f : SpanRelMor (spanRel {M} ∼) (DepRel→SpanRel (SpanRel→DepRel (spanRel {M} ∼)))
f = SpanRel→SpanRel-mor (spanRel {M} ∼)
g : SpanRelMor (DepRel→SpanRel (SpanRel→DepRel (spanRel {M} ∼))) (DepRel→SpanRel ≡-depRel)
g = DepRelMor→SpanRelMor (isFunBisim (TyBisim→FunBisim {M} ∼))
h : SpanRelMor (DepRel→SpanRel ≡-depRel) ≡-spanRel
h = ≡-DepRel→SpanRel-mor
in
-- h ∘-spanRel (g ∘-spanRel f)
_∘-spanRel_ {ℓ} {ty M} {spanRel {M} ∼} {DepRel→SpanRel ≡-depRel} {≡-spanRel}
h
(_∘-spanRel_ {ℓ} {ty M} {spanRel {M} ∼} {DepRel→SpanRel (SpanRel→DepRel (spanRel {M} ∼))} {DepRel→SpanRel ≡-depRel} g f)
|
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{-# OPTIONS --without-K #-}
module PermutationProperties where
open import Data.Nat using (ℕ; _+_)
open import Data.Fin using (Fin)
open import Relation.Binary.PropositionalEquality
using (_≡_; refl; sym; trans; cong; module ≡-Reasoning; proof-irrelevance; setoid)
open import Function.Equality using (_⟨$⟩_)
open import Data.Sum using (_⊎_)
open import Data.Product using (_,_; proj₁; proj₂)
--
import FinEquivPlusTimes using (module Plus) -- don't open, just import
import FinEquivTypeEquiv using (module PlusE) -- don't open, just import
open FinEquivPlusTimes.Plus using (⊎≃+; +≃⊎)
open FinEquivTypeEquiv.PlusE using (_+F_)
open import ConcretePermutation
open import Permutation
open import SEquivSCPermEquiv
open import Equiv using (_●_; id≃; sym≃; _⊎≃_)
open import EquivEquiv
using (id≋; sym≋; ●-assoc; _◎_; lid≋; rid≋; linv≋; rinv≋;
module ≋-Reasoning)
open import TypeEquivEquiv using (_⊎≋_)
open import FinEquivEquivPlus using ([id+id]≋id; +●≋●+)
open ≋-Reasoning
------------------------------------------------------------------------------
-- Composition
assocp : ∀ {m₁ m₂ m₃ n₁} → {p₁ : CPerm m₁ n₁} → {p₂ : CPerm m₂ m₁} →
{p₃ : CPerm m₃ m₂} →
(p₁ ●p p₂) ●p p₃ ≡ p₁ ●p (p₂ ●p p₃)
assocp {p₁ = p₁} {p₂} {p₃} =
let e₁ = p⇒e p₁ in let e₂ = p⇒e p₂ in let e₃ = p⇒e p₃ in
≋⇒≡ (begin (
p⇒e (e⇒p (e₁ ● e₂)) ● e₃
≋⟨ left-α-over-● (e₁ ● e₂) e₃ ⟩
(e₁ ● e₂) ● e₃
≋⟨ ●-assoc {f = e₃} {e₂} {e₁} ⟩
e₁ ● (e₂ ● e₃)
≋⟨ sym≋ (right-α-over-● e₁ (e₂ ● e₃)) ⟩
e₁ ● (p⇒e (e⇒p (e₂ ● e₃))) ∎))
lidp : ∀ {m₁ m₂} {p : CPerm m₂ m₁} → idp ●p p ≡ p
lidp {p = p} = trans (≋⇒≡ (begin (
(p⇒e (e⇒p id≃)) ● (p⇒e p)
≋⟨ left-α-over-● id≃ (p⇒e p) ⟩
id≃ ● (p⇒e p)
≋⟨ lid≋ ⟩
(p⇒e p) ∎))) (βu refl)
ridp : ∀ {m₁ m₂} {p : CPerm m₂ m₁} → p ●p idp ≡ p
ridp {p = p} = trans (≋⇒≡ (begin (
(p⇒e p) ● (p⇒e (e⇒p id≃))
≋⟨ right-α-over-● (p⇒e p) id≃ ⟩
(p⇒e p) ● id≃
≋⟨ rid≋ ⟩
(p⇒e p) ∎))) (βu refl)
-- Inverses
rinv : ∀ {m₁ m₂} (p : CPerm m₂ m₁) → p ●p (symp p) ≡ idp
rinv p = let e = p⇒e p in ≋⇒≡ (begin (
e ● (p⇒e (e⇒p (sym≃ e)))
≋⟨ right-α-over-● e (sym≃ e) ⟩
e ● (sym≃ e)
≋⟨ rinv≋ e ⟩
id≃ ∎))
linv : ∀ {m₁ m₂} (p : CPerm m₂ m₁) → (symp p) ●p p ≡ idp
linv p = let e = p⇒e p in ≋⇒≡ (begin (
(p⇒e (e⇒p (sym≃ e))) ● e
≋⟨ left-α-over-● (sym≃ e) e ⟩
(sym≃ e) ● e
≋⟨ linv≋ e ⟩
id≃ ∎))
-- p₁ ⊎p p₂ = e⇒p ((p⇒e p₁) +F (p⇒e p₂))
-- Fm≃Fn +F Fo≃Fp = ⊎≃+ ● Fm≃Fn ⊎≃ Fo≃Fp ● +≃⊎
⊎p●p≡●p⊎p : {m₁ m₂ n₁ n₂ o₁ o₂ : ℕ} →
{f : CPerm n₁ m₁} {g : CPerm n₂ m₂} {h : CPerm o₁ n₁} {i : CPerm o₂ n₂} →
((f ●p h) ⊎p (g ●p i)) ≡ ((f ⊎p g) ●p (h ⊎p i))
⊎p●p≡●p⊎p {f = f} {g} {h} {i} =
let e₁ = p⇒e f in let e₂ = p⇒e g in let e₃ = p⇒e h in let e₄ = p⇒e i in
let f≋ = id≋ {x = ⊎≃+} in
let g≋ = id≋ {x = +≃⊎} in
≋⇒≡ (begin -- inline ⊎p
p⇒e (e⇒p (e₁ ● e₃)) +F p⇒e (e⇒p (e₂ ● e₄))
≋⟨ id≋ ⟩ -- inline +F
⊎≃+ ● (p⇒e (e⇒p (e₁ ● e₃)) ⊎≃ p⇒e (e⇒p (e₂ ● e₄))) ● +≃⊎
≋⟨ f≋ ◎ ((α₁ ⊎≋ α₁) ◎ g≋) ⟩
⊎≃+ ● ((e₁ ● e₃) ⊎≃ (e₂ ● e₄)) ● +≃⊎
≋⟨ +●≋●+ ⟩
(e₁ +F e₂) ● (e₃ +F e₄)
≋⟨ sym≋ ((α₁ {e = e₁ +F e₂}) ◎ (α₁ {e = e₃ +F e₄})) ⟩
(p⇒e (e⇒p (e₁ +F e₂)) ● p⇒e (e⇒p (e₃ +F e₄))) ∎)
-- Additives
1p⊎1p≡1p : ∀ {m n} → idp {m} ⊎p idp {n} ≡ idp {m + n}
1p⊎1p≡1p {m} {n} =
let em = p⇒e (e⇒p (id≃ {A = Fin m})) in
let en = p⇒e (e⇒p (id≃ {A = Fin n})) in
let f≋ = id≋ {x = ⊎≃+ {m} {n}} in
let g≋ = id≋ {x = +≃⊎ {m} {n}} in
≋⇒≡ (begin (
em +F en
≋⟨ id≋ ⟩
⊎≃+ ● em ⊎≃ en ● +≃⊎
≋⟨ f≋ ◎ ((α₁ ⊎≋ α₁) ◎ g≋) ⟩
⊎≃+ ● (id≃ {A = Fin m}) ⊎≃ id≃ ● +≃⊎
≋⟨ [id+id]≋id ⟩
id≃ {A = Fin (m + n)} ∎))
-- interaction with composition
{- The underlying permutations are no longer defined!
unite+p∘[0⊎x]≡x∘unite+p : ∀ {m n} (p : CPerm m n) →
transp unite+p (0p ⊎p p) ≡ transp p unite+p
unite+p∘[0⊎x]≡x∘unite+p p = p≡ unite+∘[0⊎x]≡x∘unite+
uniti+p∘x≡[0⊎x]∘uniti+p : ∀ {m n} (p : CPerm m n) →
transp uniti+p p ≡ transp (0p ⊎p p) uniti+p
uniti+p∘x≡[0⊎x]∘uniti+p p = p≡ (uniti+∘x≡[0⊎x]∘uniti+ {x = CPerm.π p})
uniti+rp∘[x⊎0]≡x∘uniti+rp : ∀ {m n} (p : CPerm m n) →
transp uniti+rp (p ⊎p 0p) ≡ transp p uniti+rp
uniti+rp∘[x⊎0]≡x∘uniti+rp p = p≡ uniti+r∘[x⊎0]≡x∘uniti+r
-}
{-
unite+rp∘[x⊎0]≡x∘unite+rp : ∀ {m n} (p : CPerm m n) →
transp unite+rp p ≡ transp (p ⊎p 0p) unite+rp
unite+rp∘[x⊎0]≡x∘unite+rp p = p≡ unite+r∘[x⊎0]≡x∘unite+r
-}
-- Multiplicatives
{-
1p×1p≡1p : ∀ {m n} → idp {m} ×p idp {n} ≡ idp
1p×1p≡1p {m} = p≡ (1C×1C≡1C {m})
×p-distrib : ∀ {m₁ m₂ m₃ m₄ n₁ n₂} → {p₁ : CPerm m₁ n₁} → {p₂ : CPerm m₂ n₂}
→ {p₃ : CPerm m₃ m₁} → {p₄ : CPerm m₄ m₂} →
(transp p₁ p₃) ×p (transp p₂ p₄) ≡ transp (p₁ ×p p₂) (p₃ ×p p₄)
×p-distrib {p₁ = p₁} = p≡ (sym (×c-distrib {p₁ = CPerm.π p₁}))
-}
------------------------------------------------------------------------------
|
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module Logic.IntroInstances where
import Data.Tuple as Tuple
import Lvl
open import Logic.Predicate
open import Logic.Propositional
open import Type
private variable ℓ : Lvl.Level
private variable A B Obj : Type{ℓ}
private variable P : Obj → Type{ℓ}
private variable x : Obj
instance
[∧]-intro-instance : ⦃ _ : A ⦄ → ⦃ _ : B ⦄ → (A ∧ B)
[∧]-intro-instance ⦃ a ⦄ ⦃ b ⦄ = [∧]-intro a b
instance
[∨]-introₗ-instance : ⦃ _ : A ⦄ → (A ∨ B)
[∨]-introₗ-instance ⦃ a ⦄ = [∨]-introₗ a
instance
[∨]-introᵣ-instance : ⦃ _ : B ⦄ → (A ∨ B)
[∨]-introᵣ-instance ⦃ b ⦄ = [∨]-introᵣ b
instance
[∃]-intro-instance : ⦃ _ : P(x) ⦄ → ∃(P)
[∃]-intro-instance {x = x} ⦃ proof ⦄ = [∃]-intro (x) ⦃ proof ⦄
|
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------------------------------------------------------------------------
-- A container for finite binary trees with information in internal
-- nodes
------------------------------------------------------------------------
{-# OPTIONS --without-K --safe #-}
open import Equality
module Container.Tree
{c⁺} (eq : ∀ {a p} → Equality-with-J a p c⁺) where
open Derived-definitions-and-properties eq
open import Prelude hiding (id; _∘_; List; []; _∷_)
open import Bijection eq using (_↔_; module _↔_)
open import Container eq hiding (Shape; Position)
open import Container.List eq hiding (fold; fold-lemma)
open import Function-universe eq
import Tree eq as Tree
------------------------------------------------------------------------
-- The type
-- Shapes.
data Shape : Type where
lf : Shape
nd : Shape → Shape → Shape
-- Positions.
data Position : Shape → Type where
root : ∀ {l r} → Position (nd l r)
left : ∀ {l r} → Position l → Position (nd l r)
right : ∀ {l r} → Position r → Position (nd l r)
-- Trees.
Tree : Container lzero
Tree = Shape ▷ Position
------------------------------------------------------------------------
-- An isomorphism
-- The type of shapes is isomorphic to Tree.Tree ⊤.
--
-- This lemma is included because it was mentioned in the paper "Bag
-- Equivalence via a Proof-Relevant Membership Relation".
Shape↔Tree-⊤ : Shape ↔ Tree.Tree ⊤
Shape↔Tree-⊤ = record
{ surjection = record
{ logical-equivalence = record
{ to = to
; from = from
}
; right-inverse-of = to∘from
}
; left-inverse-of = from∘to
}
where
to : Shape → Tree.Tree ⊤
to lf = Tree.leaf
to (nd l r) = Tree.node (to l) tt (to r)
from : Tree.Tree ⊤ → Shape
from Tree.leaf = lf
from (Tree.node l tt r) = nd (from l) (from r)
to∘from : ∀ t → to (from t) ≡ t
to∘from Tree.leaf = refl _
to∘from (Tree.node l tt r) =
cong₂ (λ l r → Tree.node l tt r) (to∘from l) (to∘from r)
from∘to : ∀ s → from (to s) ≡ s
from∘to lf = refl _
from∘to (nd l r) = cong₂ nd (from∘to l) (from∘to r)
------------------------------------------------------------------------
-- Constructors
-- Leaves.
leaf : {A : Type} → ⟦ Tree ⟧ A
leaf = (lf , λ ())
-- Internal nodes.
node : {A : Type} → ⟦ Tree ⟧ A → A → ⟦ Tree ⟧ A → ⟦ Tree ⟧ A
node (l , lkup-l) x (r , lkup-r) =
( nd l r
, λ { root → x
; (left p) → lkup-l p
; (right p) → lkup-r p
}
)
-- Even if we don't assume extensionality we can prove that
-- intensionally distinct implementations of the constructors are bag
-- equivalent.
leaf≈ : {A : Type} {lkup : _ → A} →
_≈-bag_ {C₂ = Tree} leaf (lf , lkup)
leaf≈ _ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { (() , _) }
; from = λ { (() , _) }
}
; right-inverse-of = λ { (() , _) }
}
; left-inverse-of = λ { (() , _) }
}
node≈ : ∀ {A : Type} {l r} {lkup : _ → A} →
_≈-bag_ {C₂ = Tree}
(node (l , lkup ∘ left) (lkup root) (r , lkup ∘ right))
(nd l r , lkup)
node≈ _ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { (root , eq) → (root , eq)
; (left p , eq) → (left p , eq)
; (right p , eq) → (right p , eq)
}
; from = λ { (root , eq) → (root , eq)
; (left p , eq) → (left p , eq)
; (right p , eq) → (right p , eq)
}
}
; right-inverse-of = λ { (root , eq) → refl _
; (left p , eq) → refl _
; (right p , eq) → refl _
}
}
; left-inverse-of = λ { (root , eq) → refl _
; (left p , eq) → refl _
; (right p , eq) → refl _
}
}
-- Any lemmas for the constructors.
Any-leaf : ∀ {A : Type} (P : A → Type) →
Any P leaf ↔ ⊥₀
Any-leaf _ = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { (() , _) }
; from = λ ()
}
; right-inverse-of = λ ()
}
; left-inverse-of = λ { (() , _) }
}
Any-node : ∀ {A : Type} (P : A → Type) {l x r} →
Any P (node l x r) ↔ Any P l ⊎ P x ⊎ Any P r
Any-node _ {l = _ , _} {r = _ , _} = record
{ surjection = record
{ logical-equivalence = record
{ to = λ { (root , eq) → inj₂ (inj₁ eq)
; (left p , eq) → inj₁ (p , eq)
; (right p , eq) → inj₂ (inj₂ (p , eq))
}
; from = λ { (inj₁ (p , eq)) → (left p , eq)
; (inj₂ (inj₁ eq)) → (root , eq)
; (inj₂ (inj₂ (p , eq))) → (right p , eq)
}
}
; right-inverse-of = λ { (inj₁ (p , eq)) → refl _
; (inj₂ (inj₁ eq)) → refl _
; (inj₂ (inj₂ (p , eq))) → refl _
}
}
; left-inverse-of = λ { (root , eq) → refl _
; (left p , eq) → refl _
; (right p , eq) → refl _
}
}
------------------------------------------------------------------------
-- More functions
-- Singleton trees.
singleton : {A : Type} → A → ⟦ Tree ⟧ A
singleton x = node leaf x leaf
-- Any lemma for singleton.
Any-singleton : ∀ {A : Type} (P : A → Type) {x} →
Any P (singleton x) ↔ P x
Any-singleton P {x} =
Any P (singleton x) ↔⟨⟩
Any P (node leaf x leaf) ↔⟨ Any-node P ⟩
Any P leaf ⊎ P x ⊎ Any P leaf ↔⟨ Any-leaf P ⊎-cong id ⊎-cong Any-leaf P ⟩
⊥ ⊎ P x ⊎ ⊥ ↔⟨ ⊎-left-identity ⟩
P x ⊎ ⊥ ↔⟨ ⊎-right-identity ⟩
P x □
-- For the design considerations underlying the inclusion of fold and
-- fold-lemma, see Container.List.fold/fold-lemma.
-- A fold for trees. (Well, this is not a catamorphism, it is a
-- paramorphism.)
fold : {A B : Type} →
B → (⟦ Tree ⟧ A → A → ⟦ Tree ⟧ A → B → B → B) → ⟦ Tree ⟧ A → B
fold {A} {B} fl fn = uncurry fold′
where
fold′ : (s : Shape) → (Position s → A) → B
fold′ lf lkup = fl
fold′ (nd l r) lkup =
fn (l , lkup ∘ left )
(lkup root)
(r , lkup ∘ right)
(fold′ l (lkup ∘ left ))
(fold′ r (lkup ∘ right))
-- A lemma which can be used to prove properties about fold.
--
-- The "respects bag equivalence" argument could be omitted if
-- equality of functions were extensional.
fold-lemma : ∀ {A B : Type}
{fl : B} {fn : ⟦ Tree ⟧ A → A → ⟦ Tree ⟧ A → B → B → B}
(P : ⟦ Tree ⟧ A → B → Type) →
(∀ t₁ t₂ → t₁ ≈-bag t₂ → ∀ b → P t₁ b → P t₂ b) →
P leaf fl →
(∀ l x r b₁ b₂ →
P l b₁ → P r b₂ → P (node l x r) (fn l x r b₁ b₂)) →
∀ t → P t (fold fl fn t)
fold-lemma {A} {fl = fl} {fn} P resp P-le P-no = uncurry fold-lemma′
where
fold-lemma′ : (s : Shape) (lkup : Position s → A) →
P (s , lkup) (fold fl fn (s , lkup))
fold-lemma′ lf lkup = resp _ _ leaf≈ _ P-le
fold-lemma′ (nd l r) lkup = resp _ _ node≈ _ $
P-no _ _ _ _ _
(fold-lemma′ l (lkup ∘ left ))
(fold-lemma′ r (lkup ∘ right))
-- Inorder flattening of a tree.
flatten : {A : Type} → ⟦ Tree ⟧ A → ⟦ List ⟧ A
flatten = fold [] (λ _ x _ xs ys → xs ++ x ∷ ys)
-- Flatten does not add or remove any elements.
flatten≈ : {A : Type} (t : ⟦ Tree ⟧ A) → flatten t ≈-bag t
flatten≈ = fold-lemma
(λ t xs → xs ≈-bag t)
(λ t₁ t₂ t₁≈t₂ xs xs≈t₁ z →
z ∈ xs ↔⟨ xs≈t₁ z ⟩
z ∈ t₁ ↔⟨ t₁≈t₂ z ⟩
z ∈ t₂ □)
(λ z →
z ∈ [] ↔⟨ Any-[] (λ x → z ≡ x) ⟩
⊥ ↔⟨ inverse $ Any-leaf (λ x → z ≡ x) ⟩
z ∈ leaf □)
(λ l x r xs ys xs≈l ys≈r z →
z ∈ xs ++ x ∷ ys ↔⟨ Any-++ (λ x → z ≡ x) _ _ ⟩
z ∈ xs ⊎ z ∈ x ∷ ys ↔⟨ id ⊎-cong Any-∷ (λ x → z ≡ x) ⟩
z ∈ xs ⊎ z ≡ x ⊎ z ∈ ys ↔⟨ xs≈l z ⊎-cong id ⊎-cong ys≈r z ⟩
z ∈ l ⊎ z ≡ x ⊎ z ∈ r ↔⟨ inverse $ Any-node (λ x → z ≡ x) ⟩
z ∈ node l x r □)
|
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module Text.Greek.SBLGNT.1John where
open import Data.List
open import Text.Greek.Bible
open import Text.Greek.Script
open import Text.Greek.Script.Unicode
ΙΩΑΝΝΟΥ-Α : List (Word)
ΙΩΑΝΝΟΥ-Α =
word (Ὃ ∷ []) "1John.1.1"
∷ word (ἦ ∷ ν ∷ []) "1John.1.1"
∷ word (ἀ ∷ π ∷ []) "1John.1.1"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.1.1"
∷ word (ὃ ∷ []) "1John.1.1"
∷ word (ἀ ∷ κ ∷ η ∷ κ ∷ ό ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.1"
∷ word (ὃ ∷ []) "1John.1.1"
∷ word (ἑ ∷ ω ∷ ρ ∷ ά ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.1"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1John.1.1"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ῖ ∷ ς ∷ []) "1John.1.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.1.1"
∷ word (ὃ ∷ []) "1John.1.1"
∷ word (ἐ ∷ θ ∷ ε ∷ α ∷ σ ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.1"
∷ word (α ∷ ἱ ∷ []) "1John.1.1"
∷ word (χ ∷ ε ∷ ῖ ∷ ρ ∷ ε ∷ ς ∷ []) "1John.1.1"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.1.1"
∷ word (ἐ ∷ ψ ∷ η ∷ ∙λ ∷ ά ∷ φ ∷ η ∷ σ ∷ α ∷ ν ∷ []) "1John.1.1"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.1.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.1.1"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ υ ∷ []) "1John.1.1"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.1.1"
∷ word (ζ ∷ ω ∷ ῆ ∷ ς ∷ []) "1John.1.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.2"
∷ word (ἡ ∷ []) "1John.1.2"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "1John.1.2"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.2"
∷ word (ἑ ∷ ω ∷ ρ ∷ ά ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.2"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.2"
∷ word (ἀ ∷ π ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.2"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.1.2"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.1.2"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.1.2"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "1John.1.2"
∷ word (ἥ ∷ τ ∷ ι ∷ ς ∷ []) "1John.1.2"
∷ word (ἦ ∷ ν ∷ []) "1John.1.2"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.1.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.1.2"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.1.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.2"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.1.2"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.2"
∷ word (ὃ ∷ []) "1John.1.3"
∷ word (ἑ ∷ ω ∷ ρ ∷ ά ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.3"
∷ word (ἀ ∷ κ ∷ η ∷ κ ∷ ό ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.3"
∷ word (ἀ ∷ π ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.3"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.3"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.1.3"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1John.1.3"
∷ word (ἔ ∷ χ ∷ η ∷ τ ∷ ε ∷ []) "1John.1.3"
∷ word (μ ∷ ε ∷ θ ∷ []) "1John.1.3"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.3"
∷ word (ἡ ∷ []) "1John.1.3"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ []) "1John.1.3"
∷ word (δ ∷ ὲ ∷ []) "1John.1.3"
∷ word (ἡ ∷ []) "1John.1.3"
∷ word (ἡ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.1.3"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1John.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.3"
∷ word (μ ∷ ε ∷ τ ∷ ὰ ∷ []) "1John.1.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.4"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1John.1.4"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.4"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.1.4"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.1.4"
∷ word (ἡ ∷ []) "1John.1.4"
∷ word (χ ∷ α ∷ ρ ∷ ὰ ∷ []) "1John.1.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.1.4"
∷ word (ᾖ ∷ []) "1John.1.4"
∷ word (π ∷ ε ∷ π ∷ ∙λ ∷ η ∷ ρ ∷ ω ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "1John.1.4"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "1John.1.5"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.5"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.1.5"
∷ word (ἡ ∷ []) "1John.1.5"
∷ word (ἀ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "1John.1.5"
∷ word (ἣ ∷ ν ∷ []) "1John.1.5"
∷ word (ἀ ∷ κ ∷ η ∷ κ ∷ ό ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.5"
∷ word (ἀ ∷ π ∷ []) "1John.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.5"
∷ word (ἀ ∷ ν ∷ α ∷ γ ∷ γ ∷ έ ∷ ∙λ ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.5"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.1.5"
∷ word (ὁ ∷ []) "1John.1.5"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.1.5"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "1John.1.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.5"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ α ∷ []) "1John.1.5"
∷ word (ἐ ∷ ν ∷ []) "1John.1.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.1.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.1.5"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.5"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ μ ∷ ί ∷ α ∷ []) "1John.1.5"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.1.6"
∷ word (ε ∷ ἴ ∷ π ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.1.6"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1John.1.6"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.6"
∷ word (μ ∷ ε ∷ τ ∷ []) "1John.1.6"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.6"
∷ word (ἐ ∷ ν ∷ []) "1John.1.6"
∷ word (τ ∷ ῷ ∷ []) "1John.1.6"
∷ word (σ ∷ κ ∷ ό ∷ τ ∷ ε ∷ ι ∷ []) "1John.1.6"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.6"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.1.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.6"
∷ word (ο ∷ ὐ ∷ []) "1John.1.6"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.6"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.1.6"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "1John.1.6"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.1.7"
∷ word (δ ∷ ὲ ∷ []) "1John.1.7"
∷ word (ἐ ∷ ν ∷ []) "1John.1.7"
∷ word (τ ∷ ῷ ∷ []) "1John.1.7"
∷ word (φ ∷ ω ∷ τ ∷ ὶ ∷ []) "1John.1.7"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.7"
∷ word (ὡ ∷ ς ∷ []) "1John.1.7"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ς ∷ []) "1John.1.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.7"
∷ word (ἐ ∷ ν ∷ []) "1John.1.7"
∷ word (τ ∷ ῷ ∷ []) "1John.1.7"
∷ word (φ ∷ ω ∷ τ ∷ ί ∷ []) "1John.1.7"
∷ word (κ ∷ ο ∷ ι ∷ ν ∷ ω ∷ ν ∷ ί ∷ α ∷ ν ∷ []) "1John.1.7"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.7"
∷ word (μ ∷ ε ∷ τ ∷ []) "1John.1.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ω ∷ ν ∷ []) "1John.1.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.7"
∷ word (τ ∷ ὸ ∷ []) "1John.1.7"
∷ word (α ∷ ἷ ∷ μ ∷ α ∷ []) "1John.1.7"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1John.1.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.1.7"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.1.7"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.7"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ρ ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "1John.1.7"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.1.7"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1John.1.7"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "1John.1.7"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1John.1.7"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.1.8"
∷ word (ε ∷ ἴ ∷ π ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.1.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "1John.1.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.1.8"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.8"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1John.1.8"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.8"
∷ word (ἡ ∷ []) "1John.1.8"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "1John.1.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.1.8"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.8"
∷ word (ἐ ∷ ν ∷ []) "1John.1.8"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.8"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.1.9"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.9"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.1.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1John.1.9"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.1.9"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1John.1.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.9"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ς ∷ []) "1John.1.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.1.9"
∷ word (ἀ ∷ φ ∷ ῇ ∷ []) "1John.1.9"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.9"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.1.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1John.1.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.9"
∷ word (κ ∷ α ∷ θ ∷ α ∷ ρ ∷ ί ∷ σ ∷ ῃ ∷ []) "1John.1.9"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.1.9"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1John.1.9"
∷ word (π ∷ ά ∷ σ ∷ η ∷ ς ∷ []) "1John.1.9"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ ς ∷ []) "1John.1.9"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.1.10"
∷ word (ε ∷ ἴ ∷ π ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.1.10"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.1.10"
∷ word (ἡ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.10"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ν ∷ []) "1John.1.10"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.1.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.1.10"
∷ word (ὁ ∷ []) "1John.1.10"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1John.1.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.1.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.1.10"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.1.10"
∷ word (ἐ ∷ ν ∷ []) "1John.1.10"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.1.10"
∷ word (Τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.2.1"
∷ word (μ ∷ ο ∷ υ ∷ []) "1John.2.1"
∷ word (τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1John.2.1"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.1"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.2.1"
∷ word (μ ∷ ὴ ∷ []) "1John.2.1"
∷ word (ἁ ∷ μ ∷ ά ∷ ρ ∷ τ ∷ η ∷ τ ∷ ε ∷ []) "1John.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.1"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1John.2.1"
∷ word (τ ∷ ι ∷ ς ∷ []) "1John.2.1"
∷ word (ἁ ∷ μ ∷ ά ∷ ρ ∷ τ ∷ ῃ ∷ []) "1John.2.1"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ κ ∷ ∙λ ∷ η ∷ τ ∷ ο ∷ ν ∷ []) "1John.2.1"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.1"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.2.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.1"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.2.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "1John.2.1"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.2.1"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ο ∷ ν ∷ []) "1John.2.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.2"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.2.2"
∷ word (ἱ ∷ ∙λ ∷ α ∷ σ ∷ μ ∷ ό ∷ ς ∷ []) "1John.2.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.2"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.2.2"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.2.2"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.2.2"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.2"
∷ word (ο ∷ ὐ ∷ []) "1John.2.2"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.2.2"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.2.2"
∷ word (ἡ ∷ μ ∷ ε ∷ τ ∷ έ ∷ ρ ∷ ω ∷ ν ∷ []) "1John.2.2"
∷ word (δ ∷ ὲ ∷ []) "1John.2.2"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1John.2.2"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1John.2.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.2"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.2.2"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ υ ∷ []) "1John.2.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.2"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.2.2"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "1John.2.3"
∷ word (ἐ ∷ ν ∷ []) "1John.2.3"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.2.3"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.3"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.3"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.2.3"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.2.3"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.2.3"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.2.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.3"
∷ word (τ ∷ η ∷ ρ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.3"
∷ word (ὁ ∷ []) "1John.2.4"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ ν ∷ []) "1John.2.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.4"
∷ word (Ἔ ∷ γ ∷ ν ∷ ω ∷ κ ∷ α ∷ []) "1John.2.4"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.4"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.2.4"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.2.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.4"
∷ word (μ ∷ ὴ ∷ []) "1John.2.4"
∷ word (τ ∷ η ∷ ρ ∷ ῶ ∷ ν ∷ []) "1John.2.4"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ς ∷ []) "1John.2.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.2.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.4"
∷ word (ἐ ∷ ν ∷ []) "1John.2.4"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.2.4"
∷ word (ἡ ∷ []) "1John.2.4"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "1John.2.4"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.4"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.4"
∷ word (ὃ ∷ ς ∷ []) "1John.2.5"
∷ word (δ ∷ []) "1John.2.5"
∷ word (ἂ ∷ ν ∷ []) "1John.2.5"
∷ word (τ ∷ η ∷ ρ ∷ ῇ ∷ []) "1John.2.5"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.5"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ν ∷ []) "1John.2.5"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ῶ ∷ ς ∷ []) "1John.2.5"
∷ word (ἐ ∷ ν ∷ []) "1John.2.5"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.2.5"
∷ word (ἡ ∷ []) "1John.2.5"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.2.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.2.5"
∷ word (τ ∷ ε ∷ τ ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "1John.2.5"
∷ word (ἐ ∷ ν ∷ []) "1John.2.5"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.2.5"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.5"
∷ word (ἐ ∷ ν ∷ []) "1John.2.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.5"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.5"
∷ word (ὁ ∷ []) "1John.2.6"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ ν ∷ []) "1John.2.6"
∷ word (ἐ ∷ ν ∷ []) "1John.2.6"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.6"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "1John.2.6"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ε ∷ ι ∷ []) "1John.2.6"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.2.6"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ς ∷ []) "1John.2.6"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ ε ∷ π ∷ ά ∷ τ ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1John.2.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.6"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.2.6"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ ν ∷ []) "1John.2.6"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.2.7"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.7"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1John.2.7"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὴ ∷ ν ∷ []) "1John.2.7"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.7"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.2.7"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1John.2.7"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ὰ ∷ ν ∷ []) "1John.2.7"
∷ word (ἣ ∷ ν ∷ []) "1John.2.7"
∷ word (ε ∷ ἴ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.7"
∷ word (ἀ ∷ π ∷ []) "1John.2.7"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.2.7"
∷ word (ἡ ∷ []) "1John.2.7"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "1John.2.7"
∷ word (ἡ ∷ []) "1John.2.7"
∷ word (π ∷ α ∷ ∙λ ∷ α ∷ ι ∷ ά ∷ []) "1John.2.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.7"
∷ word (ὁ ∷ []) "1John.2.7"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1John.2.7"
∷ word (ὃ ∷ ν ∷ []) "1John.2.7"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.7"
∷ word (π ∷ ά ∷ ∙λ ∷ ι ∷ ν ∷ []) "1John.2.8"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1John.2.8"
∷ word (κ ∷ α ∷ ι ∷ ν ∷ ὴ ∷ ν ∷ []) "1John.2.8"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.8"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.8"
∷ word (ὅ ∷ []) "1John.2.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ὲ ∷ ς ∷ []) "1John.2.8"
∷ word (ἐ ∷ ν ∷ []) "1John.2.8"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.8"
∷ word (ἐ ∷ ν ∷ []) "1John.2.8"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.8"
∷ word (ἡ ∷ []) "1John.2.8"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ α ∷ []) "1John.2.8"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ γ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1John.2.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.8"
∷ word (τ ∷ ὸ ∷ []) "1John.2.8"
∷ word (φ ∷ ῶ ∷ ς ∷ []) "1John.2.8"
∷ word (τ ∷ ὸ ∷ []) "1John.2.8"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ι ∷ ν ∷ ὸ ∷ ν ∷ []) "1John.2.8"
∷ word (ἤ ∷ δ ∷ η ∷ []) "1John.2.8"
∷ word (φ ∷ α ∷ ί ∷ ν ∷ ε ∷ ι ∷ []) "1John.2.8"
∷ word (ὁ ∷ []) "1John.2.9"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ ν ∷ []) "1John.2.9"
∷ word (ἐ ∷ ν ∷ []) "1John.2.9"
∷ word (τ ∷ ῷ ∷ []) "1John.2.9"
∷ word (φ ∷ ω ∷ τ ∷ ὶ ∷ []) "1John.2.9"
∷ word (ε ∷ ἶ ∷ ν ∷ α ∷ ι ∷ []) "1John.2.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.9"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.2.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.9"
∷ word (μ ∷ ι ∷ σ ∷ ῶ ∷ ν ∷ []) "1John.2.9"
∷ word (ἐ ∷ ν ∷ []) "1John.2.9"
∷ word (τ ∷ ῇ ∷ []) "1John.2.9"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ ᾳ ∷ []) "1John.2.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.2.9"
∷ word (ἕ ∷ ω ∷ ς ∷ []) "1John.2.9"
∷ word (ἄ ∷ ρ ∷ τ ∷ ι ∷ []) "1John.2.9"
∷ word (ὁ ∷ []) "1John.2.10"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.2.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.10"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.2.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.10"
∷ word (ἐ ∷ ν ∷ []) "1John.2.10"
∷ word (τ ∷ ῷ ∷ []) "1John.2.10"
∷ word (φ ∷ ω ∷ τ ∷ ὶ ∷ []) "1John.2.10"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.2.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.10"
∷ word (σ ∷ κ ∷ ά ∷ ν ∷ δ ∷ α ∷ ∙λ ∷ ο ∷ ν ∷ []) "1John.2.10"
∷ word (ἐ ∷ ν ∷ []) "1John.2.10"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.10"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.10"
∷ word (ὁ ∷ []) "1John.2.11"
∷ word (δ ∷ ὲ ∷ []) "1John.2.11"
∷ word (μ ∷ ι ∷ σ ∷ ῶ ∷ ν ∷ []) "1John.2.11"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.11"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.2.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.11"
∷ word (ἐ ∷ ν ∷ []) "1John.2.11"
∷ word (τ ∷ ῇ ∷ []) "1John.2.11"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ ᾳ ∷ []) "1John.2.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.2.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.11"
∷ word (ἐ ∷ ν ∷ []) "1John.2.11"
∷ word (τ ∷ ῇ ∷ []) "1John.2.11"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ ᾳ ∷ []) "1John.2.11"
∷ word (π ∷ ε ∷ ρ ∷ ι ∷ π ∷ α ∷ τ ∷ ε ∷ ῖ ∷ []) "1John.2.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.11"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.11"
∷ word (ο ∷ ἶ ∷ δ ∷ ε ∷ ν ∷ []) "1John.2.11"
∷ word (π ∷ ο ∷ ῦ ∷ []) "1John.2.11"
∷ word (ὑ ∷ π ∷ ά ∷ γ ∷ ε ∷ ι ∷ []) "1John.2.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.11"
∷ word (ἡ ∷ []) "1John.2.11"
∷ word (σ ∷ κ ∷ ο ∷ τ ∷ ί ∷ α ∷ []) "1John.2.11"
∷ word (ἐ ∷ τ ∷ ύ ∷ φ ∷ ∙λ ∷ ω ∷ σ ∷ ε ∷ ν ∷ []) "1John.2.11"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1John.2.11"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ο ∷ ὺ ∷ ς ∷ []) "1John.2.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.11"
∷ word (Γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.12"
∷ word (τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.2.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.12"
∷ word (ἀ ∷ φ ∷ έ ∷ ω ∷ ν ∷ τ ∷ α ∷ ι ∷ []) "1John.2.12"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.12"
∷ word (α ∷ ἱ ∷ []) "1John.2.12"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ι ∷ []) "1John.2.12"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1John.2.12"
∷ word (τ ∷ ὸ ∷ []) "1John.2.12"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "1John.2.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.12"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.13"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "1John.2.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.13"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.13"
∷ word (ἀ ∷ π ∷ []) "1John.2.13"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.2.13"
∷ word (γ ∷ ρ ∷ ά ∷ φ ∷ ω ∷ []) "1John.2.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.13"
∷ word (ν ∷ ε ∷ α ∷ ν ∷ ί ∷ σ ∷ κ ∷ ο ∷ ι ∷ []) "1John.2.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.13"
∷ word (ν ∷ ε ∷ ν ∷ ι ∷ κ ∷ ή ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.13"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.13"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ό ∷ ν ∷ []) "1John.2.13"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.2.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.14"
∷ word (π ∷ α ∷ ι ∷ δ ∷ ί ∷ α ∷ []) "1John.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.14"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.14"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.2.14"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.2.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.14"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ ε ∷ ς ∷ []) "1John.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.14"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.14"
∷ word (ἀ ∷ π ∷ []) "1John.2.14"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.2.14"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.2.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.14"
∷ word (ν ∷ ε ∷ α ∷ ν ∷ ί ∷ σ ∷ κ ∷ ο ∷ ι ∷ []) "1John.2.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.14"
∷ word (ἰ ∷ σ ∷ χ ∷ υ ∷ ρ ∷ ο ∷ ί ∷ []) "1John.2.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1John.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.14"
∷ word (ὁ ∷ []) "1John.2.14"
∷ word (∙λ ∷ ό ∷ γ ∷ ο ∷ ς ∷ []) "1John.2.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.14"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.2.14"
∷ word (ἐ ∷ ν ∷ []) "1John.2.14"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.14"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.2.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.14"
∷ word (ν ∷ ε ∷ ν ∷ ι ∷ κ ∷ ή ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.14"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ό ∷ ν ∷ []) "1John.2.14"
∷ word (Μ ∷ ὴ ∷ []) "1John.2.15"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ τ ∷ ε ∷ []) "1John.2.15"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.15"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.2.15"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1John.2.15"
∷ word (τ ∷ ὰ ∷ []) "1John.2.15"
∷ word (ἐ ∷ ν ∷ []) "1John.2.15"
∷ word (τ ∷ ῷ ∷ []) "1John.2.15"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1John.2.15"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1John.2.15"
∷ word (τ ∷ ι ∷ ς ∷ []) "1John.2.15"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "1John.2.15"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.15"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.2.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.15"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.15"
∷ word (ἡ ∷ []) "1John.2.15"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.2.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.15"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.2.15"
∷ word (ἐ ∷ ν ∷ []) "1John.2.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.16"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1John.2.16"
∷ word (τ ∷ ὸ ∷ []) "1John.2.16"
∷ word (ἐ ∷ ν ∷ []) "1John.2.16"
∷ word (τ ∷ ῷ ∷ []) "1John.2.16"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1John.2.16"
∷ word (ἡ ∷ []) "1John.2.16"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "1John.2.16"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.2.16"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὸ ∷ ς ∷ []) "1John.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.16"
∷ word (ἡ ∷ []) "1John.2.16"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "1John.2.16"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.2.16"
∷ word (ὀ ∷ φ ∷ θ ∷ α ∷ ∙λ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.16"
∷ word (ἡ ∷ []) "1John.2.16"
∷ word (ἀ ∷ ∙λ ∷ α ∷ ζ ∷ ο ∷ ν ∷ ε ∷ ί ∷ α ∷ []) "1John.2.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.16"
∷ word (β ∷ ί ∷ ο ∷ υ ∷ []) "1John.2.16"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.16"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.16"
∷ word (ἐ ∷ κ ∷ []) "1John.2.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.16"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ό ∷ ς ∷ []) "1John.2.16"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1John.2.16"
∷ word (ἐ ∷ κ ∷ []) "1John.2.16"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.16"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.2.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.2.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.17"
∷ word (ὁ ∷ []) "1John.2.17"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1John.2.17"
∷ word (π ∷ α ∷ ρ ∷ ά ∷ γ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1John.2.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.17"
∷ word (ἡ ∷ []) "1John.2.17"
∷ word (ἐ ∷ π ∷ ι ∷ θ ∷ υ ∷ μ ∷ ί ∷ α ∷ []) "1John.2.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.17"
∷ word (ὁ ∷ []) "1John.2.17"
∷ word (δ ∷ ὲ ∷ []) "1John.2.17"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.2.17"
∷ word (τ ∷ ὸ ∷ []) "1John.2.17"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "1John.2.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.2.17"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.2.17"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.2.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.17"
∷ word (α ∷ ἰ ∷ ῶ ∷ ν ∷ α ∷ []) "1John.2.17"
∷ word (Π ∷ α ∷ ι ∷ δ ∷ ί ∷ α ∷ []) "1John.2.18"
∷ word (ἐ ∷ σ ∷ χ ∷ ά ∷ τ ∷ η ∷ []) "1John.2.18"
∷ word (ὥ ∷ ρ ∷ α ∷ []) "1John.2.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.2.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.18"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.2.18"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.18"
∷ word (ἀ ∷ ν ∷ τ ∷ ί ∷ χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1John.2.18"
∷ word (ἔ ∷ ρ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1John.2.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.18"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1John.2.18"
∷ word (ἀ ∷ ν ∷ τ ∷ ί ∷ χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ι ∷ []) "1John.2.18"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1John.2.18"
∷ word (γ ∷ ε ∷ γ ∷ ό ∷ ν ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "1John.2.18"
∷ word (ὅ ∷ θ ∷ ε ∷ ν ∷ []) "1John.2.18"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.18"
∷ word (ἐ ∷ σ ∷ χ ∷ ά ∷ τ ∷ η ∷ []) "1John.2.18"
∷ word (ὥ ∷ ρ ∷ α ∷ []) "1John.2.18"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.2.18"
∷ word (ἐ ∷ ξ ∷ []) "1John.2.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.19"
∷ word (ἐ ∷ ξ ∷ ῆ ∷ ∙λ ∷ θ ∷ α ∷ ν ∷ []) "1John.2.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.2.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.19"
∷ word (ἦ ∷ σ ∷ α ∷ ν ∷ []) "1John.2.19"
∷ word (ἐ ∷ ξ ∷ []) "1John.2.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.19"
∷ word (ε ∷ ἰ ∷ []) "1John.2.19"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1John.2.19"
∷ word (ἐ ∷ ξ ∷ []) "1John.2.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.19"
∷ word (ἦ ∷ σ ∷ α ∷ ν ∷ []) "1John.2.19"
∷ word (μ ∷ ε ∷ μ ∷ ε ∷ ν ∷ ή ∷ κ ∷ ε ∷ ι ∷ σ ∷ α ∷ ν ∷ []) "1John.2.19"
∷ word (ἂ ∷ ν ∷ []) "1John.2.19"
∷ word (μ ∷ ε ∷ θ ∷ []) "1John.2.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.19"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.2.19"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.2.19"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῶ ∷ σ ∷ ι ∷ ν ∷ []) "1John.2.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.19"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.19"
∷ word (ε ∷ ἰ ∷ σ ∷ ὶ ∷ ν ∷ []) "1John.2.19"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1John.2.19"
∷ word (ἐ ∷ ξ ∷ []) "1John.2.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.2.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.20"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.2.20"
∷ word (χ ∷ ρ ∷ ῖ ∷ σ ∷ μ ∷ α ∷ []) "1John.2.20"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.20"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1John.2.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.2.20"
∷ word (ἁ ∷ γ ∷ ί ∷ ο ∷ υ ∷ []) "1John.2.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.20"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1John.2.20"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.21"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.2.21"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.21"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.21"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.2.21"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ ν ∷ []) "1John.2.21"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.2.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.21"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.21"
∷ word (α ∷ ὐ ∷ τ ∷ ή ∷ ν ∷ []) "1John.2.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.21"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.21"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1John.2.21"
∷ word (ψ ∷ ε ∷ ῦ ∷ δ ∷ ο ∷ ς ∷ []) "1John.2.21"
∷ word (ἐ ∷ κ ∷ []) "1John.2.21"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.2.21"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1John.2.21"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.21"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.21"
∷ word (τ ∷ ί ∷ ς ∷ []) "1John.2.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.22"
∷ word (ὁ ∷ []) "1John.2.22"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ς ∷ []) "1John.2.22"
∷ word (ε ∷ ἰ ∷ []) "1John.2.22"
∷ word (μ ∷ ὴ ∷ []) "1John.2.22"
∷ word (ὁ ∷ []) "1John.2.22"
∷ word (ἀ ∷ ρ ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1John.2.22"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.22"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1John.2.22"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.22"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.22"
∷ word (ὁ ∷ []) "1John.2.22"
∷ word (χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1John.2.22"
∷ word (ο ∷ ὗ ∷ τ ∷ ό ∷ ς ∷ []) "1John.2.22"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.22"
∷ word (ὁ ∷ []) "1John.2.22"
∷ word (ἀ ∷ ν ∷ τ ∷ ί ∷ χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ς ∷ []) "1John.2.22"
∷ word (ὁ ∷ []) "1John.2.22"
∷ word (ἀ ∷ ρ ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1John.2.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.22"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.2.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.22"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.22"
∷ word (υ ∷ ἱ ∷ ό ∷ ν ∷ []) "1John.2.22"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.2.23"
∷ word (ὁ ∷ []) "1John.2.23"
∷ word (ἀ ∷ ρ ∷ ν ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1John.2.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1John.2.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.2.23"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.2.23"
∷ word (ὁ ∷ []) "1John.2.23"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ῶ ∷ ν ∷ []) "1John.2.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.23"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.2.23"
∷ word (π ∷ α ∷ τ ∷ έ ∷ ρ ∷ α ∷ []) "1John.2.23"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.2.23"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.2.24"
∷ word (ὃ ∷ []) "1John.2.24"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.24"
∷ word (ἀ ∷ π ∷ []) "1John.2.24"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.2.24"
∷ word (ἐ ∷ ν ∷ []) "1John.2.24"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.24"
∷ word (μ ∷ ε ∷ ν ∷ έ ∷ τ ∷ ω ∷ []) "1John.2.24"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.2.24"
∷ word (ἐ ∷ ν ∷ []) "1John.2.24"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.24"
∷ word (μ ∷ ε ∷ ί ∷ ν ∷ ῃ ∷ []) "1John.2.24"
∷ word (ὃ ∷ []) "1John.2.24"
∷ word (ἀ ∷ π ∷ []) "1John.2.24"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.2.24"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1John.2.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.24"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.2.24"
∷ word (ἐ ∷ ν ∷ []) "1John.2.24"
∷ word (τ ∷ ῷ ∷ []) "1John.2.24"
∷ word (υ ∷ ἱ ∷ ῷ ∷ []) "1John.2.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.24"
∷ word (ἐ ∷ ν ∷ []) "1John.2.24"
∷ word (τ ∷ ῷ ∷ []) "1John.2.24"
∷ word (π ∷ α ∷ τ ∷ ρ ∷ ὶ ∷ []) "1John.2.24"
∷ word (μ ∷ ε ∷ ν ∷ ε ∷ ῖ ∷ τ ∷ ε ∷ []) "1John.2.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.25"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.2.25"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.2.25"
∷ word (ἡ ∷ []) "1John.2.25"
∷ word (ἐ ∷ π ∷ α ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "1John.2.25"
∷ word (ἣ ∷ ν ∷ []) "1John.2.25"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.2.25"
∷ word (ἐ ∷ π ∷ η ∷ γ ∷ γ ∷ ε ∷ ί ∷ ∙λ ∷ α ∷ τ ∷ ο ∷ []) "1John.2.25"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.2.25"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.2.25"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.2.25"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "1John.2.25"
∷ word (Τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1John.2.26"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.2.26"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.26"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.2.26"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.2.26"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ώ ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1John.2.26"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.2.26"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.27"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.2.27"
∷ word (τ ∷ ὸ ∷ []) "1John.2.27"
∷ word (χ ∷ ρ ∷ ῖ ∷ σ ∷ μ ∷ α ∷ []) "1John.2.27"
∷ word (ὃ ∷ []) "1John.2.27"
∷ word (ἐ ∷ ∙λ ∷ ά ∷ β ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.27"
∷ word (ἀ ∷ π ∷ []) "1John.2.27"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.27"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.2.27"
∷ word (ἐ ∷ ν ∷ []) "1John.2.27"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.2.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.27"
∷ word (ο ∷ ὐ ∷ []) "1John.2.27"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1John.2.27"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.27"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.2.27"
∷ word (τ ∷ ι ∷ ς ∷ []) "1John.2.27"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ῃ ∷ []) "1John.2.27"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.2.27"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.2.27"
∷ word (ὡ ∷ ς ∷ []) "1John.2.27"
∷ word (τ ∷ ὸ ∷ []) "1John.2.27"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.27"
∷ word (χ ∷ ρ ∷ ῖ ∷ σ ∷ μ ∷ α ∷ []) "1John.2.27"
∷ word (δ ∷ ι ∷ δ ∷ ά ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1John.2.27"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.2.27"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.2.27"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ ω ∷ ν ∷ []) "1John.2.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.27"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ έ ∷ ς ∷ []) "1John.2.27"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.27"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.2.27"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.27"
∷ word (ψ ∷ ε ∷ ῦ ∷ δ ∷ ο ∷ ς ∷ []) "1John.2.27"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.27"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.2.27"
∷ word (ἐ ∷ δ ∷ ί ∷ δ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "1John.2.27"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.2.27"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.27"
∷ word (ἐ ∷ ν ∷ []) "1John.2.27"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.27"
∷ word (Κ ∷ α ∷ ὶ ∷ []) "1John.2.28"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1John.2.28"
∷ word (τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.2.28"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.28"
∷ word (ἐ ∷ ν ∷ []) "1John.2.28"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.2.28"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.2.28"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.2.28"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "1John.2.28"
∷ word (σ ∷ χ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.28"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1John.2.28"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.2.28"
∷ word (μ ∷ ὴ ∷ []) "1John.2.28"
∷ word (α ∷ ἰ ∷ σ ∷ χ ∷ υ ∷ ν ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.2.28"
∷ word (ἀ ∷ π ∷ []) "1John.2.28"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.28"
∷ word (ἐ ∷ ν ∷ []) "1John.2.28"
∷ word (τ ∷ ῇ ∷ []) "1John.2.28"
∷ word (π ∷ α ∷ ρ ∷ ο ∷ υ ∷ σ ∷ ί ∷ ᾳ ∷ []) "1John.2.28"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.28"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.2.29"
∷ word (ε ∷ ἰ ∷ δ ∷ ῆ ∷ τ ∷ ε ∷ []) "1John.2.29"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.29"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ό ∷ ς ∷ []) "1John.2.29"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.2.29"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "1John.2.29"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.2.29"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.2.29"
∷ word (ὁ ∷ []) "1John.2.29"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.2.29"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.2.29"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "1John.2.29"
∷ word (ἐ ∷ ξ ∷ []) "1John.2.29"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.2.29"
∷ word (γ ∷ ε ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1John.2.29"
∷ word (ἴ ∷ δ ∷ ε ∷ τ ∷ ε ∷ []) "1John.3.1"
∷ word (π ∷ ο ∷ τ ∷ α ∷ π ∷ ὴ ∷ ν ∷ []) "1John.3.1"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1John.3.1"
∷ word (δ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.1"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.3.1"
∷ word (ὁ ∷ []) "1John.3.1"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "1John.3.1"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.3.1"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1John.3.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.1"
∷ word (κ ∷ ∙λ ∷ η ∷ θ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.1"
∷ word (ἐ ∷ σ ∷ μ ∷ έ ∷ ν ∷ []) "1John.3.1"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1John.3.1"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1John.3.1"
∷ word (ὁ ∷ []) "1John.3.1"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1John.3.1"
∷ word (ο ∷ ὐ ∷ []) "1John.3.1"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1John.3.1"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.3.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.1"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.3.1"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "1John.3.1"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.3.1"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.3.2"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1John.3.2"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1John.3.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.2"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.2"
∷ word (ο ∷ ὔ ∷ π ∷ ω ∷ []) "1John.3.2"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.3.2"
∷ word (τ ∷ ί ∷ []) "1John.3.2"
∷ word (ἐ ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.3.2"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.2"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.3.2"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ω ∷ θ ∷ ῇ ∷ []) "1John.3.2"
∷ word (ὅ ∷ μ ∷ ο ∷ ι ∷ ο ∷ ι ∷ []) "1John.3.2"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.2"
∷ word (ἐ ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.3.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.2"
∷ word (ὀ ∷ ψ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.3.2"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.3.2"
∷ word (κ ∷ α ∷ θ ∷ ώ ∷ ς ∷ []) "1John.3.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.3"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.3"
∷ word (ὁ ∷ []) "1John.3.3"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1John.3.3"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.3"
∷ word (ἐ ∷ ∙λ ∷ π ∷ ί ∷ δ ∷ α ∷ []) "1John.3.3"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "1John.3.3"
∷ word (ἐ ∷ π ∷ []) "1John.3.3"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.3"
∷ word (ἁ ∷ γ ∷ ν ∷ ί ∷ ζ ∷ ε ∷ ι ∷ []) "1John.3.3"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.3.3"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.3.3"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.3"
∷ word (ἁ ∷ γ ∷ ν ∷ ό ∷ ς ∷ []) "1John.3.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.3"
∷ word (Π ∷ ᾶ ∷ ς ∷ []) "1John.3.4"
∷ word (ὁ ∷ []) "1John.3.4"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.3.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.4"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.4"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.4"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.4"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ []) "1John.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.4"
∷ word (ἡ ∷ []) "1John.3.4"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1John.3.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.3.4"
∷ word (ἡ ∷ []) "1John.3.4"
∷ word (ἀ ∷ ν ∷ ο ∷ μ ∷ ί ∷ α ∷ []) "1John.3.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.5"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1John.3.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.5"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.5"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.3.5"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.3.5"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.3.5"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ς ∷ []) "1John.3.5"
∷ word (ἄ ∷ ρ ∷ ῃ ∷ []) "1John.3.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.5"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1John.3.5"
∷ word (ἐ ∷ ν ∷ []) "1John.3.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.5"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.3.5"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.5"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.6"
∷ word (ὁ ∷ []) "1John.3.6"
∷ word (ἐ ∷ ν ∷ []) "1John.3.6"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.6"
∷ word (μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1John.3.6"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.3.6"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.6"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.6"
∷ word (ὁ ∷ []) "1John.3.6"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ω ∷ ν ∷ []) "1John.3.6"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.3.6"
∷ word (ἑ ∷ ώ ∷ ρ ∷ α ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.6"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.3.6"
∷ word (ο ∷ ὐ ∷ δ ∷ ὲ ∷ []) "1John.3.6"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.6"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.3.6"
∷ word (τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.3.7"
∷ word (μ ∷ η ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1John.3.7"
∷ word (π ∷ ∙λ ∷ α ∷ ν ∷ ά ∷ τ ∷ ω ∷ []) "1John.3.7"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.3.7"
∷ word (ὁ ∷ []) "1John.3.7"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.3.7"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.7"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "1John.3.7"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ό ∷ ς ∷ []) "1John.3.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.7"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.3.7"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.7"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ ό ∷ ς ∷ []) "1John.3.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.7"
∷ word (ὁ ∷ []) "1John.3.8"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.3.8"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.8"
∷ word (ἐ ∷ κ ∷ []) "1John.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.8"
∷ word (δ ∷ ι ∷ α ∷ β ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "1John.3.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.3.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.8"
∷ word (ἀ ∷ π ∷ []) "1John.3.8"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.3.8"
∷ word (ὁ ∷ []) "1John.3.8"
∷ word (δ ∷ ι ∷ ά ∷ β ∷ ο ∷ ∙λ ∷ ο ∷ ς ∷ []) "1John.3.8"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1John.3.8"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.3.8"
∷ word (ὁ ∷ []) "1John.3.8"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "1John.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.8"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.8"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.3.8"
∷ word (∙λ ∷ ύ ∷ σ ∷ ῃ ∷ []) "1John.3.8"
∷ word (τ ∷ ὰ ∷ []) "1John.3.8"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "1John.3.8"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.8"
∷ word (δ ∷ ι ∷ α ∷ β ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "1John.3.8"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.9"
∷ word (ὁ ∷ []) "1John.3.9"
∷ word (γ ∷ ε ∷ γ ∷ ε ∷ ν ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.9"
∷ word (ἐ ∷ κ ∷ []) "1John.3.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.9"
∷ word (ο ∷ ὐ ∷ []) "1John.3.9"
∷ word (π ∷ ο ∷ ι ∷ ε ∷ ῖ ∷ []) "1John.3.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.9"
∷ word (σ ∷ π ∷ έ ∷ ρ ∷ μ ∷ α ∷ []) "1John.3.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.9"
∷ word (ἐ ∷ ν ∷ []) "1John.3.9"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.9"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.9"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.9"
∷ word (ο ∷ ὐ ∷ []) "1John.3.9"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1John.3.9"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ ν ∷ []) "1John.3.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.9"
∷ word (ἐ ∷ κ ∷ []) "1John.3.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.9"
∷ word (γ ∷ ε ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1John.3.9"
∷ word (ἐ ∷ ν ∷ []) "1John.3.10"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.3.10"
∷ word (φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ά ∷ []) "1John.3.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.10"
∷ word (τ ∷ ὰ ∷ []) "1John.3.10"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1John.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.10"
∷ word (τ ∷ ὰ ∷ []) "1John.3.10"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1John.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (δ ∷ ι ∷ α ∷ β ∷ ό ∷ ∙λ ∷ ο ∷ υ ∷ []) "1John.3.10"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.10"
∷ word (ὁ ∷ []) "1John.3.10"
∷ word (μ ∷ ὴ ∷ []) "1John.3.10"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.3.10"
∷ word (δ ∷ ι ∷ κ ∷ α ∷ ι ∷ ο ∷ σ ∷ ύ ∷ ν ∷ η ∷ ν ∷ []) "1John.3.10"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.3.10"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.3.10"
∷ word (ἐ ∷ κ ∷ []) "1John.3.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.10"
∷ word (ὁ ∷ []) "1John.3.10"
∷ word (μ ∷ ὴ ∷ []) "1John.3.10"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.3.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.10"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.3.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.10"
∷ word (Ὅ ∷ τ ∷ ι ∷ []) "1John.3.11"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.3.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.3.11"
∷ word (ἡ ∷ []) "1John.3.11"
∷ word (ἀ ∷ γ ∷ γ ∷ ε ∷ ∙λ ∷ ί ∷ α ∷ []) "1John.3.11"
∷ word (ἣ ∷ ν ∷ []) "1John.3.11"
∷ word (ἠ ∷ κ ∷ ο ∷ ύ ∷ σ ∷ α ∷ τ ∷ ε ∷ []) "1John.3.11"
∷ word (ἀ ∷ π ∷ []) "1John.3.11"
∷ word (ἀ ∷ ρ ∷ χ ∷ ῆ ∷ ς ∷ []) "1John.3.11"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.3.11"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1John.3.11"
∷ word (ο ∷ ὐ ∷ []) "1John.3.12"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.3.12"
∷ word (Κ ∷ ά ∷ ϊ ∷ ν ∷ []) "1John.3.12"
∷ word (ἐ ∷ κ ∷ []) "1John.3.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (ἦ ∷ ν ∷ []) "1John.3.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.12"
∷ word (ἔ ∷ σ ∷ φ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "1John.3.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.12"
∷ word (χ ∷ ά ∷ ρ ∷ ι ∷ ν ∷ []) "1John.3.12"
∷ word (τ ∷ ί ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.12"
∷ word (ἔ ∷ σ ∷ φ ∷ α ∷ ξ ∷ ε ∷ ν ∷ []) "1John.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.3.12"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.12"
∷ word (τ ∷ ὰ ∷ []) "1John.3.12"
∷ word (ἔ ∷ ρ ∷ γ ∷ α ∷ []) "1John.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ὰ ∷ []) "1John.3.12"
∷ word (ἦ ∷ ν ∷ []) "1John.3.12"
∷ word (τ ∷ ὰ ∷ []) "1John.3.12"
∷ word (δ ∷ ὲ ∷ []) "1John.3.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.12"
∷ word (δ ∷ ί ∷ κ ∷ α ∷ ι ∷ α ∷ []) "1John.3.12"
∷ word (μ ∷ ὴ ∷ []) "1John.3.13"
∷ word (θ ∷ α ∷ υ ∷ μ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "1John.3.13"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ί ∷ []) "1John.3.13"
∷ word (ε ∷ ἰ ∷ []) "1John.3.13"
∷ word (μ ∷ ι ∷ σ ∷ ε ∷ ῖ ∷ []) "1John.3.13"
∷ word (ὑ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.3.13"
∷ word (ὁ ∷ []) "1John.3.13"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1John.3.13"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.3.14"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.14"
∷ word (μ ∷ ε ∷ τ ∷ α ∷ β ∷ ε ∷ β ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.14"
∷ word (ἐ ∷ κ ∷ []) "1John.3.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.14"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ο ∷ υ ∷ []) "1John.3.14"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.3.14"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.14"
∷ word (ζ ∷ ω ∷ ή ∷ ν ∷ []) "1John.3.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.14"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.14"
∷ word (τ ∷ ο ∷ ὺ ∷ ς ∷ []) "1John.3.14"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ο ∷ ύ ∷ ς ∷ []) "1John.3.14"
∷ word (ὁ ∷ []) "1John.3.14"
∷ word (μ ∷ ὴ ∷ []) "1John.3.14"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.3.14"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.14"
∷ word (ἐ ∷ ν ∷ []) "1John.3.14"
∷ word (τ ∷ ῷ ∷ []) "1John.3.14"
∷ word (θ ∷ α ∷ ν ∷ ά ∷ τ ∷ ῳ ∷ []) "1John.3.14"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.15"
∷ word (ὁ ∷ []) "1John.3.15"
∷ word (μ ∷ ι ∷ σ ∷ ῶ ∷ ν ∷ []) "1John.3.15"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.15"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.3.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.15"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ κ ∷ τ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.3.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.15"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ τ ∷ ε ∷ []) "1John.3.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.15"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.3.15"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ω ∷ π ∷ ο ∷ κ ∷ τ ∷ ό ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.15"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.3.15"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.3.15"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.3.15"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "1John.3.15"
∷ word (ἐ ∷ ν ∷ []) "1John.3.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.15"
∷ word (μ ∷ έ ∷ ν ∷ ο ∷ υ ∷ σ ∷ α ∷ ν ∷ []) "1John.3.15"
∷ word (ἐ ∷ ν ∷ []) "1John.3.16"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.3.16"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.16"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1John.3.16"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.16"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ο ∷ ς ∷ []) "1John.3.16"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1John.3.16"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.3.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.16"
∷ word (ψ ∷ υ ∷ χ ∷ ὴ ∷ ν ∷ []) "1John.3.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.16"
∷ word (ἔ ∷ θ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.3.16"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.16"
∷ word (ὑ ∷ π ∷ ὲ ∷ ρ ∷ []) "1John.3.16"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.3.16"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ῶ ∷ ν ∷ []) "1John.3.16"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.3.16"
∷ word (ψ ∷ υ ∷ χ ∷ ὰ ∷ ς ∷ []) "1John.3.16"
∷ word (θ ∷ ε ∷ ῖ ∷ ν ∷ α ∷ ι ∷ []) "1John.3.16"
∷ word (ὃ ∷ ς ∷ []) "1John.3.17"
∷ word (δ ∷ []) "1John.3.17"
∷ word (ἂ ∷ ν ∷ []) "1John.3.17"
∷ word (ἔ ∷ χ ∷ ῃ ∷ []) "1John.3.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.17"
∷ word (β ∷ ί ∷ ο ∷ ν ∷ []) "1John.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.3.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.17"
∷ word (θ ∷ ε ∷ ω ∷ ρ ∷ ῇ ∷ []) "1John.3.17"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.17"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.3.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (χ ∷ ρ ∷ ε ∷ ί ∷ α ∷ ν ∷ []) "1John.3.17"
∷ word (ἔ ∷ χ ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "1John.3.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.17"
∷ word (κ ∷ ∙λ ∷ ε ∷ ί ∷ σ ∷ ῃ ∷ []) "1John.3.17"
∷ word (τ ∷ ὰ ∷ []) "1John.3.17"
∷ word (σ ∷ π ∷ ∙λ ∷ ά ∷ γ ∷ χ ∷ ν ∷ α ∷ []) "1John.3.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (ἀ ∷ π ∷ []) "1John.3.17"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (π ∷ ῶ ∷ ς ∷ []) "1John.3.17"
∷ word (ἡ ∷ []) "1John.3.17"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.3.17"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.3.17"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.17"
∷ word (ἐ ∷ ν ∷ []) "1John.3.17"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.17"
∷ word (Τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.3.18"
∷ word (μ ∷ ὴ ∷ []) "1John.3.18"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.18"
∷ word (∙λ ∷ ό ∷ γ ∷ ῳ ∷ []) "1John.3.18"
∷ word (μ ∷ η ∷ δ ∷ ὲ ∷ []) "1John.3.18"
∷ word (τ ∷ ῇ ∷ []) "1John.3.18"
∷ word (γ ∷ ∙λ ∷ ώ ∷ σ ∷ σ ∷ ῃ ∷ []) "1John.3.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1John.3.18"
∷ word (ἐ ∷ ν ∷ []) "1John.3.18"
∷ word (ἔ ∷ ρ ∷ γ ∷ ῳ ∷ []) "1John.3.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.18"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ ᾳ ∷ []) "1John.3.18"
∷ word (ἐ ∷ ν ∷ []) "1John.3.19"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.3.19"
∷ word (γ ∷ ν ∷ ω ∷ σ ∷ ό ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.3.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.19"
∷ word (ἐ ∷ κ ∷ []) "1John.3.19"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.3.19"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1John.3.19"
∷ word (ἐ ∷ σ ∷ μ ∷ έ ∷ ν ∷ []) "1John.3.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.19"
∷ word (ἔ ∷ μ ∷ π ∷ ρ ∷ ο ∷ σ ∷ θ ∷ ε ∷ ν ∷ []) "1John.3.19"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.19"
∷ word (π ∷ ε ∷ ί ∷ σ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.19"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.3.19"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.19"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.3.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.20"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.3.20"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ῃ ∷ []) "1John.3.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.3.20"
∷ word (ἡ ∷ []) "1John.3.20"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ []) "1John.3.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.20"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1John.3.20"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.3.20"
∷ word (ὁ ∷ []) "1John.3.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.3.20"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.3.20"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ ς ∷ []) "1John.3.20"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.3.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.20"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1John.3.20"
∷ word (π ∷ ά ∷ ν ∷ τ ∷ α ∷ []) "1John.3.20"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.3.21"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.3.21"
∷ word (ἡ ∷ []) "1John.3.21"
∷ word (κ ∷ α ∷ ρ ∷ δ ∷ ί ∷ α ∷ []) "1John.3.21"
∷ word (μ ∷ ὴ ∷ []) "1John.3.21"
∷ word (κ ∷ α ∷ τ ∷ α ∷ γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ῃ ∷ []) "1John.3.21"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.3.21"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1John.3.21"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.21"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.3.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.3.21"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1John.3.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.22"
∷ word (ὃ ∷ []) "1John.3.22"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.3.22"
∷ word (α ∷ ἰ ∷ τ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.22"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.22"
∷ word (ἀ ∷ π ∷ []) "1John.3.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.22"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.22"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.3.22"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.3.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.22"
∷ word (τ ∷ η ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.22"
∷ word (τ ∷ ὰ ∷ []) "1John.3.22"
∷ word (ἀ ∷ ρ ∷ ε ∷ σ ∷ τ ∷ ὰ ∷ []) "1John.3.22"
∷ word (ἐ ∷ ν ∷ ώ ∷ π ∷ ι ∷ ο ∷ ν ∷ []) "1John.3.22"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.22"
∷ word (π ∷ ο ∷ ι ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.22"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.23"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.3.23"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.3.23"
∷ word (ἡ ∷ []) "1John.3.23"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ []) "1John.3.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.3.23"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.23"
∷ word (τ ∷ ῷ ∷ []) "1John.3.23"
∷ word (ὀ ∷ ν ∷ ό ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1John.3.23"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.23"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.23"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1John.3.23"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.3.23"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.23"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1John.3.23"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.3.23"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.24"
∷ word (ὁ ∷ []) "1John.3.24"
∷ word (τ ∷ η ∷ ρ ∷ ῶ ∷ ν ∷ []) "1John.3.24"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.3.24"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.3.24"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.3.24"
∷ word (ἐ ∷ ν ∷ []) "1John.3.24"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.24"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.24"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.3.24"
∷ word (ἐ ∷ ν ∷ []) "1John.3.24"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.3.24"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.3.24"
∷ word (ἐ ∷ ν ∷ []) "1John.3.24"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.3.24"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.3.24"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.3.24"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.3.24"
∷ word (ἐ ∷ ν ∷ []) "1John.3.24"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.3.24"
∷ word (ἐ ∷ κ ∷ []) "1John.3.24"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.3.24"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1John.3.24"
∷ word (ο ∷ ὗ ∷ []) "1John.3.24"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.3.24"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.3.24"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.4.1"
∷ word (μ ∷ ὴ ∷ []) "1John.4.1"
∷ word (π ∷ α ∷ ν ∷ τ ∷ ὶ ∷ []) "1John.4.1"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1John.4.1"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ε ∷ τ ∷ ε ∷ []) "1John.4.1"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ὰ ∷ []) "1John.4.1"
∷ word (δ ∷ ο ∷ κ ∷ ι ∷ μ ∷ ά ∷ ζ ∷ ε ∷ τ ∷ ε ∷ []) "1John.4.1"
∷ word (τ ∷ ὰ ∷ []) "1John.4.1"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1John.4.1"
∷ word (ε ∷ ἰ ∷ []) "1John.4.1"
∷ word (ἐ ∷ κ ∷ []) "1John.4.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.1"
∷ word (π ∷ ο ∷ ∙λ ∷ ∙λ ∷ ο ∷ ὶ ∷ []) "1John.4.1"
∷ word (ψ ∷ ε ∷ υ ∷ δ ∷ ο ∷ π ∷ ρ ∷ ο ∷ φ ∷ ῆ ∷ τ ∷ α ∷ ι ∷ []) "1John.4.1"
∷ word (ἐ ∷ ξ ∷ ε ∷ ∙λ ∷ η ∷ ∙λ ∷ ύ ∷ θ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "1John.4.1"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.4.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.1"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.4.1"
∷ word (ἐ ∷ ν ∷ []) "1John.4.2"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.2"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ τ ∷ ε ∷ []) "1John.4.2"
∷ word (τ ∷ ὸ ∷ []) "1John.4.2"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.4.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.2"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1John.4.2"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.4.2"
∷ word (ὃ ∷ []) "1John.4.2"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ []) "1John.4.2"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "1John.4.2"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ν ∷ []) "1John.4.2"
∷ word (ἐ ∷ ν ∷ []) "1John.4.2"
∷ word (σ ∷ α ∷ ρ ∷ κ ∷ ὶ ∷ []) "1John.4.2"
∷ word (ἐ ∷ ∙λ ∷ η ∷ ∙λ ∷ υ ∷ θ ∷ ό ∷ τ ∷ α ∷ []) "1John.4.2"
∷ word (ἐ ∷ κ ∷ []) "1John.4.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.2"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.3"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1John.4.3"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.4.3"
∷ word (ὃ ∷ []) "1John.4.3"
∷ word (μ ∷ ὴ ∷ []) "1John.4.3"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ε ∷ ῖ ∷ []) "1John.4.3"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.3"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ν ∷ []) "1John.4.3"
∷ word (ἐ ∷ κ ∷ []) "1John.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.4.3"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ό ∷ []) "1John.4.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.3"
∷ word (τ ∷ ὸ ∷ []) "1John.4.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.3"
∷ word (ἀ ∷ ν ∷ τ ∷ ι ∷ χ ∷ ρ ∷ ί ∷ σ ∷ τ ∷ ο ∷ υ ∷ []) "1John.4.3"
∷ word (ὃ ∷ []) "1John.4.3"
∷ word (ἀ ∷ κ ∷ η ∷ κ ∷ ό ∷ α ∷ τ ∷ ε ∷ []) "1John.4.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.3"
∷ word (ἔ ∷ ρ ∷ χ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1John.4.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.3"
∷ word (ν ∷ ῦ ∷ ν ∷ []) "1John.4.3"
∷ word (ἐ ∷ ν ∷ []) "1John.4.3"
∷ word (τ ∷ ῷ ∷ []) "1John.4.3"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1John.4.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.4.3"
∷ word (ἤ ∷ δ ∷ η ∷ []) "1John.4.3"
∷ word (ὑ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.4"
∷ word (ἐ ∷ κ ∷ []) "1John.4.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ε ∷ []) "1John.4.4"
∷ word (τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.4.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.4"
∷ word (ν ∷ ε ∷ ν ∷ ι ∷ κ ∷ ή ∷ κ ∷ α ∷ τ ∷ ε ∷ []) "1John.4.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ύ ∷ ς ∷ []) "1John.4.4"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.4"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1John.4.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.4.4"
∷ word (ὁ ∷ []) "1John.4.4"
∷ word (ἐ ∷ ν ∷ []) "1John.4.4"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.4"
∷ word (ἢ ∷ []) "1John.4.4"
∷ word (ὁ ∷ []) "1John.4.4"
∷ word (ἐ ∷ ν ∷ []) "1John.4.4"
∷ word (τ ∷ ῷ ∷ []) "1John.4.4"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1John.4.4"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ὶ ∷ []) "1John.4.5"
∷ word (ἐ ∷ κ ∷ []) "1John.4.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.5"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.4.5"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "1John.4.5"
∷ word (δ ∷ ι ∷ ὰ ∷ []) "1John.4.5"
∷ word (τ ∷ ο ∷ ῦ ∷ τ ∷ ο ∷ []) "1John.4.5"
∷ word (ἐ ∷ κ ∷ []) "1John.4.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.5"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.4.5"
∷ word (∙λ ∷ α ∷ ∙λ ∷ ο ∷ ῦ ∷ σ ∷ ι ∷ ν ∷ []) "1John.4.5"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.5"
∷ word (ὁ ∷ []) "1John.4.5"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1John.4.5"
∷ word (α ∷ ὐ ∷ τ ∷ ῶ ∷ ν ∷ []) "1John.4.5"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1John.4.5"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.6"
∷ word (ἐ ∷ κ ∷ []) "1John.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.6"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.6"
∷ word (ὁ ∷ []) "1John.4.6"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ω ∷ ν ∷ []) "1John.4.6"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.6"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1John.4.6"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1John.4.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.4.6"
∷ word (ὃ ∷ ς ∷ []) "1John.4.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.4.6"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.6"
∷ word (ἐ ∷ κ ∷ []) "1John.4.6"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.6"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.4.6"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1John.4.6"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.4.6"
∷ word (ἐ ∷ κ ∷ []) "1John.4.6"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ο ∷ υ ∷ []) "1John.4.6"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.6"
∷ word (τ ∷ ὸ ∷ []) "1John.4.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.4.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.4.6"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ε ∷ ί ∷ α ∷ ς ∷ []) "1John.4.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.6"
∷ word (τ ∷ ὸ ∷ []) "1John.4.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.4.6"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.4.6"
∷ word (π ∷ ∙λ ∷ ά ∷ ν ∷ η ∷ ς ∷ []) "1John.4.6"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.4.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.7"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1John.4.7"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.7"
∷ word (ἡ ∷ []) "1John.4.7"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.7"
∷ word (ἐ ∷ κ ∷ []) "1John.4.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.7"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.7"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.4.7"
∷ word (ὁ ∷ []) "1John.4.7"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.4.7"
∷ word (ἐ ∷ κ ∷ []) "1John.4.7"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.7"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.7"
∷ word (γ ∷ ε ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1John.4.7"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.7"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ε ∷ ι ∷ []) "1John.4.7"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.7"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1John.4.7"
∷ word (ὁ ∷ []) "1John.4.8"
∷ word (μ ∷ ὴ ∷ []) "1John.4.8"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.4.8"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.4.8"
∷ word (ἔ ∷ γ ∷ ν ∷ ω ∷ []) "1John.4.8"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.8"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1John.4.8"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.8"
∷ word (ὁ ∷ []) "1John.4.8"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.8"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.8"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.4.8"
∷ word (ἐ ∷ ν ∷ []) "1John.4.9"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.9"
∷ word (ἐ ∷ φ ∷ α ∷ ν ∷ ε ∷ ρ ∷ ώ ∷ θ ∷ η ∷ []) "1John.4.9"
∷ word (ἡ ∷ []) "1John.4.9"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.9"
∷ word (ἐ ∷ ν ∷ []) "1John.4.9"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.9"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.4.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.9"
∷ word (μ ∷ ο ∷ ν ∷ ο ∷ γ ∷ ε ∷ ν ∷ ῆ ∷ []) "1John.4.9"
∷ word (ἀ ∷ π ∷ έ ∷ σ ∷ τ ∷ α ∷ ∙λ ∷ κ ∷ ε ∷ ν ∷ []) "1John.4.9"
∷ word (ὁ ∷ []) "1John.4.9"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.9"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.4.9"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.9"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.4.9"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.4.9"
∷ word (ζ ∷ ή ∷ σ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.9"
∷ word (δ ∷ ι ∷ []) "1John.4.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.9"
∷ word (ἐ ∷ ν ∷ []) "1John.4.10"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.10"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.4.10"
∷ word (ἡ ∷ []) "1John.4.10"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.10"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.4.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.10"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.10"
∷ word (ἠ ∷ γ ∷ α ∷ π ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.10"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1John.4.10"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.4.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.10"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.4.10"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1John.4.10"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.4.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.10"
∷ word (ἀ ∷ π ∷ έ ∷ σ ∷ τ ∷ ε ∷ ι ∷ ∙λ ∷ ε ∷ ν ∷ []) "1John.4.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.10"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.4.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.10"
∷ word (ἱ ∷ ∙λ ∷ α ∷ σ ∷ μ ∷ ὸ ∷ ν ∷ []) "1John.4.10"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.4.10"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.4.10"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ι ∷ ῶ ∷ ν ∷ []) "1John.4.10"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.4.10"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ η ∷ τ ∷ ο ∷ ί ∷ []) "1John.4.11"
∷ word (ε ∷ ἰ ∷ []) "1John.4.11"
∷ word (ο ∷ ὕ ∷ τ ∷ ω ∷ ς ∷ []) "1John.4.11"
∷ word (ὁ ∷ []) "1John.4.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.11"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1John.4.11"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.4.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.11"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.11"
∷ word (ὀ ∷ φ ∷ ε ∷ ί ∷ ∙λ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.11"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1John.4.11"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ ν ∷ []) "1John.4.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1John.4.12"
∷ word (ο ∷ ὐ ∷ δ ∷ ε ∷ ὶ ∷ ς ∷ []) "1John.4.12"
∷ word (π ∷ ώ ∷ π ∷ ο ∷ τ ∷ ε ∷ []) "1John.4.12"
∷ word (τ ∷ ε ∷ θ ∷ έ ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1John.4.12"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.4.12"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.12"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ ή ∷ ∙λ ∷ ο ∷ υ ∷ ς ∷ []) "1John.4.12"
∷ word (ὁ ∷ []) "1John.4.12"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.12"
∷ word (ἐ ∷ ν ∷ []) "1John.4.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.12"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.4.12"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.12"
∷ word (ἡ ∷ []) "1John.4.12"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.12"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.12"
∷ word (ἐ ∷ ν ∷ []) "1John.4.12"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.12"
∷ word (τ ∷ ε ∷ τ ∷ ε ∷ ∙λ ∷ ε ∷ ι ∷ ω ∷ μ ∷ έ ∷ ν ∷ η ∷ []) "1John.4.12"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.12"
∷ word (Ἐ ∷ ν ∷ []) "1John.4.13"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.13"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.13"
∷ word (ἐ ∷ ν ∷ []) "1John.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.4.13"
∷ word (μ ∷ έ ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.4.13"
∷ word (ἐ ∷ ν ∷ []) "1John.4.13"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.13"
∷ word (ἐ ∷ κ ∷ []) "1John.4.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.13"
∷ word (π ∷ ν ∷ ε ∷ ύ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1John.4.13"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.13"
∷ word (δ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.4.13"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.14"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.14"
∷ word (τ ∷ ε ∷ θ ∷ ε ∷ ά ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.4.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.14"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.14"
∷ word (ὁ ∷ []) "1John.4.14"
∷ word (π ∷ α ∷ τ ∷ ὴ ∷ ρ ∷ []) "1John.4.14"
∷ word (ἀ ∷ π ∷ έ ∷ σ ∷ τ ∷ α ∷ ∙λ ∷ κ ∷ ε ∷ ν ∷ []) "1John.4.14"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.14"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.4.14"
∷ word (σ ∷ ω ∷ τ ∷ ῆ ∷ ρ ∷ α ∷ []) "1John.4.14"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.14"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ υ ∷ []) "1John.4.14"
∷ word (ὃ ∷ ς ∷ []) "1John.4.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.4.15"
∷ word (ὁ ∷ μ ∷ ο ∷ ∙λ ∷ ο ∷ γ ∷ ή ∷ σ ∷ ῃ ∷ []) "1John.4.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.15"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1John.4.15"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.15"
∷ word (ὁ ∷ []) "1John.4.15"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "1John.4.15"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.4.15"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.4.15"
∷ word (ὁ ∷ []) "1John.4.15"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.15"
∷ word (ἐ ∷ ν ∷ []) "1John.4.15"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.4.15"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.4.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.15"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.4.15"
∷ word (ἐ ∷ ν ∷ []) "1John.4.15"
∷ word (τ ∷ ῷ ∷ []) "1John.4.15"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1John.4.15"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.16"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.16"
∷ word (ἐ ∷ γ ∷ ν ∷ ώ ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.16"
∷ word (π ∷ ε ∷ π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.16"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.4.16"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ ν ∷ []) "1John.4.16"
∷ word (ἣ ∷ ν ∷ []) "1John.4.16"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.4.16"
∷ word (ὁ ∷ []) "1John.4.16"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.16"
∷ word (ἐ ∷ ν ∷ []) "1John.4.16"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.4.16"
∷ word (Ὁ ∷ []) "1John.4.16"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.16"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.16"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.16"
∷ word (ὁ ∷ []) "1John.4.16"
∷ word (μ ∷ έ ∷ ν ∷ ω ∷ ν ∷ []) "1John.4.16"
∷ word (ἐ ∷ ν ∷ []) "1John.4.16"
∷ word (τ ∷ ῇ ∷ []) "1John.4.16"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "1John.4.16"
∷ word (ἐ ∷ ν ∷ []) "1John.4.16"
∷ word (τ ∷ ῷ ∷ []) "1John.4.16"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1John.4.16"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.4.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.16"
∷ word (ὁ ∷ []) "1John.4.16"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.4.16"
∷ word (ἐ ∷ ν ∷ []) "1John.4.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.4.16"
∷ word (μ ∷ έ ∷ ν ∷ ε ∷ ι ∷ []) "1John.4.16"
∷ word (ἐ ∷ ν ∷ []) "1John.4.17"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.17"
∷ word (τ ∷ ε ∷ τ ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "1John.4.17"
∷ word (ἡ ∷ []) "1John.4.17"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.17"
∷ word (μ ∷ ε ∷ θ ∷ []) "1John.4.17"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.4.17"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.4.17"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ ν ∷ []) "1John.4.17"
∷ word (ἔ ∷ χ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.17"
∷ word (ἐ ∷ ν ∷ []) "1John.4.17"
∷ word (τ ∷ ῇ ∷ []) "1John.4.17"
∷ word (ἡ ∷ μ ∷ έ ∷ ρ ∷ ᾳ ∷ []) "1John.4.17"
∷ word (τ ∷ ῆ ∷ ς ∷ []) "1John.4.17"
∷ word (κ ∷ ρ ∷ ί ∷ σ ∷ ε ∷ ω ∷ ς ∷ []) "1John.4.17"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.17"
∷ word (κ ∷ α ∷ θ ∷ ὼ ∷ ς ∷ []) "1John.4.17"
∷ word (ἐ ∷ κ ∷ ε ∷ ῖ ∷ ν ∷ ό ∷ ς ∷ []) "1John.4.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.17"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.17"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.17"
∷ word (ἐ ∷ ν ∷ []) "1John.4.17"
∷ word (τ ∷ ῷ ∷ []) "1John.4.17"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ῳ ∷ []) "1John.4.17"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.4.17"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ς ∷ []) "1John.4.18"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.4.18"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.4.18"
∷ word (ἐ ∷ ν ∷ []) "1John.4.18"
∷ word (τ ∷ ῇ ∷ []) "1John.4.18"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "1John.4.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.4.18"
∷ word (ἡ ∷ []) "1John.4.18"
∷ word (τ ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ α ∷ []) "1John.4.18"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.4.18"
∷ word (ἔ ∷ ξ ∷ ω ∷ []) "1John.4.18"
∷ word (β ∷ ά ∷ ∙λ ∷ ∙λ ∷ ε ∷ ι ∷ []) "1John.4.18"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.18"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ν ∷ []) "1John.4.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.18"
∷ word (ὁ ∷ []) "1John.4.18"
∷ word (φ ∷ ό ∷ β ∷ ο ∷ ς ∷ []) "1John.4.18"
∷ word (κ ∷ ό ∷ ∙λ ∷ α ∷ σ ∷ ι ∷ ν ∷ []) "1John.4.18"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.4.18"
∷ word (ὁ ∷ []) "1John.4.18"
∷ word (δ ∷ ὲ ∷ []) "1John.4.18"
∷ word (φ ∷ ο ∷ β ∷ ο ∷ ύ ∷ μ ∷ ε ∷ ν ∷ ο ∷ ς ∷ []) "1John.4.18"
∷ word (ο ∷ ὐ ∷ []) "1John.4.18"
∷ word (τ ∷ ε ∷ τ ∷ ε ∷ ∙λ ∷ ε ∷ ί ∷ ω ∷ τ ∷ α ∷ ι ∷ []) "1John.4.18"
∷ word (ἐ ∷ ν ∷ []) "1John.4.18"
∷ word (τ ∷ ῇ ∷ []) "1John.4.18"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ ῃ ∷ []) "1John.4.18"
∷ word (ἡ ∷ μ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.4.19"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.19"
∷ word (α ∷ ὐ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.4.19"
∷ word (π ∷ ρ ∷ ῶ ∷ τ ∷ ο ∷ ς ∷ []) "1John.4.19"
∷ word (ἠ ∷ γ ∷ ά ∷ π ∷ η ∷ σ ∷ ε ∷ ν ∷ []) "1John.4.19"
∷ word (ἡ ∷ μ ∷ ᾶ ∷ ς ∷ []) "1John.4.19"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1John.4.20"
∷ word (τ ∷ ι ∷ ς ∷ []) "1John.4.20"
∷ word (ε ∷ ἴ ∷ π ∷ ῃ ∷ []) "1John.4.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.4.20"
∷ word (Ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ []) "1John.4.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (θ ∷ ε ∷ ό ∷ ν ∷ []) "1John.4.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.20"
∷ word (μ ∷ ι ∷ σ ∷ ῇ ∷ []) "1John.4.20"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ς ∷ []) "1John.4.20"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.4.20"
∷ word (ὁ ∷ []) "1John.4.20"
∷ word (γ ∷ ὰ ∷ ρ ∷ []) "1John.4.20"
∷ word (μ ∷ ὴ ∷ []) "1John.4.20"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.4.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.20"
∷ word (ὃ ∷ ν ∷ []) "1John.4.20"
∷ word (ἑ ∷ ώ ∷ ρ ∷ α ∷ κ ∷ ε ∷ ν ∷ []) "1John.4.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1John.4.20"
∷ word (ὃ ∷ ν ∷ []) "1John.4.20"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.4.20"
∷ word (ἑ ∷ ώ ∷ ρ ∷ α ∷ κ ∷ ε ∷ ν ∷ []) "1John.4.20"
∷ word (ο ∷ ὐ ∷ []) "1John.4.20"
∷ word (δ ∷ ύ ∷ ν ∷ α ∷ τ ∷ α ∷ ι ∷ []) "1John.4.20"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾶ ∷ ν ∷ []) "1John.4.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.21"
∷ word (τ ∷ α ∷ ύ ∷ τ ∷ η ∷ ν ∷ []) "1John.4.21"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.4.21"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὴ ∷ ν ∷ []) "1John.4.21"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.4.21"
∷ word (ἀ ∷ π ∷ []) "1John.4.21"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.21"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.4.21"
∷ word (ὁ ∷ []) "1John.4.21"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.4.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.21"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1John.4.21"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "1John.4.21"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.4.21"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.4.21"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.4.21"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.4.21"
∷ word (Π ∷ ᾶ ∷ ς ∷ []) "1John.5.1"
∷ word (ὁ ∷ []) "1John.5.1"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1John.5.1"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.1"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1John.5.1"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.1"
∷ word (ὁ ∷ []) "1John.5.1"
∷ word (χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ὸ ∷ ς ∷ []) "1John.5.1"
∷ word (ἐ ∷ κ ∷ []) "1John.5.1"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.1"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.1"
∷ word (γ ∷ ε ∷ γ ∷ έ ∷ ν ∷ ν ∷ η ∷ τ ∷ α ∷ ι ∷ []) "1John.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.1"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.5.1"
∷ word (ὁ ∷ []) "1John.5.1"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ ν ∷ []) "1John.5.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.1"
∷ word (γ ∷ ε ∷ ν ∷ ν ∷ ή ∷ σ ∷ α ∷ ν ∷ τ ∷ α ∷ []) "1John.5.1"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ᾷ ∷ []) "1John.5.1"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.1"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.1"
∷ word (γ ∷ ε ∷ γ ∷ ε ∷ ν ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1John.5.1"
∷ word (ἐ ∷ ξ ∷ []) "1John.5.1"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.1"
∷ word (ἐ ∷ ν ∷ []) "1John.5.2"
∷ word (τ ∷ ο ∷ ύ ∷ τ ∷ ῳ ∷ []) "1John.5.2"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.2"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.2"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.2"
∷ word (τ ∷ ὰ ∷ []) "1John.5.2"
∷ word (τ ∷ έ ∷ κ ∷ ν ∷ α ∷ []) "1John.5.2"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.2"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.2"
∷ word (ὅ ∷ τ ∷ α ∷ ν ∷ []) "1John.5.2"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.2"
∷ word (θ ∷ ε ∷ ὸ ∷ ν ∷ []) "1John.5.2"
∷ word (ἀ ∷ γ ∷ α ∷ π ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.2"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.2"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.5.2"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.5.2"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.2"
∷ word (π ∷ ο ∷ ι ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.2"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.3"
∷ word (γ ∷ ά ∷ ρ ∷ []) "1John.5.3"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.3"
∷ word (ἡ ∷ []) "1John.5.3"
∷ word (ἀ ∷ γ ∷ ά ∷ π ∷ η ∷ []) "1John.5.3"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.3"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.3"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.5.3"
∷ word (τ ∷ ὰ ∷ ς ∷ []) "1John.5.3"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ ὰ ∷ ς ∷ []) "1John.5.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.3"
∷ word (τ ∷ η ∷ ρ ∷ ῶ ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.3"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.3"
∷ word (α ∷ ἱ ∷ []) "1John.5.3"
∷ word (ἐ ∷ ν ∷ τ ∷ ο ∷ ∙λ ∷ α ∷ ὶ ∷ []) "1John.5.3"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.3"
∷ word (β ∷ α ∷ ρ ∷ ε ∷ ῖ ∷ α ∷ ι ∷ []) "1John.5.3"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.5.3"
∷ word (ε ∷ ἰ ∷ σ ∷ ί ∷ ν ∷ []) "1John.5.3"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.4"
∷ word (π ∷ ᾶ ∷ ν ∷ []) "1John.5.4"
∷ word (τ ∷ ὸ ∷ []) "1John.5.4"
∷ word (γ ∷ ε ∷ γ ∷ ε ∷ ν ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ν ∷ []) "1John.5.4"
∷ word (ἐ ∷ κ ∷ []) "1John.5.4"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.4"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.4"
∷ word (ν ∷ ι ∷ κ ∷ ᾷ ∷ []) "1John.5.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.4"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.5.4"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.4"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.4"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.5.4"
∷ word (ἡ ∷ []) "1John.5.4"
∷ word (ν ∷ ί ∷ κ ∷ η ∷ []) "1John.5.4"
∷ word (ἡ ∷ []) "1John.5.4"
∷ word (ν ∷ ι ∷ κ ∷ ή ∷ σ ∷ α ∷ σ ∷ α ∷ []) "1John.5.4"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.4"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.5.4"
∷ word (ἡ ∷ []) "1John.5.4"
∷ word (π ∷ ί ∷ σ ∷ τ ∷ ι ∷ ς ∷ []) "1John.5.4"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.5.4"
∷ word (τ ∷ ί ∷ ς ∷ []) "1John.5.5"
∷ word (δ ∷ έ ∷ []) "1John.5.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.5"
∷ word (ὁ ∷ []) "1John.5.5"
∷ word (ν ∷ ι ∷ κ ∷ ῶ ∷ ν ∷ []) "1John.5.5"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.5"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ν ∷ []) "1John.5.5"
∷ word (ε ∷ ἰ ∷ []) "1John.5.5"
∷ word (μ ∷ ὴ ∷ []) "1John.5.5"
∷ word (ὁ ∷ []) "1John.5.5"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1John.5.5"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.5"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1John.5.5"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.5"
∷ word (ὁ ∷ []) "1John.5.5"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "1John.5.5"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.5"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.5"
∷ word (Ο ∷ ὗ ∷ τ ∷ ό ∷ ς ∷ []) "1John.5.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.6"
∷ word (ὁ ∷ []) "1John.5.6"
∷ word (ἐ ∷ ∙λ ∷ θ ∷ ὼ ∷ ν ∷ []) "1John.5.6"
∷ word (δ ∷ ι ∷ []) "1John.5.6"
∷ word (ὕ ∷ δ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1John.5.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.6"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ο ∷ ς ∷ []) "1John.5.6"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ ς ∷ []) "1John.5.6"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ό ∷ ς ∷ []) "1John.5.6"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.5.6"
∷ word (ἐ ∷ ν ∷ []) "1John.5.6"
∷ word (τ ∷ ῷ ∷ []) "1John.5.6"
∷ word (ὕ ∷ δ ∷ α ∷ τ ∷ ι ∷ []) "1John.5.6"
∷ word (μ ∷ ό ∷ ν ∷ ο ∷ ν ∷ []) "1John.5.6"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.5.6"
∷ word (ἐ ∷ ν ∷ []) "1John.5.6"
∷ word (τ ∷ ῷ ∷ []) "1John.5.6"
∷ word (ὕ ∷ δ ∷ α ∷ τ ∷ ι ∷ []) "1John.5.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.6"
∷ word (ἐ ∷ ν ∷ []) "1John.5.6"
∷ word (τ ∷ ῷ ∷ []) "1John.5.6"
∷ word (α ∷ ἵ ∷ μ ∷ α ∷ τ ∷ ι ∷ []) "1John.5.6"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.6"
∷ word (τ ∷ ὸ ∷ []) "1John.5.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ ά ∷ []) "1John.5.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.6"
∷ word (τ ∷ ὸ ∷ []) "1John.5.6"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ []) "1John.5.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.6"
∷ word (τ ∷ ὸ ∷ []) "1John.5.6"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ ά ∷ []) "1John.5.6"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.6"
∷ word (ἡ ∷ []) "1John.5.6"
∷ word (ἀ ∷ ∙λ ∷ ή ∷ θ ∷ ε ∷ ι ∷ α ∷ []) "1John.5.6"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.7"
∷ word (τ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.5.7"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1John.5.7"
∷ word (ο ∷ ἱ ∷ []) "1John.5.7"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ο ∷ ῦ ∷ ν ∷ τ ∷ ε ∷ ς ∷ []) "1John.5.7"
∷ word (τ ∷ ὸ ∷ []) "1John.5.8"
∷ word (π ∷ ν ∷ ε ∷ ῦ ∷ μ ∷ α ∷ []) "1John.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.8"
∷ word (τ ∷ ὸ ∷ []) "1John.5.8"
∷ word (ὕ ∷ δ ∷ ω ∷ ρ ∷ []) "1John.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.8"
∷ word (τ ∷ ὸ ∷ []) "1John.5.8"
∷ word (α ∷ ἷ ∷ μ ∷ α ∷ []) "1John.5.8"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.8"
∷ word (ο ∷ ἱ ∷ []) "1John.5.8"
∷ word (τ ∷ ρ ∷ ε ∷ ῖ ∷ ς ∷ []) "1John.5.8"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.5.8"
∷ word (τ ∷ ὸ ∷ []) "1John.5.8"
∷ word (ἕ ∷ ν ∷ []) "1John.5.8"
∷ word (ε ∷ ἰ ∷ σ ∷ ι ∷ ν ∷ []) "1John.5.8"
∷ word (ε ∷ ἰ ∷ []) "1John.5.9"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.5.9"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "1John.5.9"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.5.9"
∷ word (ἀ ∷ ν ∷ θ ∷ ρ ∷ ώ ∷ π ∷ ω ∷ ν ∷ []) "1John.5.9"
∷ word (∙λ ∷ α ∷ μ ∷ β ∷ ά ∷ ν ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.9"
∷ word (ἡ ∷ []) "1John.5.9"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ []) "1John.5.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (μ ∷ ε ∷ ί ∷ ζ ∷ ω ∷ ν ∷ []) "1John.5.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.5.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.9"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.9"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.5.9"
∷ word (ἡ ∷ []) "1John.5.9"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ []) "1John.5.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.9"
∷ word (μ ∷ ε ∷ μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.9"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.5.9"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.9"
∷ word (ὁ ∷ []) "1John.5.10"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1John.5.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.5.10"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.10"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.10"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.10"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.5.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.5.10"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "1John.5.10"
∷ word (ἐ ∷ ν ∷ []) "1John.5.10"
∷ word (α ∷ ὑ ∷ τ ∷ ῷ ∷ []) "1John.5.10"
∷ word (ὁ ∷ []) "1John.5.10"
∷ word (μ ∷ ὴ ∷ []) "1John.5.10"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ω ∷ ν ∷ []) "1John.5.10"
∷ word (τ ∷ ῷ ∷ []) "1John.5.10"
∷ word (θ ∷ ε ∷ ῷ ∷ []) "1John.5.10"
∷ word (ψ ∷ ε ∷ ύ ∷ σ ∷ τ ∷ η ∷ ν ∷ []) "1John.5.10"
∷ word (π ∷ ε ∷ π ∷ ο ∷ ί ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.10"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.5.10"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.10"
∷ word (ο ∷ ὐ ∷ []) "1John.5.10"
∷ word (π ∷ ε ∷ π ∷ ί ∷ σ ∷ τ ∷ ε ∷ υ ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.10"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.5.10"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.5.10"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ ν ∷ []) "1John.5.10"
∷ word (ἣ ∷ ν ∷ []) "1John.5.10"
∷ word (μ ∷ ε ∷ μ ∷ α ∷ ρ ∷ τ ∷ ύ ∷ ρ ∷ η ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.10"
∷ word (ὁ ∷ []) "1John.5.10"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.5.10"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.5.10"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.10"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.5.10"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.10"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.11"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.5.11"
∷ word (ἡ ∷ []) "1John.5.11"
∷ word (μ ∷ α ∷ ρ ∷ τ ∷ υ ∷ ρ ∷ ί ∷ α ∷ []) "1John.5.11"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.11"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.5.11"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "1John.5.11"
∷ word (ἔ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.11"
∷ word (ὁ ∷ []) "1John.5.11"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.5.11"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.5.11"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.11"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.11"
∷ word (ἡ ∷ []) "1John.5.11"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "1John.5.11"
∷ word (ἐ ∷ ν ∷ []) "1John.5.11"
∷ word (τ ∷ ῷ ∷ []) "1John.5.11"
∷ word (υ ∷ ἱ ∷ ῷ ∷ []) "1John.5.11"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.11"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.11"
∷ word (ὁ ∷ []) "1John.5.12"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1John.5.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.12"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.5.12"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.5.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.5.12"
∷ word (ζ ∷ ω ∷ ή ∷ ν ∷ []) "1John.5.12"
∷ word (ὁ ∷ []) "1John.5.12"
∷ word (μ ∷ ὴ ∷ []) "1John.5.12"
∷ word (ἔ ∷ χ ∷ ω ∷ ν ∷ []) "1John.5.12"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.12"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ν ∷ []) "1John.5.12"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.12"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.12"
∷ word (τ ∷ ὴ ∷ ν ∷ []) "1John.5.12"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.5.12"
∷ word (ο ∷ ὐ ∷ κ ∷ []) "1John.5.12"
∷ word (ἔ ∷ χ ∷ ε ∷ ι ∷ []) "1John.5.12"
∷ word (Τ ∷ α ∷ ῦ ∷ τ ∷ α ∷ []) "1John.5.13"
∷ word (ἔ ∷ γ ∷ ρ ∷ α ∷ ψ ∷ α ∷ []) "1John.5.13"
∷ word (ὑ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.5.13"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.5.13"
∷ word (ε ∷ ἰ ∷ δ ∷ ῆ ∷ τ ∷ ε ∷ []) "1John.5.13"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.13"
∷ word (ζ ∷ ω ∷ ὴ ∷ ν ∷ []) "1John.5.13"
∷ word (ἔ ∷ χ ∷ ε ∷ τ ∷ ε ∷ []) "1John.5.13"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ν ∷ []) "1John.5.13"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1John.5.13"
∷ word (π ∷ ι ∷ σ ∷ τ ∷ ε ∷ ύ ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1John.5.13"
∷ word (ε ∷ ἰ ∷ ς ∷ []) "1John.5.13"
∷ word (τ ∷ ὸ ∷ []) "1John.5.13"
∷ word (ὄ ∷ ν ∷ ο ∷ μ ∷ α ∷ []) "1John.5.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.13"
∷ word (υ ∷ ἱ ∷ ο ∷ ῦ ∷ []) "1John.5.13"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.13"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.13"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.14"
∷ word (α ∷ ὕ ∷ τ ∷ η ∷ []) "1John.5.14"
∷ word (ἐ ∷ σ ∷ τ ∷ ὶ ∷ ν ∷ []) "1John.5.14"
∷ word (ἡ ∷ []) "1John.5.14"
∷ word (π ∷ α ∷ ρ ∷ ρ ∷ η ∷ σ ∷ ί ∷ α ∷ []) "1John.5.14"
∷ word (ἣ ∷ ν ∷ []) "1John.5.14"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.14"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.14"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.5.14"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.14"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1John.5.14"
∷ word (τ ∷ ι ∷ []) "1John.5.14"
∷ word (α ∷ ἰ ∷ τ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.5.14"
∷ word (κ ∷ α ∷ τ ∷ ὰ ∷ []) "1John.5.14"
∷ word (τ ∷ ὸ ∷ []) "1John.5.14"
∷ word (θ ∷ έ ∷ ∙λ ∷ η ∷ μ ∷ α ∷ []) "1John.5.14"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.14"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1John.5.14"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.5.14"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.5.15"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.15"
∷ word (ἀ ∷ κ ∷ ο ∷ ύ ∷ ε ∷ ι ∷ []) "1John.5.15"
∷ word (ἡ ∷ μ ∷ ῶ ∷ ν ∷ []) "1John.5.15"
∷ word (ὃ ∷ []) "1John.5.15"
∷ word (ἐ ∷ ὰ ∷ ν ∷ []) "1John.5.15"
∷ word (α ∷ ἰ ∷ τ ∷ ώ ∷ μ ∷ ε ∷ θ ∷ α ∷ []) "1John.5.15"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.15"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.15"
∷ word (ἔ ∷ χ ∷ ο ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.15"
∷ word (τ ∷ ὰ ∷ []) "1John.5.15"
∷ word (α ∷ ἰ ∷ τ ∷ ή ∷ μ ∷ α ∷ τ ∷ α ∷ []) "1John.5.15"
∷ word (ἃ ∷ []) "1John.5.15"
∷ word (ᾐ ∷ τ ∷ ή ∷ κ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.15"
∷ word (ἀ ∷ π ∷ []) "1John.5.15"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.15"
∷ word (ἐ ∷ ά ∷ ν ∷ []) "1John.5.16"
∷ word (τ ∷ ι ∷ ς ∷ []) "1John.5.16"
∷ word (ἴ ∷ δ ∷ ῃ ∷ []) "1John.5.16"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.16"
∷ word (ἀ ∷ δ ∷ ε ∷ ∙λ ∷ φ ∷ ὸ ∷ ν ∷ []) "1John.5.16"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ο ∷ ν ∷ τ ∷ α ∷ []) "1John.5.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ ν ∷ []) "1John.5.16"
∷ word (μ ∷ ὴ ∷ []) "1John.5.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.16"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1John.5.16"
∷ word (α ∷ ἰ ∷ τ ∷ ή ∷ σ ∷ ε ∷ ι ∷ []) "1John.5.16"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.16"
∷ word (δ ∷ ώ ∷ σ ∷ ε ∷ ι ∷ []) "1John.5.16"
∷ word (α ∷ ὐ ∷ τ ∷ ῷ ∷ []) "1John.5.16"
∷ word (ζ ∷ ω ∷ ή ∷ ν ∷ []) "1John.5.16"
∷ word (τ ∷ ο ∷ ῖ ∷ ς ∷ []) "1John.5.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ο ∷ υ ∷ σ ∷ ι ∷ ν ∷ []) "1John.5.16"
∷ word (μ ∷ ὴ ∷ []) "1John.5.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.16"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1John.5.16"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.16"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1John.5.16"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.16"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1John.5.16"
∷ word (ο ∷ ὐ ∷ []) "1John.5.16"
∷ word (π ∷ ε ∷ ρ ∷ ὶ ∷ []) "1John.5.16"
∷ word (ἐ ∷ κ ∷ ε ∷ ί ∷ ν ∷ η ∷ ς ∷ []) "1John.5.16"
∷ word (∙λ ∷ έ ∷ γ ∷ ω ∷ []) "1John.5.16"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.5.16"
∷ word (ἐ ∷ ρ ∷ ω ∷ τ ∷ ή ∷ σ ∷ ῃ ∷ []) "1John.5.16"
∷ word (π ∷ ᾶ ∷ σ ∷ α ∷ []) "1John.5.17"
∷ word (ἀ ∷ δ ∷ ι ∷ κ ∷ ί ∷ α ∷ []) "1John.5.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1John.5.17"
∷ word (ἐ ∷ σ ∷ τ ∷ ί ∷ ν ∷ []) "1John.5.17"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.17"
∷ word (ἔ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.17"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ί ∷ α ∷ []) "1John.5.17"
∷ word (ο ∷ ὐ ∷ []) "1John.5.17"
∷ word (π ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.17"
∷ word (θ ∷ ά ∷ ν ∷ α ∷ τ ∷ ο ∷ ν ∷ []) "1John.5.17"
∷ word (Ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.18"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.18"
∷ word (π ∷ ᾶ ∷ ς ∷ []) "1John.5.18"
∷ word (ὁ ∷ []) "1John.5.18"
∷ word (γ ∷ ε ∷ γ ∷ ε ∷ ν ∷ ν ∷ η ∷ μ ∷ έ ∷ ν ∷ ο ∷ ς ∷ []) "1John.5.18"
∷ word (ἐ ∷ κ ∷ []) "1John.5.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.18"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.5.18"
∷ word (ἁ ∷ μ ∷ α ∷ ρ ∷ τ ∷ ά ∷ ν ∷ ε ∷ ι ∷ []) "1John.5.18"
∷ word (ἀ ∷ ∙λ ∷ ∙λ ∷ []) "1John.5.18"
∷ word (ὁ ∷ []) "1John.5.18"
∷ word (γ ∷ ε ∷ ν ∷ ν ∷ η ∷ θ ∷ ε ∷ ὶ ∷ ς ∷ []) "1John.5.18"
∷ word (ἐ ∷ κ ∷ []) "1John.5.18"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.18"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.18"
∷ word (τ ∷ η ∷ ρ ∷ ε ∷ ῖ ∷ []) "1John.5.18"
∷ word (α ∷ ὐ ∷ τ ∷ ό ∷ ν ∷ []) "1John.5.18"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.18"
∷ word (ὁ ∷ []) "1John.5.18"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ὸ ∷ ς ∷ []) "1John.5.18"
∷ word (ο ∷ ὐ ∷ χ ∷ []) "1John.5.18"
∷ word (ἅ ∷ π ∷ τ ∷ ε ∷ τ ∷ α ∷ ι ∷ []) "1John.5.18"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.18"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.19"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.19"
∷ word (ἐ ∷ κ ∷ []) "1John.5.19"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.19"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.19"
∷ word (ἐ ∷ σ ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.19"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.19"
∷ word (ὁ ∷ []) "1John.5.19"
∷ word (κ ∷ ό ∷ σ ∷ μ ∷ ο ∷ ς ∷ []) "1John.5.19"
∷ word (ὅ ∷ ∙λ ∷ ο ∷ ς ∷ []) "1John.5.19"
∷ word (ἐ ∷ ν ∷ []) "1John.5.19"
∷ word (τ ∷ ῷ ∷ []) "1John.5.19"
∷ word (π ∷ ο ∷ ν ∷ η ∷ ρ ∷ ῷ ∷ []) "1John.5.19"
∷ word (κ ∷ ε ∷ ῖ ∷ τ ∷ α ∷ ι ∷ []) "1John.5.19"
∷ word (ο ∷ ἴ ∷ δ ∷ α ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.20"
∷ word (δ ∷ ὲ ∷ []) "1John.5.20"
∷ word (ὅ ∷ τ ∷ ι ∷ []) "1John.5.20"
∷ word (ὁ ∷ []) "1John.5.20"
∷ word (υ ∷ ἱ ∷ ὸ ∷ ς ∷ []) "1John.5.20"
∷ word (τ ∷ ο ∷ ῦ ∷ []) "1John.5.20"
∷ word (θ ∷ ε ∷ ο ∷ ῦ ∷ []) "1John.5.20"
∷ word (ἥ ∷ κ ∷ ε ∷ ι ∷ []) "1John.5.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.20"
∷ word (δ ∷ έ ∷ δ ∷ ω ∷ κ ∷ ε ∷ ν ∷ []) "1John.5.20"
∷ word (ἡ ∷ μ ∷ ῖ ∷ ν ∷ []) "1John.5.20"
∷ word (δ ∷ ι ∷ ά ∷ ν ∷ ο ∷ ι ∷ α ∷ ν ∷ []) "1John.5.20"
∷ word (ἵ ∷ ν ∷ α ∷ []) "1John.5.20"
∷ word (γ ∷ ι ∷ ν ∷ ώ ∷ σ ∷ κ ∷ ω ∷ μ ∷ ε ∷ ν ∷ []) "1John.5.20"
∷ word (τ ∷ ὸ ∷ ν ∷ []) "1John.5.20"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ι ∷ ν ∷ ό ∷ ν ∷ []) "1John.5.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.20"
∷ word (ἐ ∷ σ ∷ μ ∷ ὲ ∷ ν ∷ []) "1John.5.20"
∷ word (ἐ ∷ ν ∷ []) "1John.5.20"
∷ word (τ ∷ ῷ ∷ []) "1John.5.20"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ι ∷ ν ∷ ῷ ∷ []) "1John.5.20"
∷ word (ἐ ∷ ν ∷ []) "1John.5.20"
∷ word (τ ∷ ῷ ∷ []) "1John.5.20"
∷ word (υ ∷ ἱ ∷ ῷ ∷ []) "1John.5.20"
∷ word (α ∷ ὐ ∷ τ ∷ ο ∷ ῦ ∷ []) "1John.5.20"
∷ word (Ἰ ∷ η ∷ σ ∷ ο ∷ ῦ ∷ []) "1John.5.20"
∷ word (Χ ∷ ρ ∷ ι ∷ σ ∷ τ ∷ ῷ ∷ []) "1John.5.20"
∷ word (ο ∷ ὗ ∷ τ ∷ ό ∷ ς ∷ []) "1John.5.20"
∷ word (ἐ ∷ σ ∷ τ ∷ ι ∷ ν ∷ []) "1John.5.20"
∷ word (ὁ ∷ []) "1John.5.20"
∷ word (ἀ ∷ ∙λ ∷ η ∷ θ ∷ ι ∷ ν ∷ ὸ ∷ ς ∷ []) "1John.5.20"
∷ word (θ ∷ ε ∷ ὸ ∷ ς ∷ []) "1John.5.20"
∷ word (κ ∷ α ∷ ὶ ∷ []) "1John.5.20"
∷ word (ζ ∷ ω ∷ ὴ ∷ []) "1John.5.20"
∷ word (α ∷ ἰ ∷ ώ ∷ ν ∷ ι ∷ ο ∷ ς ∷ []) "1John.5.20"
∷ word (Τ ∷ ε ∷ κ ∷ ν ∷ ί ∷ α ∷ []) "1John.5.21"
∷ word (φ ∷ υ ∷ ∙λ ∷ ά ∷ ξ ∷ α ∷ τ ∷ ε ∷ []) "1John.5.21"
∷ word (ἑ ∷ α ∷ υ ∷ τ ∷ ὰ ∷ []) "1John.5.21"
∷ word (ἀ ∷ π ∷ ὸ ∷ []) "1John.5.21"
∷ word (τ ∷ ῶ ∷ ν ∷ []) "1John.5.21"
∷ word (ε ∷ ἰ ∷ δ ∷ ώ ∷ ∙λ ∷ ω ∷ ν ∷ []) "1John.5.21"
∷ []
|
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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Categories.Category.Monoidal using (Monoidal)
module Categories.Category.Monoidal.Reasoning {o ℓ e} {C : Category o ℓ e} (M : Monoidal C) where
open import Data.Product using (_,_)
open import Categories.Functor using (Functor)
open Category C
private
variable
X Y : Obj
f g h i : X ⇒ Y
open Monoidal M using (_⊗₀_; _⊗₁_; ⊗)
open Functor ⊗ using (F-resp-≈; homomorphism)
open HomReasoning public
infixr 6 _⟩⊗⟨_ refl⟩⊗⟨_
infixl 7 _⟩⊗⟨refl
⊗-resp-≈ : f ≈ h → g ≈ i → (f ⊗₁ g) ≈ (h ⊗₁ i)
⊗-resp-≈ p q = F-resp-≈ (p , q)
⊗-resp-≈ˡ : f ≈ h → (f ⊗₁ g) ≈ (h ⊗₁ g)
⊗-resp-≈ˡ p = ⊗-resp-≈ p Equiv.refl
⊗-resp-≈ʳ : g ≈ i → (f ⊗₁ g) ≈ (f ⊗₁ i)
⊗-resp-≈ʳ p = ⊗-resp-≈ Equiv.refl p
_⟩⊗⟨_ : f ≈ h → g ≈ i → (f ⊗₁ g) ≈ (h ⊗₁ i)
_⟩⊗⟨_ = ⊗-resp-≈
refl⟩⊗⟨_ : g ≈ i → (f ⊗₁ g) ≈ (f ⊗₁ i)
refl⟩⊗⟨_ = ⊗-resp-≈ʳ
_⟩⊗⟨refl : f ≈ h → (f ⊗₁ g) ≈ (h ⊗₁ g)
_⟩⊗⟨refl = ⊗-resp-≈ˡ
-- This corresponds to the graphical coherence property of diagrams
-- modelling monoidal categories:
--
-- | | | |
-- [h] [i] [h] [i]
-- | | ≈ | |
-- [f] [g] | |
-- | | | |
-- [f] [g]
-- | |
⊗-distrib-over-∘ : ((f ∘ h) ⊗₁ (g ∘ i)) ≈ ((f ⊗₁ g) ∘ (h ⊗₁ i))
⊗-distrib-over-∘ = homomorphism
-- Parallel commutation
parallel : ∀ {X₁ X₂ Y₁ Y₂ Z₁ Z₂ W₁ W₂}
{f₁ : Y₁ ⇒ W₁} {f₂ : Z₁ ⇒ W₁} {g₁ : Y₂ ⇒ W₂} {g₂ : Z₂ ⇒ W₂}
{h₁ : X₁ ⇒ Y₁} {h₂ : X₁ ⇒ Z₁} {i₁ : X₂ ⇒ Y₂} {i₂ : X₂ ⇒ Z₂} →
f₁ ∘ h₁ ≈ f₂ ∘ h₂ → g₁ ∘ i₁ ≈ g₂ ∘ i₂ →
f₁ ⊗₁ g₁ ∘ h₁ ⊗₁ i₁ ≈ f₂ ⊗₁ g₂ ∘ h₂ ⊗₁ i₂
parallel {f₁ = f₁} {f₂} {g₁} {g₂} {h₁} {h₂} {i₁} {i₂} hyp₁ hyp₂ = begin
f₁ ⊗₁ g₁ ∘ h₁ ⊗₁ i₁ ≈˘⟨ ⊗-distrib-over-∘ ⟩
(f₁ ∘ h₁) ⊗₁ (g₁ ∘ i₁) ≈⟨ hyp₁ ⟩⊗⟨ hyp₂ ⟩
(f₂ ∘ h₂) ⊗₁ (g₂ ∘ i₂) ≈⟨ ⊗-distrib-over-∘ ⟩
f₂ ⊗₁ g₂ ∘ h₂ ⊗₁ i₂ ∎
-- Parallel-to-serial conversions
--
-- | | | | | |
-- | | | [g] [f] |
-- [f] [g] = | | = | |
-- | | [f] | | [g]
-- | | | | | |
serialize₁₂ : ∀ {X₁ Y₁ X₂ Y₂} {f : X₁ ⇒ Y₁} {g : X₂ ⇒ Y₂} →
f ⊗₁ g ≈ f ⊗₁ id ∘ id ⊗₁ g
serialize₁₂ {f = f} {g} = begin
f ⊗₁ g ≈˘⟨ identityʳ ⟩⊗⟨ identityˡ ⟩
(f ∘ id) ⊗₁ (id ∘ g) ≈⟨ ⊗-distrib-over-∘ ⟩
f ⊗₁ id ∘ id ⊗₁ g ∎
serialize₂₁ : ∀ {X₁ Y₁ X₂ Y₂} {f : X₁ ⇒ Y₁} {g : X₂ ⇒ Y₂} →
f ⊗₁ g ≈ id ⊗₁ g ∘ f ⊗₁ id
serialize₂₁ {f = f} {g} = begin
f ⊗₁ g ≈˘⟨ identityˡ ⟩⊗⟨ identityʳ ⟩
(id ∘ f) ⊗₁ (g ∘ id) ≈⟨ ⊗-distrib-over-∘ ⟩
id ⊗₁ g ∘ f ⊗₁ id ∎
-- Split a composite in the first component
--
-- | | | | | |
-- [g] | [g] | [g] [h]
-- | [h] = | | = | |
-- [f] | [f] [h] [f] |
-- | | | | | |
split₁ʳ : ∀ {X₁ Y₁ Z₁ X₂ Y₂} {f : Y₁ ⇒ Z₁} {g : X₁ ⇒ Y₁} {h : X₂ ⇒ Y₂} →
(f ∘ g) ⊗₁ h ≈ f ⊗₁ h ∘ g ⊗₁ id
split₁ʳ {f = f} {g} {h} = begin
(f ∘ g) ⊗₁ h ≈˘⟨ refl⟩⊗⟨ identityʳ ⟩
(f ∘ g) ⊗₁ (h ∘ id) ≈⟨ ⊗-distrib-over-∘ ⟩
f ⊗₁ h ∘ g ⊗₁ id ∎
split₁ˡ : ∀ {X₁ Y₁ Z₁ X₂ Y₂} {f : Y₁ ⇒ Z₁} {g : X₁ ⇒ Y₁} {h : X₂ ⇒ Y₂} →
(f ∘ g) ⊗₁ h ≈ f ⊗₁ id ∘ g ⊗₁ h
split₁ˡ {f = f} {g} {h} = begin
(f ∘ g) ⊗₁ h ≈˘⟨ refl⟩⊗⟨ identityˡ ⟩
(f ∘ g) ⊗₁ (id ∘ h) ≈⟨ ⊗-distrib-over-∘ ⟩
f ⊗₁ id ∘ g ⊗₁ h ∎
-- Split a composite in the second component
--
-- | | | | | |
-- | [h] | [h] [f] [h]
-- [f] | = | | = | |
-- | [g] [f] [g] | [g]
-- | | | | | |
split₂ʳ : ∀ {X₁ Y₁ X₂ Y₂ Z₂} {f : X₁ ⇒ Y₁} {g : Y₂ ⇒ Z₂} {h : X₂ ⇒ Y₂} →
f ⊗₁ (g ∘ h) ≈ f ⊗₁ g ∘ id ⊗₁ h
split₂ʳ {f = f} {g} {h} = begin
f ⊗₁ (g ∘ h) ≈˘⟨ identityʳ ⟩⊗⟨refl ⟩
(f ∘ id) ⊗₁ (g ∘ h) ≈⟨ ⊗-distrib-over-∘ ⟩
f ⊗₁ g ∘ id ⊗₁ h ∎
split₂ˡ : ∀ {X₁ Y₁ X₂ Y₂ Z₂} {f : X₁ ⇒ Y₁} {g : Y₂ ⇒ Z₂} {h : X₂ ⇒ Y₂} →
f ⊗₁ (g ∘ h) ≈ id ⊗₁ g ∘ f ⊗₁ h
split₂ˡ {f = f} {g} {h} = begin
f ⊗₁ (g ∘ h) ≈˘⟨ identityˡ ⟩⊗⟨refl ⟩
(id ∘ f) ⊗₁ (g ∘ h) ≈⟨ ⊗-distrib-over-∘ ⟩
id ⊗₁ g ∘ f ⊗₁ h ∎
-- Combined splitting and re-association.
module _ {X Y Z} {f : X ⇒ Z} {g : Y ⇒ Z} {h : X ⇒ Y} (f≈gh : f ≈ g ∘ h) where
infixr 4 split₁_⟩∘⟨_ split₂_⟩∘⟨_
infixl 5 _⟩∘⟨split₁_ _⟩∘⟨split₂_
split₁_⟩∘⟨_ : ∀ {V W} {i j : V ⇒ X ⊗₀ W} → i ≈ j →
f ⊗₁ id ∘ i ≈ g ⊗₁ id ∘ h ⊗₁ id ∘ j
split₁_⟩∘⟨_ {_} {_} {i} {j} i≈j = begin
f ⊗₁ id ∘ i ≈⟨ f≈gh ⟩⊗⟨refl ⟩∘⟨ i≈j ⟩
(g ∘ h) ⊗₁ id ∘ j ≈⟨ split₁ˡ ⟩∘⟨refl ⟩
(g ⊗₁ id ∘ h ⊗₁ id) ∘ j ≈⟨ assoc ⟩
g ⊗₁ id ∘ (h ⊗₁ id ∘ j) ∎
split₂_⟩∘⟨_ : ∀ {V W} {i j : V ⇒ W ⊗₀ X} → i ≈ j →
id ⊗₁ f ∘ i ≈ id ⊗₁ g ∘ id ⊗₁ h ∘ j
split₂_⟩∘⟨_ {_} {_} {i} {j} i≈j = begin
id ⊗₁ f ∘ i ≈⟨ refl⟩⊗⟨ f≈gh ⟩∘⟨ i≈j ⟩
id ⊗₁ (g ∘ h) ∘ j ≈⟨ split₂ˡ ⟩∘⟨refl ⟩
(id ⊗₁ g ∘ id ⊗₁ h) ∘ j ≈⟨ assoc ⟩
id ⊗₁ g ∘ (id ⊗₁ h ∘ j) ∎
_⟩∘⟨split₁_ : ∀ {V W} {i j : Z ⊗₀ W ⇒ V} → i ≈ j →
i ∘ f ⊗₁ id ≈ (j ∘ g ⊗₁ id) ∘ h ⊗₁ id
_⟩∘⟨split₁_ {_} {_} {i} {j} i≈j = begin
i ∘ f ⊗₁ id ≈⟨ i≈j ⟩∘⟨ f≈gh ⟩⊗⟨refl ⟩
j ∘ (g ∘ h) ⊗₁ id ≈⟨ refl⟩∘⟨ split₁ˡ ⟩
j ∘ (g ⊗₁ id ∘ h ⊗₁ id) ≈⟨ sym-assoc ⟩
(j ∘ g ⊗₁ id) ∘ h ⊗₁ id ∎
_⟩∘⟨split₂_ : ∀ {V W} {i j : W ⊗₀ Z ⇒ V} → i ≈ j →
i ∘ id ⊗₁ f ≈ (j ∘ id ⊗₁ g) ∘ id ⊗₁ h
_⟩∘⟨split₂_ {_} {_} {i} {j} i≈j = begin
i ∘ id ⊗₁ f ≈⟨ i≈j ⟩∘⟨ refl⟩⊗⟨ f≈gh ⟩
j ∘ id ⊗₁ (g ∘ h) ≈⟨ refl⟩∘⟨ split₂ˡ ⟩
j ∘ (id ⊗₁ g ∘ id ⊗₁ h) ≈⟨ sym-assoc ⟩
(j ∘ id ⊗₁ g) ∘ id ⊗₁ h ∎
-- Combined merging and re-association.
module _ {X Y Z} {f : Y ⇒ Z} {g : X ⇒ Y} {h : X ⇒ Z} (fg≈h : f ∘ g ≈ h) where
infixr 4 merge₁_⟩∘⟨_ merge₂_⟩∘⟨_
infixl 5 _⟩∘⟨merge₁_ _⟩∘⟨merge₂_
merge₁_⟩∘⟨_ : ∀ {V W} {i j : V ⇒ X ⊗₀ W} → i ≈ j →
f ⊗₁ id ∘ g ⊗₁ id ∘ i ≈ h ⊗₁ id ∘ j
merge₁_⟩∘⟨_ i≈j = ⟺ (split₁ ⟺ fg≈h ⟩∘⟨ ⟺ i≈j)
merge₂_⟩∘⟨_ : ∀ {V W} {i j : V ⇒ W ⊗₀ X} → i ≈ j →
id ⊗₁ f ∘ id ⊗₁ g ∘ i ≈ id ⊗₁ h ∘ j
merge₂_⟩∘⟨_ i≈j = ⟺ (split₂ ⟺ fg≈h ⟩∘⟨ ⟺ i≈j)
_⟩∘⟨merge₁_ : ∀ {V W} {i j : Z ⊗₀ W ⇒ V} → i ≈ j →
(i ∘ f ⊗₁ id) ∘ g ⊗₁ id ≈ j ∘ h ⊗₁ id
_⟩∘⟨merge₁_ i≈j = ⟺ (⟺ fg≈h ⟩∘⟨split₁ ⟺ i≈j)
_⟩∘⟨merge₂_ : ∀ {V W} {i j : W ⊗₀ Z ⇒ V} → i ≈ j →
(i ∘ id ⊗₁ f) ∘ id ⊗₁ g ≈ j ∘ id ⊗₁ h
_⟩∘⟨merge₂_ i≈j = ⟺ (⟺ fg≈h ⟩∘⟨split₂ ⟺ i≈j)
|
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module Issue254 where
data Unit : Set where
* : Unit
data Nat : Set where
zero : Nat
suc : Nat → Nat
data Vec : Nat → Set where
cons : ∀ n → Vec (suc n)
remove : ∀ n → Nat → Vec (suc n) → Unit
remove n x (cons .n) with *
remove n x (cons .n) | * = {!!}
|
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{-
This file contains:
- Definition of functions of the equivalence between FreeGroup and the FundamentalGroup
- Definition of encode decode functions
- Proof that for all x : Bouquet A → p : base ≡ x → decode x (encode x p) ≡ p (no truncations)
- Proof of the truncated versions of encodeDecode and of right-homotopy
- Definition of truncated encode decode functions
- Proof of the truncated versions of decodeEncode and encodeDecode
- Proof that π₁Bouquet ≡ FreeGroup A
-}
{-# OPTIONS --safe #-}
module Cubical.HITs.Bouquet.FundamentalGroupProof where
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Pointed
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws renaming (assoc to pathAssoc)
open import Cubical.HITs.SetTruncation hiding (rec2)
open import Cubical.HITs.PropositionalTruncation hiding (map ; elim) renaming (rec to propRec)
open import Cubical.Algebra.Group
open import Cubical.Homotopy.Group.Base
open import Cubical.Homotopy.Loopspace
open import Cubical.HITs.Bouquet.Base
open import Cubical.HITs.FreeGroup.Base
open import Cubical.HITs.FreeGroupoid
private
variable
ℓ : Level
A : Type ℓ
-- Pointed versions of the non truncated types
ΩBouquet : {A : Type ℓ} → Pointed ℓ
ΩBouquet {A = A} = Ω (Bouquet∙ A)
FreeGroupoid∙ : {A : Type ℓ} → Pointed ℓ
FreeGroupoid∙ {A = A} = FreeGroupoid A , ε
-- Functions without using the truncated forms of types
looping : FreeGroupoid A → typ ΩBouquet
looping (η a) = loop a
looping (g1 · g2) = looping g1 ∙ looping g2
looping ε = refl
looping (inv g) = sym (looping g)
looping (assoc g1 g2 g3 i) = pathAssoc (looping g1) (looping g2) (looping g3) i
looping (idr g i) = rUnit (looping g) i
looping (idl g i) = lUnit (looping g) i
looping (invr g i) = rCancel (looping g) i
looping (invl g i) = lCancel (looping g) i
looping∙ : FreeGroupoid∙ →∙ ΩBouquet {A = A}
looping∙ = looping , refl
code : {A : Type ℓ} → (Bouquet A) → Type ℓ
code {A = A} base = (FreeGroupoid A)
code (loop a i) = pathsInU (η a) i
winding : typ ΩBouquet → FreeGroupoid A
winding l = subst code l ε
winding∙ : ΩBouquet →∙ FreeGroupoid∙ {A = A}
winding∙ = winding , refl
-- Functions using the truncated forms of types
π₁Bouquet : {A : Type ℓ} → Type ℓ
π₁Bouquet {A = A} = π 1 (Bouquet∙ A)
loopingT : ∥ FreeGroupoid A ∥₂ → π₁Bouquet
loopingT = map looping
windingT : π₁Bouquet → ∥ FreeGroupoid A ∥₂
windingT = map winding
-- Utility proofs
substPathsR : {C : Type ℓ}{y z : C} → (x : C) → (p : y ≡ z) → subst (λ y → x ≡ y) p ≡ λ q → q ∙ p
substPathsR {y = y} x p = funExt homotopy where
homotopy : ∀ q → subst (λ y → x ≡ y) p q ≡ q ∙ p
homotopy q = J P d p where
P : ∀ z' → y ≡ z' → Type _
P z' p' = subst (λ y → x ≡ y) p' q ≡ q ∙ p'
d : P y refl
d =
subst (λ y → x ≡ y) refl q
≡⟨ substRefl {B = λ y → x ≡ y} q ⟩
q
≡⟨ rUnit q ⟩
q ∙ refl ∎
substFunctions : {B C : A → Type ℓ}{x y : A}
→ (p : x ≡ y)
→ (f : B x → C x)
→ subst (λ z → (B z → C z)) p f ≡ subst C p ∘ f ∘ subst B (sym p)
substFunctions {B = B} {C = C} {x = x} p f = J P d p where
auxC : idfun (C x) ≡ subst C refl
auxC = funExt (λ c → sym (substRefl {B = C} c))
auxB : idfun (B x) ≡ subst B refl
auxB = funExt (λ b → sym (substRefl {B = B} b))
P : ∀ y' → x ≡ y' → Type _
P y' p' = subst (λ z → (B z → C z)) p' f ≡ subst C p' ∘ f ∘ subst B (sym p')
d : P x refl
d =
subst (λ z → (B z → C z)) refl f
≡⟨ substRefl {B = λ z → (B z → C z)} f ⟩
f
≡⟨ cong (λ h → h ∘ f ∘ idfun (B x)) auxC ⟩
subst C refl ∘ f ∘ idfun (B x)
≡⟨ cong (λ h → subst C refl ∘ f ∘ h) auxB ⟩
subst C refl ∘ f ∘ subst B (sym refl) ∎
-- Definition of the encode - decode functions over the family of types Π(x : W A) → code x
encode : (x : Bouquet A) → base ≡ x → code x
encode x l = subst code l ε
decode : {A : Type ℓ}(x : Bouquet A) → code x → base ≡ x
decode {A = A} base = looping
decode {A = A} (loop a i) = path i where
pathover : PathP (λ i → (code (loop a i) → base ≡ (loop a i))) looping (subst (λ z → (code z → base ≡ z)) (loop a) looping)
pathover = subst-filler (λ z → (code z → base ≡ z)) (loop a) looping
aux : (a : A) → subst code (sym (loop a)) ≡ action (inv (η a))
aux a = funExt homotopy where
homotopy : ∀ (g : FreeGroupoid A) → subst code (sym (loop a)) g ≡ action (inv (η a)) g
homotopy g =
subst code (sym (loop a)) g
≡⟨ cong (λ x → transport x g) (sym (invPathsInUNaturality (η a))) ⟩
transport (pathsInU (inv (η a))) g
≡⟨ uaβ (equivs (inv (η a))) g ⟩
action (inv (η a)) g ∎
calculations : ∀ (a : A) → ∀ g → looping (g · (inv (η a))) ∙ loop a ≡ looping g
calculations a g =
looping (g · (inv (η a))) ∙ loop a
≡⟨ sym (pathAssoc (looping g) (sym (loop a)) (loop a)) ⟩
looping g ∙ (sym (loop a) ∙ loop a)
≡⟨ cong (λ x → looping g ∙ x) (lCancel (loop a)) ⟩
looping g ∙ refl
≡⟨ sym (rUnit (looping g)) ⟩
looping g ∎
path' : subst (λ z → (code z → base ≡ z)) (loop a) looping ≡ looping
path' =
subst (λ z → (code z → base ≡ z)) (loop a) looping
≡⟨ substFunctions {B = code} {C = λ z → base ≡ z} (loop a) looping ⟩
subst (λ z → base ≡ z) (loop a) ∘ looping ∘ subst code (sym (loop a))
≡⟨ cong (λ x → x ∘ looping ∘ subst code (sym (loop a))) (substPathsR base (loop a)) ⟩
(λ p → p ∙ loop a) ∘ looping ∘ subst code (sym (loop a))
≡⟨ cong (λ x → (λ p → p ∙ loop a) ∘ looping ∘ x) (aux a) ⟩
(λ p → p ∙ loop a) ∘ looping ∘ action (inv (η a))
≡⟨ funExt (calculations a) ⟩
looping ∎
path'' : PathP (λ i → code ((loop a ∙ refl) i) → base ≡ ((loop a ∙ refl) i)) looping looping
path'' = compPathP' {A = Bouquet A} {B = λ z → code z → base ≡ z} pathover path'
p''≡p : PathP (λ i → code ((loop a ∙ refl) i) → base ≡ ((loop a ∙ refl) i)) looping looping ≡
PathP (λ i → code (loop a i) → base ≡ (loop a i)) looping looping
p''≡p = cong (λ x → PathP (λ i → code (x i) → base ≡ (x i)) looping looping) (sym (rUnit (loop a)))
path : PathP (λ i → code (loop a i) → base ≡ (loop a i)) looping looping
path = transport p''≡p path''
-- Non truncated Left Homotopy
decodeEncode : (x : Bouquet A) → (p : base ≡ x) → decode x (encode x p) ≡ p
decodeEncode x p = J P d p where
P : (x' : Bouquet A) → base ≡ x' → Type _
P x' p' = decode x' (encode x' p') ≡ p'
d : P base refl
d =
decode base (encode base refl)
≡⟨ cong (λ e' → looping e') (transportRefl ε) ⟩
refl ∎
left-homotopy : ∀ (l : typ (ΩBouquet {A = A})) → looping (winding l) ≡ l
left-homotopy l = decodeEncode base l
-- Truncated proofs of right homotopy of winding/looping functions
truncatedPathEquality : (g : FreeGroupoid A) → ∥ cong code (looping g) ≡ ua (equivs g) ∥₁
truncatedPathEquality = elimProp
Bprop
(λ a → ∣ η-ind a ∣₁)
(λ g1 g2 → λ ∣ind1∣₁ ∣ind2∣₁ → rec2 squash₁ (λ ind1 ind2 → ∣ ·-ind g1 g2 ind1 ind2 ∣₁) ∣ind1∣₁ ∣ind2∣₁)
∣ ε-ind ∣₁
(λ g → λ ∣ind∣₁ → propRec squash₁ (λ ind → ∣ inv-ind g ind ∣₁) ∣ind∣₁) where
B : ∀ g → Type _
B g = cong code (looping g) ≡ ua (equivs g)
Bprop : ∀ g → isProp ∥ B g ∥₁
Bprop g = squash₁
η-ind : ∀ a → B (η a)
η-ind a = refl
·-ind : ∀ g1 g2 → B g1 → B g2 → B (g1 · g2)
·-ind g1 g2 ind1 ind2 =
cong code (looping (g1 · g2))
≡⟨ cong (λ x → x ∙ cong code (looping g2)) ind1 ⟩
pathsInU g1 ∙ cong code (looping g2)
≡⟨ cong (λ x → pathsInU g1 ∙ x) ind2 ⟩
pathsInU g1 ∙ pathsInU g2
≡⟨ sym (multPathsInUNaturality g1 g2) ⟩
pathsInU (g1 · g2) ∎
ε-ind : B ε
ε-ind =
cong code (looping ε)
≡⟨ sym idPathsInUNaturality ⟩
pathsInU ε ∎
inv-ind : ∀ g → B g → B (inv g)
inv-ind g ind =
cong code (looping (inv g))
≡⟨ cong sym ind ⟩
sym (pathsInU g)
≡⟨ sym (invPathsInUNaturality g) ⟩
ua (equivs (inv g)) ∎
truncatedRight-homotopy : (g : FreeGroupoid A) → ∥ winding (looping g) ≡ g ∥₁
truncatedRight-homotopy g = propRec squash₁ recursion (truncatedPathEquality g) where
recursion : cong code (looping g) ≡ ua (equivs g) → ∥ winding (looping g) ≡ g ∥₁
recursion hyp = ∣ aux ∣₁ where
aux : winding (looping g) ≡ g
aux =
winding (looping g)
≡⟨ cong (λ x → transport x ε) hyp ⟩
transport (ua (equivs g)) ε
≡⟨ uaβ (equivs g) ε ⟩
ε · g
≡⟨ sym (idl g) ⟩
g ∎
right-homotopyInTruncatedGroupoid : (g : FreeGroupoid A) → ∣ winding (looping g) ∣₂ ≡ ∣ g ∣₂
right-homotopyInTruncatedGroupoid g = Iso.inv PathIdTrunc₀Iso (truncatedRight-homotopy g)
-- Truncated encodeDecode over all fibrations
truncatedEncodeDecode : (x : Bouquet A) → (g : code x) → ∥ encode x (decode x g) ≡ g ∥₁
truncatedEncodeDecode base = truncatedRight-homotopy
truncatedEncodeDecode (loop a i) = isProp→PathP prop truncatedRight-homotopy truncatedRight-homotopy i where
prop : ∀ i → isProp (∀ (g : code (loop a i)) → ∥ encode (loop a i) (decode (loop a i) g) ≡ g ∥₁)
prop i f g = funExt pointwise where
pointwise : (x : code (loop a i)) → PathP (λ _ → ∥ encode (loop a i) (decode (loop a i) x) ≡ x ∥₁) (f x) (g x)
pointwise x = isProp→PathP (λ i → squash₁) (f x) (g x)
encodeDecodeInTruncatedGroupoid : (x : Bouquet A) → (g : code x) → ∣ encode x (decode x g) ∣₂ ≡ ∣ g ∣₂
encodeDecodeInTruncatedGroupoid x g = Iso.inv PathIdTrunc₀Iso (truncatedEncodeDecode x g)
-- Encode Decode over the truncated versions of the types
encodeT : (x : Bouquet A) → ∥ base ≡ x ∥₂ → ∥ code x ∥₂
encodeT x = map (encode x)
decodeT : (x : Bouquet A) → ∥ code x ∥₂ → ∥ base ≡ x ∥₂
decodeT x = map (decode x)
decodeEncodeT : (x : Bouquet A) → (p : ∥ base ≡ x ∥₂) → decodeT x (encodeT x p) ≡ p
decodeEncodeT x g = elim sethood induction g where
sethood : (q : ∥ base ≡ x ∥₂) → isSet (decodeT x (encodeT x q) ≡ q)
sethood q = isProp→isSet (squash₂ (decodeT x (encodeT x q)) q)
induction : (l : base ≡ x) → ∣ decode x (encode x l) ∣₂ ≡ ∣ l ∣₂
induction l = cong (λ l' → ∣ l' ∣₂) (decodeEncode x l)
encodeDecodeT : (x : Bouquet A) → (g : ∥ code x ∥₂) → encodeT x (decodeT x g) ≡ g
encodeDecodeT x g = elim sethood induction g where
sethood : (z : ∥ code x ∥₂) → isSet (encodeT x (decodeT x z) ≡ z)
sethood z = isProp→isSet (squash₂ (encodeT x (decodeT x z)) z)
induction : (a : code x) → ∣ encode x (decode x a) ∣₂ ≡ ∣ a ∣₂
induction a = encodeDecodeInTruncatedGroupoid x a
-- Final equivalences
TruncatedFamiliesIso : (x : Bouquet A) → Iso ∥ base ≡ x ∥₂ ∥ code x ∥₂
Iso.fun (TruncatedFamiliesIso x) = encodeT x
Iso.inv (TruncatedFamiliesIso x) = decodeT x
Iso.rightInv (TruncatedFamiliesIso x) = encodeDecodeT x
Iso.leftInv (TruncatedFamiliesIso x) = decodeEncodeT x
TruncatedFamiliesEquiv : (x : Bouquet A) → ∥ base ≡ x ∥₂ ≃ ∥ code x ∥₂
TruncatedFamiliesEquiv x = isoToEquiv (TruncatedFamiliesIso x)
TruncatedFamilies≡ : (x : Bouquet A) → ∥ base ≡ x ∥₂ ≡ ∥ code x ∥₂
TruncatedFamilies≡ x = ua (TruncatedFamiliesEquiv x)
π₁Bouquet≡∥FreeGroupoid∥₂ : π₁Bouquet ≡ ∥ FreeGroupoid A ∥₂
π₁Bouquet≡∥FreeGroupoid∥₂ = TruncatedFamilies≡ base
π₁Bouquet≡FreeGroup : {A : Type ℓ} → π₁Bouquet ≡ FreeGroup A
π₁Bouquet≡FreeGroup {A = A} =
π₁Bouquet
≡⟨ π₁Bouquet≡∥FreeGroupoid∥₂ ⟩
∥ FreeGroupoid A ∥₂
≡⟨ sym freeGroupTruncIdempotent ⟩
FreeGroup A ∎
|
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open import Agda.Builtin.Reflection
open import Agda.Builtin.Unit
open import Agda.Builtin.List
open import Agda.Builtin.Equality
infixl 4 _>>=_
_>>=_ = bindTC
data Tm : Set where
[_] : Term → Tm
macro
qType : Term → Term → TC ⊤
qType t hole = inferType t >>= quoteTC >>= unify hole
qTerm : Term → Term → TC ⊤
qTerm t hole = quoteTC t >>= unify hole
unQ : Tm → Term → TC ⊤
unQ [ t ] hole = unify hole t
postulate
X : Set
x y z : X
id : (A : Set) → A → A
id _ x = x
record R (A B : Set) : Set₁ where
field
F : X → X → X → Set
bar : F x y z → Term
bar fx = qType fx -- result: F z (x y are dropped)
check-bar : F x y z → F x y z
check-bar fx = id (unQ [ bar fx ]) fx
baz : ∀ {A B} (r : R A B) → R.F r x y z → Term
baz r fx = qType fx
check-baz : ∀ {A B} (r : R A B) → R.F r x y z → R.F r x y z
check-baz r fx = id (unQ [ baz r fx ]) fx
module M (A B : Set) where
data D : Set where
d : D
`d = qTerm d
d′ = unquote (unify `d)
`Md = qTerm (M.d {X} {X})
Md : M.D X X
Md = unquote (unify `Md)
|
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{-# OPTIONS --without-K --rewriting --allow-unsolved-metas #-}
open import HoTT renaming (pt to pt⊙)
open import homotopy.DisjointlyPointedSet
open import lib.types.Nat
open import lib.types.Vec
module simplicial.SCequivCW where
open import cw.CW public
open import cw.examples.Sphere public
open import simplicial.Base
SC-to-CW : {dim : ℕ} → SC (S dim) → Skeleton {lzero} dim
SC-to-CW {dim} sc = SC-to-CW' dim dim ltS sc
where
SC-to-CW' : (n : ℕ) → (predim : ℕ) → (p : n < (S predim)) → SC (S predim) → Skeleton {lzero} n
SC-to-CW' 0 predim _ (complex ss _) = (Fin (length (lookup ss (0 , O<S predim)))) , Fin-is-set
SC-to-CW' (S n) predim p (complex ss c) =
attached-skeleton
(SC-to-CW' n predim (<-cancel-S-left p) (complex ss c))
((Fin (length (lookup ss (n , <-cancel-S-left p)))) , Fin-is-set)
λ cells sph → {!!}
|
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.ZCohomology.KcompPrelims where
open import Cubical.ZCohomology.Base
open import Cubical.Homotopy.Connected
open import Cubical.HITs.Hopf
open import Cubical.Homotopy.Freudenthal hiding (encode)
open import Cubical.HITs.Sn
open import Cubical.HITs.S1
open import Cubical.HITs.Truncation renaming (elim to trElim ; rec to trRec ; map to trMap)
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Transport
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Univalence
open import Cubical.Foundations.Equiv.HalfAdjoint
open import Cubical.Data.Int renaming (_+_ to +Int)
open import Cubical.Data.Nat hiding (_·_)
open import Cubical.Data.Unit
open import Cubical.HITs.Susp
open import Cubical.HITs.Nullification
open import Cubical.Data.Prod.Base
open import Cubical.Homotopy.Loopspace
open import Cubical.Data.Bool
open import Cubical.Data.Sum.Base
open import Cubical.Data.Sigma hiding (_×_)
open import Cubical.Foundations.Function
open import Cubical.Foundations.Pointed
open import Cubical.HITs.S3
private
variable
ℓ : Level
A : Type ℓ
{- We want to prove that Kn≃ΩKn+1. For this we use the map ϕ-}
ϕ : (pt a : A) → typ (Ω (Susp A , north))
ϕ pt a = (merid a) ∙ sym (merid pt)
private
Kn→ΩKn+1 : (n : ℕ) → coHomK n → typ (Ω (coHomK-ptd (suc n)))
Kn→ΩKn+1 zero x i = ∣ intLoop x i ∣
Kn→ΩKn+1 (suc zero) = trRec (isOfHLevelTrunc 4 ∣ north ∣ ∣ north ∣)
λ a i → ∣ ϕ base a i ∣
Kn→ΩKn+1 (suc (suc n)) = trRec (isOfHLevelTrunc (2 + (3 + n)) ∣ north ∣ ∣ north ∣)
λ a i → ∣ ϕ north a i ∣
d-map : typ (Ω ((Susp S¹) , north)) → S¹
d-map p = subst HopfSuspS¹ p base
d-mapId : (r : S¹) → d-map (ϕ base r) ≡ r
d-mapId r = substComposite HopfSuspS¹ (merid r) (sym (merid base)) base ∙
rotLemma r
where
rotLemma : (r : S¹) → r · base ≡ r
rotLemma base = refl
rotLemma (loop i) = refl
sphereConnectedSpecCase : isConnected 4 (Susp (Susp S¹))
sphereConnectedSpecCase = sphereConnected 3
d-mapComp : Iso (fiber d-map base) (Path (S₊ 3) north north)
d-mapComp = compIso (IsoΣPathTransportPathΣ {B = HopfSuspS¹} _ _)
(congIso (invIso IsoS³TotalHopf))
is1Connected-dmap : isConnectedFun 3 d-map
is1Connected-dmap = toPropElim (λ _ → isPropIsOfHLevel 0)
(isConnectedRetractFromIso 3 d-mapComp
(isOfHLevelRetractFromIso 0 (invIso (PathIdTruncIso 3))
contrHelper))
where
contrHelper : isContr (Path (∥ Susp (Susp S¹) ∥ 4) ∣ north ∣ ∣ north ∣)
fst contrHelper = refl
snd contrHelper = isOfHLevelPlus {n = 0} 2 (sphereConnected 3) ∣ north ∣ ∣ north ∣ refl
d-Iso : Iso (∥ Path (S₊ 2) north north ∥ 3) (coHomK 1)
d-Iso = connectedTruncIso _ d-map is1Connected-dmap
d-mapId2 : Iso.fun d-Iso ∘ trMap (ϕ base) ≡ idfun (coHomK 1)
d-mapId2 = funExt (trElim (λ _ → isOfHLevelPath 3 (isOfHLevelTrunc 3) _ _) λ a i → ∣ d-mapId a i ∣)
Iso∥ϕ₁∥ : Iso (coHomK 1) (∥ Path (S₊ 2) north north ∥ 3)
Iso∥ϕ₁∥ = composesToId→Iso d-Iso (trMap (ϕ base)) d-mapId2
Iso-Kn-ΩKn+1 : (n : HLevel) → Iso (coHomK n) (typ (Ω (coHomK-ptd (suc n))))
Iso-Kn-ΩKn+1 zero = invIso (compIso (congIso (truncIdempotentIso _ isGroupoidS¹)) ΩS¹IsoInt)
Iso-Kn-ΩKn+1 (suc zero) = compIso Iso∥ϕ₁∥ (invIso (PathIdTruncIso 3))
Iso-Kn-ΩKn+1 (suc (suc n)) = compIso (stabSpheres-n≥2 n)
(invIso (PathIdTruncIso (4 + n)))
where
helper : n + (4 + n) ≡ 2 + (n + (2 + n))
helper = +-suc n (3 + n) ∙ (λ i → suc (+-suc n (2 + n) i))
|
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module Simple where
open import Data.Nat renaming (_≟_ to _≟ℕ_)
open import Data.Fin hiding (_+_; inject)
open import Data.String hiding (_++_) renaming (_≟_ to _≟S_)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Binary.PropositionalEquality using (refl; _≡_)
open import Data.List
open import Data.Sum using (_⊎_; inj₁; inj₂)
FName : Set
FName = String
FNames : Set
FNames = List FName
data Expr : ℕ → Set where
num : ∀ {n} → ℕ → Expr n
bv : ∀ {n} → (i : Fin n) → Expr n
fv : ∀ {n} → (x : FName) → Expr n
ƛ : ∀ {n} → (e : Expr (suc n)) → Expr n
_·_ : ∀ {n} → (f : Expr n) → (e : Expr n) → Expr n
-- This idx indicates the amount of existence of lambda
-- out of current express
Expr0 : Set
Expr0 = Expr zero
↓ℕ≠ℕ : ∀ {n m} {i : Fin m}
→ ¬ (suc n ≡ toℕ (suc i))
→ ¬ (n ≡ toℕ i)
↓ℕ≠ℕ {n} {m} {i} sn≠si n≡i rewrite n≡i = sn≠si refl
-- ↓ℕ≠ℕ {n} {m} {i} sn≠si n≡i with n≡i
-- ↓ℕ≠ℕ sn≠si n≡i | refl = sn≠si refl
↓fin : ∀ {n} → (i : Fin (suc n)) → ¬ (n ≡ toℕ i) → Fin n
↓fin {zero} zero 0≠0 with 0≠0 refl
↓fin {zero} zero 0≠0 | ()
↓fin {zero} (suc ()) 0≠n
↓fin {suc n} zero i≠0 = zero
↓fin {suc n} (suc i) sn≠si = suc (↓fin i (↓ℕ≠ℕ sn≠si))
↑expr : ∀ {n} → Expr n → Expr (suc n)
↑expr (num i) = num i
↑expr (bv i) = bv (inject₁ i)
-- inject₁ : ∀ {m} → Fin m → Fin (suc m)
↑expr (fv x) = fv x
↑expr (ƛ e) = ƛ (↑expr e)
↑expr (e · e₁) = ↑expr e · ↑expr e₁
{- substitution for bounded and free variable-}
[_↦_] : ∀ n → Expr n → Expr (suc n) → Expr n
[ m ↦ t ] (num i) = num i
[ m ↦ t ] (bv i) with m ≟ℕ toℕ i
... | yes m=i = t
... | no m≠i = bv (↓fin i m≠i)
[ m ↦ t ] (fv x) = fv x
[ m ↦ t ] (ƛ e) = ƛ ([ suc m ↦ ↑expr t ] e)
[ m ↦ t ] (e · e₁) = [ m ↦ t ] e · [ m ↦ t ] e₁
[_↤_] : ∀ n → FName → Expr n → Expr (suc n)
[ m ↤ name ] (num x) = num x
[ m ↤ name ] (bv i) = ↑expr (bv i)
[ m ↤ name ] (fv x) with x ≟S name
[ m ↤ name ] (fv x) | yes p = bv (fromℕ m)
[ m ↤ name ] (fv x) | no ¬p = fv x
[ m ↤ name ] (ƛ t) = ƛ ([ suc m ↤ name ] t)
[ m ↤ name ] (t · t₁) = [ m ↤ name ] t · [ m ↤ name ] t₁
[_↝_] : ∀ {n} → FName → Expr n → Expr n → Expr n
[ n ↝ t ] (num i) = num i
[ n ↝ t ] (bv i) = bv i
[ n ↝ t ] (fv x) with n ≟S x
[ n ↝ t ] (fv x) | yes p = t
[ n ↝ t ] (fv x) | no ¬p = fv x
[ n ↝ t ] (ƛ x) = ƛ ([ n ↝ ↑expr t ] x)
[ n ↝ t ] (x · y) = [ n ↝ t ] x · [ n ↝ t ] y
_₀↦_ : Expr 1 → Expr 0 → Expr 0
m ₀↦ t = [ 0 ↦ t ] m
_↦₀_ : FName → Expr 0 → Expr 1
name ↦₀ t = [ 0 ↤ name ] t
_₀↤_ : Expr 0 → FName → Expr 1
t ₀↤ x = [ 0 ↤ x ] t
_₀↝_ : Expr 1 → FName → Expr 0
x ₀↝ s = x ₀↦ (fv s)
fvars : ∀ {n} → Expr n → FNames
fvars (num x) = []
fvars (bv i) = []
fvars (fv x) = x ∷ []
fvars (ƛ f) = fvars f
fvars (f · x) = fvars f ++ fvars x
-- ############################## --
{- locally closed -}
-- ############################## --
open import Data.List.Any as Any
open Any.Membership-≡ using (_∈_; _∉_)
open import Data.Product
data LC : ∀ {n} → Expr n → Set where
numᶜ : ∀ {n} → (i : ℕ) → LC {n} (num i)
fvᶜ : ∀ {n}
→ (x : FName)
→ LC {n} (fv x)
_·ᶜ_ : ∀ {n} {f x}
→ LC {n} f
→ LC {n} x
→ LC {n} (f · x)
ƛᶜ : ∀ {e}
→ (ns : FNames)
→ ( ∀ {x} → x ∉ ns → LC {0} (e ₀↦ fv x) )
→ LC {0} (ƛ e)
postulate fresh-gen : FNames → FName
postulate fresh-gen-spec : ∀ ns → fresh-gen ns ∉ ns
genName : (ns : FNames) → ∃ (λ x → x ∉ ns)
genName ns = fresh-gen ns , fresh-gen-spec ns
orz : ∀ {e} {nn} → nn ↦₀ (e ₀↦ fv nn) ≡ e
orz = {! !}
foo : ∀ {nn} → (e : Expr 1) → LC {0} (e ₀↦ fv nn) → LC {1} e
foo (num i) lcp = {! !}
foo (bv i) lcp = {! !}
foo (fv y) lcp = {! !}
foo (ƛ e) lcp = {! !}
foo (f · x) lcp = {! !}
absᶜ : ∀ {e} → LC {0} (ƛ e) → LC {1} e
absᶜ (ƛᶜ ns lcex) = {! !}
where nn = proj₁ (genName ns)
nn∉ns = proj₂ (genName ns)
x = lcex {nn} nn∉ns
appᶜ₁ : ∀ {n} {f x} → LC {n} (f · x) → LC {n} f
appᶜ₁ (lcf ·ᶜ lcx) = lcf
appᶜ₂ : ∀ {n} {f x} → LC {n} (f · x) → LC {n} x
appᶜ₂ (lcf ·ᶜ lcx) = lcx
lc? : ∀ {n} → (e : Expr n) → (LC {n} e ⊎ ¬ LC {n} e)
lc? (num x) = inj₁ (numᶜ x)
lc? (bv i) = inj₂ (λ ())
lc? (fv x) = inj₁ (fvᶜ x)
lc? (ƛ e) with lc? e
lc? (ƛ e) | inj₁ x = inj₁ {! !}
lc? (ƛ e) | inj₂ y = inj₂ {! !}
lc? (f · x) with lc? f | lc? x
lc? (f · x) | inj₁ f' | inj₁ x' = inj₁ (f' ·ᶜ x')
lc? (f · x) | inj₁ f' | inj₂ x' = inj₂ (λ p → x' (appᶜ₂ p))
lc? (f · x) | inj₂ f' | _ = inj₂ (λ p → f' (appᶜ₁ p))
-- ############################## --
{- value and semantics -}
-- ############################## --
data Val : Expr0 → Set where
num⁰ : ∀ n → Val (num n)
ƛ⁰ : ∀ e → Val (ƛ e)
var? : (e : Expr0) → (Val e ⊎ ¬ (Val e))
var? (num i) = inj₁ (num⁰ i)
var? (bv i) = inj₂ (λ ())
var? (fv x) = inj₂ (λ ())
var? (ƛ x) = inj₁ (ƛ⁰ x)
var? (x · y) = inj₂ (λ ())
-- don't know what the hell this shit is
data _⟼_ : Expr 0 → Expr 0 → Set where
app : ∀ {body para}
→ ((ƛ body) · para) ⟼ (body ₀↦ para)
|
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{-# OPTIONS --cubical --safe --postfix-projections #-}
module Data.Fin.Properties where
open import Prelude
open import Data.Fin.Base
import Data.Nat.Properties as ℕ
open import Data.Nat.Properties using (+-comm)
open import Data.Nat
open import Function.Injective
open import Agda.Builtin.Nat renaming (_<_ to _<ᵇ_)
private
variable
n m : ℕ
suc-natfin : Σ[ m ⦂ ℕ ] (m ℕ.< n) → Σ[ m ⦂ ℕ ] (m ℕ.< suc n)
suc-natfin (m , p) = suc m , p
Fin-to-Nat-lt : Fin n → Σ[ m ⦂ ℕ ] (m ℕ.< n)
Fin-to-Nat-lt {n = suc n} f0 = zero , tt
Fin-to-Nat-lt {n = suc n} (fs x) = suc-natfin (Fin-to-Nat-lt x)
Fin-from-Nat-lt : ∀ m → m ℕ.< n → Fin n
Fin-from-Nat-lt {n = suc n} zero p = f0
Fin-from-Nat-lt {n = suc n} (suc m) p = fs (Fin-from-Nat-lt m p)
Fin-Nat-lt-rightInv : ∀ m → (p : m ℕ.< n) → Fin-to-Nat-lt {n = n} (Fin-from-Nat-lt m p) ≡ (m , p)
Fin-Nat-lt-rightInv {n = suc n} zero p = refl
Fin-Nat-lt-rightInv {n = suc n} (suc m) p = cong (suc-natfin {n = n}) (Fin-Nat-lt-rightInv {n = n} m p)
Fin-Nat-lt-leftInv : (x : Fin n) → uncurry Fin-from-Nat-lt (Fin-to-Nat-lt x) ≡ x
Fin-Nat-lt-leftInv {n = suc n} f0 = refl
Fin-Nat-lt-leftInv {n = suc n} (fs x) = cong fs (Fin-Nat-lt-leftInv x)
Fin-Nat-lt : Fin n ⇔ Σ[ m ⦂ ℕ ] (m ℕ.< n)
Fin-Nat-lt .fun = Fin-to-Nat-lt
Fin-Nat-lt .inv = uncurry Fin-from-Nat-lt
Fin-Nat-lt .rightInv = uncurry Fin-Nat-lt-rightInv
Fin-Nat-lt .leftInv = Fin-Nat-lt-leftInv
FinToℕ : Fin n → ℕ
FinToℕ {n = suc n} f0 = zero
FinToℕ {n = suc n} (fs x) = suc (FinToℕ x)
FinToℕ-injective : ∀ {k} {m n : Fin k} → FinToℕ m ≡ FinToℕ n → m ≡ n
FinToℕ-injective {suc k} {f0} {f0} _ = refl
FinToℕ-injective {suc k} {f0} {fs x} p = ⊥-elim (ℕ.znots p)
FinToℕ-injective {suc k} {fs m} {f0} p = ⊥-elim (ℕ.snotz p)
FinToℕ-injective {suc k} {fs m} {fs x} p = cong fs (FinToℕ-injective (ℕ.injSuc p))
pred : Fin (suc n) → Fin (suc (ℕ.pred n))
pred f0 = f0
pred {n = suc n} (fs m) = m
discreteFin : ∀ {k} → Discrete (Fin k)
discreteFin {k = suc _} f0 f0 = yes refl
discreteFin {k = suc _} f0 (fs fk) = no (ℕ.znots ∘ cong FinToℕ)
discreteFin {k = suc _} (fs fj) f0 = no (ℕ.snotz ∘ cong FinToℕ)
discreteFin {k = suc _} (fs fj) (fs fk) =
⟦yes cong fs ,no cong (λ { f0 → fk ; (fs x) → x}) ⟧ (discreteFin fj fk)
isSetFin : isSet (Fin n)
isSetFin = Discrete→isSet discreteFin
FinFromℕ : (n m : ℕ) → T (n <ᵇ m) → Fin m
FinFromℕ zero (suc m) p = f0
FinFromℕ (suc n) (suc m) p = fs (FinFromℕ n m p)
infix 4 _≢ᶠ_ _≡ᶠ_
_≢ᶠ_ _≡ᶠ_ : Fin n → Fin n → Type _
n ≢ᶠ m = T (not (discreteFin n m .does))
n ≡ᶠ m = T (discreteFin n m .does)
_F↣_ : ℕ → ℕ → Type₀
n F↣ m = Σ[ f ⦂ (Fin n → Fin m) ] ∀ {x y} → x ≢ᶠ y → f x ≢ᶠ f y
shift : (x y : Fin (suc n)) → x ≢ᶠ y → Fin n
shift f0 (fs y) x≢y = y
shift {suc _} (fs x) f0 x≢y = f0
shift {suc _} (fs x) (fs y) x≢y = fs (shift x y x≢y)
shift-inj : ∀ (x y z : Fin (suc n)) x≢y x≢z → y ≢ᶠ z → shift x y x≢y ≢ᶠ shift x z x≢z
shift-inj f0 (fs y) (fs z) x≢y x≢z x+y≢x+z = x+y≢x+z
shift-inj {suc _} (fs x) f0 (fs z) x≢y x≢z x+y≢x+z = tt
shift-inj {suc _} (fs x) (fs y) f0 x≢y x≢z x+y≢x+z = tt
shift-inj {suc _} (fs x) (fs y) (fs z) x≢y x≢z x+y≢x+z = shift-inj x y z x≢y x≢z x+y≢x+z
shrink : suc n F↣ suc m → n F↣ m
shrink (f , inj) .fst x = shift (f f0) (f (fs x)) (inj tt)
shrink (f , inj) .snd p = shift-inj (f f0) (f (fs _)) (f (fs _)) (inj tt) (inj tt) (inj p)
¬plus-inj : ∀ n m → ¬ (suc (n + m) F↣ m)
¬plus-inj zero zero (f , _) = f f0
¬plus-inj zero (suc m) inj = ¬plus-inj zero m (shrink inj)
¬plus-inj (suc n) m (f , p) = ¬plus-inj n m (f ∘ fs , p)
toFin-inj : (Fin n ↣ Fin m) → n F↣ m
toFin-inj f .fst = f .fst
toFin-inj (f , inj) .snd {x} {y} x≢ᶠy with discreteFin x y | discreteFin (f x) (f y)
... | no ¬p | yes p = ¬p (inj _ _ p)
... | no _ | no _ = tt
n≢sn+m : ∀ n m → Fin n ≢ Fin (suc (n + m))
n≢sn+m n m n≡m =
¬plus-inj m n
(toFin-inj
(subst
(_↣ Fin n)
(n≡m ; cong (Fin ∘ suc) (+-comm n m))
refl-↣))
Fin-inj : Injective Fin
Fin-inj n m eq with compare n m
... | equal _ = refl
... | less n k = ⊥-elim (n≢sn+m n k eq)
... | greater m k = ⊥-elim (n≢sn+m m k (sym eq))
|
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module Data.MoreNatProp where
open import Relation.Binary.PropositionalEquality
using (_≡_ ; refl ; cong)
open import Data.Nat as Nat
using (ℕ ; suc ; zero ; _≤′_ ; _≤_ ; _+_ ; s≤s ; z≤n ; ≤′-refl ;
≤′-step ; _⊔_)
open import Data.Nat.Properties as NatP
using (≤⇒≤′ ; ≤′⇒≤ ; m≤m+n ; s≤′s ; ≤-stepsˡ ; ≤⇒pred≤)
open import Data.Nat.Properties.Simple as NatPS
using (+-comm ; +-suc)
≤-trans : ∀ {x y z} -> x ≤ y -> y ≤ z -> x ≤ z
≤-trans z≤n _ = z≤n
≤-trans (s≤s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)
≤′-trans : ∀ {x y z} -> x ≤′ y -> y ≤′ z -> x ≤′ z
≤′-trans xy yz = ≤⇒≤′ (≤-trans (≤′⇒≤ xy) (≤′⇒≤ yz))
≤′+r : ∀ {x y z} -> x ≤′ y -> z + x ≤′ z + y
≤′+r {x} {y} {zero} x≤′y = x≤′y
≤′+r {x} {y} {suc z} x≤′y = s≤′s (≤′+r {x} {y} {z} x≤′y)
≤′+l : ∀ {x y z} -> x ≤′ y -> x + z ≤′ y + z
≤′+l {x} {y} {z} x≤′y rewrite +-comm x z | +-comm y z = ≤′+r{x}{y}{z} x≤′y
≡is≤′ : ∀ {p q} -> p ≡ q -> p ≤′ q
≡is≤′ p≡q rewrite p≡q = ≤′-refl
≤+b : ∀ x y z w -> x ≤ z -> y ≤ w -> x + y ≤ z + w
≤+b .0 y z w Nat.z≤n y≤w = ≤-stepsˡ z y≤w
≤+b .(suc x) y .(suc z) w (Nat.s≤s{x}{z} x≤z) y≤w = s≤s (≤+b x y z w x≤z y≤w)
≤′+b : ∀ x y z w -> x ≤′ z -> y ≤′ w -> x + y ≤′ z + w
≤′+b x y z w x≤′z y≤′w = ≤⇒≤′ (≤+b x y z w (≤′⇒≤ x≤′z) (≤′⇒≤ y≤′w))
suc≤′⇒≤′ : ∀ x y -> suc x ≤′ y -> x ≤′ y
suc≤′⇒≤′ x .(suc x) ≤′-refl = ≤′-step ≤′-refl
suc≤′⇒≤′ x (suc n) (≤′-step sucx≤′y) = ≤′-step (suc≤′⇒≤′ x n sucx≤′y)
⊔-sym : ∀ n m -> n ⊔ m ≡ m ⊔ n
⊔-sym zero zero = refl
⊔-sym zero (suc m) = refl
⊔-sym (suc n) zero = refl
⊔-sym (suc n) (suc m) = cong suc (⊔-sym n m)
|
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|
-- {-# OPTIONS -v extendedlambda:100 -v int2abs.reifyterm.def:100 #-}
module Issue435 where
data Bool : Set where
true false : Bool
record Unit : Set where
postulate
Dh : ({ x : Bool } → Bool) → Set
Di : ({{x : Bool}} → Bool) → Set
noth : Set
noth = Dh (\ { {true} → false ; {false} → true})
noti : Set
noti = Di (\ { {{true}} → false ; {{false}} → true})
-- Testing absurd patterns
data ⊥ : Set where
data T : Set where
expl : (⊥ → ⊥) → T
impl : ({_ : ⊥} → ⊥) → T
inst : ({{_ : ⊥}} → ⊥) → T
explicit : T
explicit = expl (λ ())
implicit : T
implicit = impl (λ {})
instance : T
instance = inst (λ {{ }})
explicit-match : T
explicit-match = expl (λ { () })
implicit-match : T
implicit-match = impl (λ { {} })
implicit-match′ : T
implicit-match′ = impl (λ { { () } })
instance-match : T
instance-match = inst (λ { {{}} })
instance-match′ : T
instance-match′ = inst (λ { {{ () }} })
|
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{-# OPTIONS --allow-unsolved-metas #-}
open import Agda.Builtin.Equality
postulate
cong₂ : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c} {x y : A} {u v : B}
→ (f : A → B → C) → x ≡ y → u ≡ v → f x u ≡ f y v
test = cong₂ (λ A B → A → B)
|
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module Ex6App where -- git indicates that only commenting out of imports has happened 0/15 + 0/5 gives 0/20
open import CS410-Prelude
open import CS410-Nat
open import CS410-Vec
open import CS410-Indexed
open import CS410-Monoid
open import Ex6AgdaSetup
-- open import Ex5Sol
-- open import Ex5
--open MonadIx TilingMonadIx
--open FunctorIx (monadFunctorIx TilingMonadIx)
---------------------------------------------------------------------------
-- CURSES DISPLAY FOR APPLICATIONS (5 marks) --
---------------------------------------------------------------------------
-- You may need to look at the Ex6AgdaSetup file to find some of the relevant
-- kit for this episode, and it's worth looking there for goodies, anyway.
-- We start from the idea of a main loop.
{- This is the bit of code I wrote in Haskell to animate your code. -}
postulate
mainAppLoop : {S : Set} -> -- program state
-- INITIALIZER
S -> -- initial state
-- EVENT HANDLER
(Event -> S -> -- event and state in
S ** List Action) -> -- new state and screen actions out
-- PUT 'EM TOGETHER AND YOU'VE GOT AN APPLICATION!
IO One
{-# COMPILED mainAppLoop (\ _ -> HaskellSetup.mainAppLoop) #-}
-- The type S ** T is a type of pairs that the compiler can share with
-- Haskell. Its constructor is _,_ just as for S * T.
-- To make a thing you can run, you need to
-- (i) choose a type to represent the program's internal state S
-- (ii) give the initial state
-- (iii) explain how, given an event and the current state, to
-- produce a new state and a list of actions to update the
-- display.
-- Let me show you an example...
-- To run this program, start a terminal, then
--
-- make app
--
-- You should be able to press keys and resize the thing and see sensible
-- stuff happen. Ctrl-C quits.
-- When you're bored of green rectangles, comment out the above main
-- function, so you can move on to the actual work. There are other
-- main functions further down the file which you can comment in as you
-- need them. Of course, you can have at most one main at a time.
-- Now your turn. Making use of the equipment you developed in exercise 5,
-- get us from a Painting to a List Action in two hops. Note that you will
-- have to decide how to render a Hole: some bland background stuff, please.
-- (1 mark)
layerMatrix : [ Layer -:> Matrix Cell ]
layerMatrix p = paste matrixPaste (mapIx fill p) where
fill : [ HoleOr (Matrix Cell) -:> Matrix Cell ]
fill h = {!!}
vecFoldR : {X Y : Set} -> (X -> Y -> Y) -> Y -> {n : Nat} -> Vec X n -> Y
vecFoldR c n [] = n
vecFoldR c n (x :: xs) = c x (vecFoldR c n xs)
matrixAction : forall {wh} -> Matrix Cell wh -> List Action
matrixAction = vecFoldR (vecFoldR {!!} id) []
---------------------------------------------------------------------------
-- APPLICATIONS --
---------------------------------------------------------------------------
-- Here's a general idea of what it means to be an "application".
-- You need to choose some sort of size-dependent state, then provide these
-- bits and pieces. We need to know how the state is updated according to
-- events, with resizing potentially affecting the state's type. We must
-- be able to paint the state. The state should propose a cursor position.
-- (Keen students may modify this definition to ensure the cursor must be
-- within the bounds of the application.)
record Application (S : Nat * Nat -> Set) : Set where
field
handleKey : Key -> [ S -:> S ]
handleResize : {w h : Nat}(w' h' : Nat) -> S (w , h) -> S (w' , h')
paintMe : [ S -:> Layer ]
cursorMe : {w h : Nat} -> S (w , h) -> Nat * Nat -- x,y coords
open Application public
-- Now your turn. Build the appropriate handler to connect these
-- applications with mainLoop. Again, work in two stages, first
-- figuring out how to do the right actions, then managing the
-- state properly. (1 mark)
AppState : (S : Nat * Nat -> Set) -> Set
AppState S = Sg Nat \ w -> Sg Nat \ h -> S (w , h)
appPaint : {S : Nat * Nat -> Set}{w h : Nat} ->
Application S -> S (w , h) -> List Action
appPaint app s =
goRowCol 0 0 :: matrixAction (layerMatrix p)
++ (goRowCol (snd xy) (fst xy) :: [])
where
p = paintMe app s
xy = cursorMe app s
appHandler : {S : Nat * Nat -> Set} ->
Application S ->
Event -> AppState S -> AppState S ** List Action
appHandler app (key k) (w , h , s) = (w , h , s') , appPaint app s'
where s' = handleKey app k s
appHandler app (resize w' h') (w , h , s) = (w' , h' , s') , appPaint app s'
where s' = handleResize app w' h' s
-- Your code turns into a main function, as follows.
appMain : {S : Nat * Nat -> Set} -> Application S -> S (0 , 0) -> IO One
appMain app s = mainAppLoop (0 , 0 , s) (appHandler app)
---------------------------------------------------------------------------
-- A DEMO APPLICATION --
---------------------------------------------------------------------------
sillyChar : Char -> {w h : Nat} -> Layer (w , h)
sillyChar c = ! (block (vec (vec (green - c / black))))
sillyApp : Application \ _ -> Char
sillyApp = record
{ handleKey = \ { (char c) _ -> c ; _ c -> c }
; handleResize = \ _ _ c -> c
; paintMe = \
{ {suc (suc w) , suc (suc h)} c ->
joinV 1 (suc h) refl
(sillyChar c)
(joinV h 1 (sym (plusCommFact 1 h))
(joinH 1 (suc w) refl (sillyChar c)
(joinH w 1 (sym (plusCommFact 1 w)) (sillyChar ' ') (sillyChar c))
)
(sillyChar c) )
; c -> sillyChar c
}
; cursorMe = \ _ -> 0 , 0
}
{- -}
main : IO One
main = appMain sillyApp '*'
{- -}
---------------------------------------------------------------------------
-- COMPARING TWO NUMBERS --
---------------------------------------------------------------------------
-- You've done the tricky part in exercise 5, comparing two splittings of
-- the same number. Here's an easy way to reuse that code just to compare
-- two numbers. It may help in what is to come.
Compare : Nat -> Nat -> Set
Compare x y = CutComparable x y y x (x +N y)
compare : (x y : Nat) -> Compare x y
compare x y = cutCompare x y y x (x +N y) refl (sym (plusCommFact x y))
-- To make sure you've got the message, try writing these things
-- "with compare" to resize paintings. If you need to make a painting
-- bigger, pad its right or bottom with a hole. If you need to make it
-- smaller, trim off the right or bottom excess. You have all the gadgets
-- you need! I'm not giving marks for these, but they'll be useful in
-- the next bit.
cropPadLR : (w h w' : Nat) -> Layer (w , h) -> Layer (w' , h)
cropPadLR w h w' p = {!!}
cropPadTB : (w h h' : Nat) -> Layer (w , h) -> Layer (w , h')
cropPadTB w h h' p = {!!}
---------------------------------------------------------------------------
-- THE MOVING RECTANGLE --
---------------------------------------------------------------------------
-- This is the crux of this episode. Please build me an application which
-- displays a movable resizeable rectangle drawn with ascii art as follows
--
-- +--------------+
-- | |
-- | |
-- +--------------+
--
-- The "size" of the application is the terminal size: the rectangle should
-- be freely resizable *within* the terminal, so you should pad out the
-- rectangle to the size of the screen using Hole.
-- That is, only the rectangle is opaque; the rest is transparent.
-- The background colour of the rectangle should be given as an argument.
-- The foreground colour of the rectangle should be white.
-- The rectangle should have an interior consisting of opaque space with
-- the given background colour.
--
-- The arrow keys should move the rectangle around inside the terminal
-- window, preserving its size (like when you drag a window around).
-- Shifted arrow keys should resize the rectangle by moving its bottom
-- right corner (again, like you might do with a mouse).
-- You do not need to ensure that the rectangle fits inside the terminal.
-- The cursor should sit at the bottom right corner of the rectangle.
--
-- Mac users: the Terminal application which ships with OS X does NOT
-- give the correct treatment to shift-up and shift-down. You can get a
-- suitable alternative from http://iterm2.com/ (Thank @sigfpe for the tip!)
--
-- (2 marks, one for key handling, one for painting)
record RectState : Set where
constructor rect
field
gapL rectW : Nat
gapT rectH : Nat
rectKey : Key -> RectState -> RectState
rectKey k s = {!!}
rectApp : Colour -> Application \ _ -> RectState
rectApp c = record
{ handleKey = \ k -> rectKey k
; handleResize = \ _ _ -> id
; paintMe = {!!}
; cursorMe = {!!}
} where
-- helper functions can go here
{- -
main : IO One
main = appMain (rectApp blue) (rect 10 40 3 15)
- -}
---------------------------------------------------------------------------
-- TWO BECOME ONE --
---------------------------------------------------------------------------
-- Write a function which turns two sub-applications into one main
-- application by layering them.
--
-- For some S and T, you get an Application S and an Application T
-- You should choose a suitable state representation so that you know
-- (i) which of the two applications is at the front, and which behind
-- (ii) the states of both.
--
-- The Tab key should swap which sub-application is at the front, as if you had
-- clicked on the one at the back. All other keys should be handled by
-- whichever action is in front at the time. Also, the cursor position
-- should be chosen by the sub-application at the front.
--
-- The overall application size will be used as the size for both
-- sub-application sizes, which means you should be able to compute the
-- Layer, using equipment from exercise 5. Crucially, we should be
-- able to see through the holes in the front sub-application to stuff from
-- the back sub-application.
--
-- (1 mark)
frontBack : {S T : Nat * Nat -> Set} ->
Application S ->
Application T ->
Application \ wh -> {!!}
frontBack appS appT = record
{ handleKey = {!!}
; handleResize = {!!}
; paintMe = {!!}
; cursorMe = {!!}
}
-- By way of example, let's have a blue rectangle and a red rectangle.
{- -
main : IO One
main = appMain (frontBack (rectApp blue) (rectApp red))
(inl (rect 10 40 3 15 , rect 20 40 8 15))
- -}
---------------------------------------------------------------------------
-- IF YOU WANT MORE... --
---------------------------------------------------------------------------
-- Figure out how to reduce flicker.
-- Put editors in the rectangles.
|
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{-# OPTIONS --without-K --rewriting #-}
open import lib.Basics
module lib.types.Empty where
Empty-rec : ∀ {i} {A : Type i} → (Empty → A)
Empty-rec = Empty-elim
⊥-rec : ∀ {i} {A : Type i} → (⊥ → A)
⊥-rec = Empty-rec
abstract
Empty-is-prop : is-prop Empty
Empty-is-prop = Empty-elim
Empty-is-set : is-set Empty
Empty-is-set = raise-level -1 Empty-is-prop
Empty-level = Empty-is-prop
⊥-is-prop = Empty-is-prop
⊥-is-set = Empty-is-set
⊥-level = Empty-level
|
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{-# OPTIONS --without-K #-}
open import library.Basics hiding (Type ; Σ)
open import library.types.Sigma
open import library.NType2
open import Sec2preliminaries
open import Sec3hedberg
open import Sec4hasConstToSplit
module Sec5factorConst where
-- Definition 5.1
factors-through : {X Y : Type} → Type → (X → Y) → Type
factors-through {X} {Y} Z f = Σ (X → Z) λ f₁ → Σ (Z → Y) λ f₂ → (x : X) → f₂ (f₁ x) == f x
-- Definition 5.1 addendum: "factors through the truncation"
factors : {X Y : Type} → (X → Y) → Type
factors {X} = factors-through (Trunc X)
-- Principle 5.2
factor-helper : {X Y : Type} → (f : X → Y) → (P : Type) → (X → is-prop P) → (factors-through P f) → factors f
factor-helper {X} {Y} f P xpp (f₁ , f₂ , q) = f₁' , f₂' , q' where
f₁' : X → Trunc X
f₁' = ∣_∣
f₂' : Trunc X → Y
f₂' z = f₂ (rec {X} (rec is-prop-is-prop xpp z) f₁ z)
q' : (x : X) → f₂' (f₁' x) == f x
q' x =
f₂' (f₁' x) =⟨ idp ⟩
f₂' ∣ x ∣ =⟨ idp ⟩
f₂ (rec _ f₁ ∣ x ∣ ) =⟨ ap f₂ (trunc-β _ f₁ x) ⟩
f₂ (f₁ x) =⟨ q x ⟩
f x ∎
-- Theorem 5.3
module thm53 {X Y : Type} (f : X → Y) (c : const f) where
One = ¬ X
Two = X
Three = Trunc X → X
Four = hasConst X
Five = Y → X
One→Three : One → Three
One→Three nx z = Empty-elim {A = λ _ → X} (rec Empty-elim nx z)
Two→Three : Two → Three
Two→Three x = λ _ → x
Five→Four : Five → Four
Five→Four g = g ∘ f , (λ x₁ x₂ → ap g (c x₁ x₂))
Three↔Four : Three ↔ Four
Three↔Four = snd hasConst↔splitSup , fst hasConst↔splitSup
Three→factors : Three → factors f
Three→factors s = ∣_∣ , f ∘ s , (λ x → c _ _)
-- Theorem 5.4
-- as a small preparation, let us define a function that applies the truncation recursor twice:
double-rec : {X₁ X₂ P : Type} → (is-prop P) → (X₁ → X₂ → P) → Trunc X₁ → Trunc X₂ → P
double-rec {X₁} {X₂} {P} pp f z₁ z₂ = next-step z₂ z₁ where
step : X₁ → Trunc X₂ → P
step x₁ = rec {X = X₂} {P = P} pp (f x₁)
step-switch : Trunc X₂ → X₁ → P
step-switch z₂ x₁ = step x₁ z₂
next-step : Trunc X₂ → Trunc X₁ → P
next-step z₂ = rec pp (step-switch z₂)
-- now, the actual Theorem 5.4
factor-codomain-set : {X Y : Type} → (f : X → Y) → const f → is-set Y → factors f
factor-codomain-set {X} {Y} f c h = factor-helper f P (λ _ → pp) fact-through-p where
P : Type
P = Σ Y λ y → Trunc(Σ X λ x → f x == y)
pp : is-prop P
pp = all-paths-is-prop all-paths where
all-paths : has-all-paths P
all-paths (y₁ , t₁) (y₂ , t₂) = pair= ys ts where
ys = double-rec {P = y₁ == y₂} (h _ _) (λ {(x₁ , p₁) (x₂ , p₂) → ! p₁ ∙ c _ _ ∙ p₂}) t₁ t₂
ts = from-transp _ ys (prop-has-all-paths (h-tr _) _ _)
fact-through-p : factors-through P f
fact-through-p = f₁ , f₂ , q where
f₁ : X → P
f₁ x = f x , ∣ x , idp ∣
f₂ : P → Y
f₂ = fst
q : (x : X) → f₂ (f₁ x) == f x
q x = idp
-- Theorem 5.5
-- Note that this lemma requires function extensionality (hidden in the use of the library.types.Pi library:
-- at one point, we use that Π X Y is propositional as soon as Y x is (for all x)).
open import library.types.Pi
-- a general auxiliary function which switches the second and third component of a Σ-type with four components
-- (provided that it is possible)
-- we formulate this explicitly as we will need it several times
switch23 : {X : Type} → {Y Z : X → Type} → {W : (x : X) → (Y x) → (Z x) → Type} →
(Σ X λ x → Σ (Y x) λ y → Σ (Z x) λ z → (W x y z)) ≃
(Σ X λ x → Σ (Z x) λ z → Σ (Y x) λ y → (W x y z))
switch23 = equiv (λ {(y , s , t , coh) → (y , t , s , coh)}) (λ {(y , t , s , coh) → (y , s , t , coh)}) (λ _ → idp) (λ _ → idp)
module thm55aux {P : Type} {Y : P → Type} (pp : is-prop P) (x₀ : P) where
neutral-contr-base-space : Σ P Y ≃ Y x₀
neutral-contr-base-space =
Σ P Y ≃⟨ (equiv-Σ-fst {A = Unit} {B = P} Y
{λ _ → x₀}
(is-eq _ (λ _ → unit) (λ _ → prop-has-all-paths pp _ _) (λ _ → idp))) ⁻¹ ⟩
Σ Unit (λ _ → Y x₀) ≃⟨ Σ₁-Unit ⟩
Y x₀ ≃∎
neutral-contr-exp : Π P Y ≃ Y x₀
neutral-contr-exp =
Π P Y ≃⟨ (equiv-Π-l {A = Unit} {B = P} Y
{λ _ → x₀}
(is-eq _ (λ _ → unit) (λ _ → prop-has-all-paths pp _ _) (λ _ → idp))) ⁻¹ ⟩
Π Unit (λ _ → Y x₀) ≃⟨ Π₁-Unit ⟩
Y x₀ ≃∎
module thm55 {Q R Y : Type} (qq : is-prop Q) (rr : is-prop R) (f : Q + R → Y) (c : const f) where
P : Type
P = Σ Y λ y →
Σ ((q : Q) → y == f(inl q)) λ s →
Σ ((r : R) → y == f(inr r)) λ t →
(q : Q) → (r : R) → ! (s q) ∙ (t r) == c (inl q) (inr r)
-- This is going to be tedious: if q₀ : Q is given, we can show that P is equivalent to the Unit type.
given-q-reduce-P : Q → P ≃ Unit
given-q-reduce-P q₀ =
P
≃⟨ switch23 ⟩
(Σ Y λ y →
Σ ((r : R) → y == f(inr r)) λ t →
Σ ((q : Q) → y == f(inl q)) λ s →
(q : Q) → (r : R) → ! (s q) ∙ (t r) == c (inl q) (inr r))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ t → choice ⁻¹)) ⟩
(Σ Y λ y →
Σ ((r : R) → y == f(inr r)) λ t →
(q : Q) → Σ (y == f(inl q)) λ s-reduced →
(r : R) → ! s-reduced ∙ (t r) == c (inl q) (inr r))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ t → thm55aux.neutral-contr-exp qq q₀)) ⟩
(Σ Y λ y →
Σ ((r : R) → y == f(inr r)) λ t →
Σ (y == f(inl q₀)) λ s-reduced →
(r : R) → ! s-reduced ∙ (t r) == c (inl q₀) (inr r))
≃⟨ switch23 ⟩
(Σ Y λ y →
Σ (y == f(inl q₀)) λ s-reduced →
Σ ((r : R) → y == f(inr r)) λ t →
(r : R) → ! s-reduced ∙ (t r) == c (inl q₀) (inr r))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ t → choice ⁻¹)) ⟩
(Σ Y λ y →
Σ (y == f(inl q₀)) λ s-reduced →
(r : R) → Σ (y == f(inr r)) λ t-reduced →
! s-reduced ∙ t-reduced == c (inl q₀) (inr r))
≃⟨ Σ-assoc ⁻¹ ⟩
(Σ (Σ Y λ y → (y == f(inl q₀))) λ {(y , s-reduced) →
(r : R) → Σ (y == f(inr r)) λ t-reduced →
! s-reduced ∙ t-reduced == c (inl q₀) (inr r)})
≃⟨ thm55aux.neutral-contr-base-space (contr-is-prop (pathto-is-contr _)) (f(inl q₀) , idp) ⟩
((r : R) → Σ (f(inl q₀) == f(inr r)) λ t-reduced →
idp ∙ t-reduced == c (inl q₀) (inr r))
≃⟨ neutral-codomain (λ r → pathto-is-contr _) ⟩
Unit ≃∎
given-r-reduce-P : R → P ≃ Unit
given-r-reduce-P r₀ =
P
≃⟨ equiv-Σ-snd (λ _ → equiv-Σ-snd (λ _ → equiv-Σ-snd (λ _ → switch-args))) ⟩
(Σ Y λ y →
Σ ((q : Q) → y == f(inl q)) λ s →
Σ ((r : R) → y == f(inr r)) λ t →
(r : R) → (q : Q) → ! (s q) ∙ (t r) == c (inl q) (inr r))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ s → choice ⁻¹)) ⟩
(Σ Y λ y →
Σ ((q : Q) → y == f(inl q)) λ s →
(r : R) → Σ (y == f(inr r)) λ t-reduced →
(q : Q) → ! (s q) ∙ t-reduced == c (inl q) (inr r))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ t → thm55aux.neutral-contr-exp rr r₀)) ⟩
(Σ Y λ y →
Σ ((q : Q) → y == f(inl q)) λ s →
Σ (y == f(inr r₀)) λ t-reduced →
(q : Q) → ! (s q) ∙ t-reduced == c (inl q) (inr r₀))
≃⟨ switch23 ⟩
(Σ Y λ y →
Σ (y == f(inr r₀)) λ t-reduced →
Σ ((q : Q) → y == f(inl q)) λ s →
(q : Q) → ! (s q) ∙ t-reduced == c (inl q) (inr r₀))
≃⟨ equiv-Σ-snd (λ y → equiv-Σ-snd (λ s → choice ⁻¹)) ⟩
(Σ Y λ y →
Σ (y == f(inr r₀)) λ t-reduced →
(q : Q) → Σ (y == f(inl q)) λ s-reduced →
! s-reduced ∙ t-reduced == c (inl q) (inr r₀))
≃⟨ Σ-assoc ⁻¹ ⟩
(Σ (Σ Y λ y → (y == f(inr r₀))) λ {(y , t-reduced) →
(q : Q) → Σ (y == f(inl q)) λ s-reduced →
! s-reduced ∙ t-reduced == c (inl q) (inr r₀)})
≃⟨ thm55aux.neutral-contr-base-space (contr-is-prop (pathto-is-contr _)) (f(inr r₀) , idp) ⟩
((q : Q) → Σ (f(inr r₀) == f(inl q)) λ s-reduced →
! s-reduced ∙ idp == c (inl q) (inr r₀))
≃⟨ equiv-Π-r (λ q → equiv-Σ-snd (λ proof →
! proof ∙ idp == c (inl q) (inr r₀) ≃⟨ delete-idp _ _ ⟩
! proof == c (inl q) (inr r₀) ≃⟨ reverse-paths _ _ ⟩
proof == ! (c (inl q) (inr r₀)) ≃∎
)) ⟩
((q : Q) → Σ (f(inr r₀) == f(inl q)) λ s-reduced →
s-reduced == ! (c (inl q) (inr r₀)))
≃⟨ neutral-codomain (λ q → pathto-is-contr _) ⟩
Unit ≃∎
given-q+r-reduce-P : Q + R → P ≃ Unit
given-q+r-reduce-P (inl q) = given-q-reduce-P q
given-q+r-reduce-P (inr r) = given-r-reduce-P r
Q+R→P : Q + R → P
Q+R→P x = <– (given-q+r-reduce-P x) _
P→Y : P → Y
P→Y = fst
open import library.types.Unit
-- Finally : the statement of Theorem 5.5
factor-f : factors f
factor-f = factor-helper f P (λ x → equiv-preserves-level ((given-q+r-reduce-P x) ⁻¹) Unit-is-prop) (Q+R→P , P→Y , proof) where
proof : (x : Q + R) → P→Y (Q+R→P x) == f x
proof (inl q) = idp
proof (inr r) = idp
-- and Theorem 5.5 again (outside of a specialized module)
Theorem55 : {Q R Y : Type} → (is-prop Q) → (is-prop R) → (f : Q + R → Y) → (const f) → factors f
Theorem55 = thm55.factor-f
|
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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary using (Rel; Setoid; IsEquivalence)
module Magma.Structures
{a ℓ} {A : Set a} -- The underlying set
(_≈_ : Rel A ℓ) -- The underlying equality relation
where
open import Algebra.Core
open import Level using (_⊔_)
open import Data.Product using (_,_; proj₁; proj₂)
open import Algebra.Definitions _≈_
open import Algebra.Structures _≈_
open import Magma.Definitions _≈_
record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
idem : Idempotent ∙
open IsMagma isMagma public
record IsAlternateMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
alter : Alternative ∙
open IsMagma isMagma public
record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
flex : Flexible ∙
open IsMagma isMagma public
record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
medial : Medial ∙
open IsMagma isMagma public
record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
semiMedial : Semimedial ∙
open IsMagma isMagma public
record IsLeftUnitalMagma (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
identity : LeftIdentity ε ∙
open IsMagma isMagma public
record IsRightUnitalMagma (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
field
isMagma : IsMagma ∙
identity : RightIdentity ε ∙
open IsMagma isMagma public
|
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{-# OPTIONS --without-K #-}
open import HoTT
open import homotopy.RibbonCover
module homotopy.CoverClassification {i} (A∙ : Ptd i)
(A-conn : is-connected 0 (fst A∙)) where
open Cover
private
A : Type i
A = fst A∙
a₁ : A
a₁ = snd A∙
π1A = fundamental-group A∙
-- A covering space constructed from a G-set.
gset-to-cover : ∀ {j} → Gset π1A j → Cover A (lmax i j)
gset-to-cover gs = Ribbon-cover A∙ gs
-- Covering spaces to G-sets.
cover-to-gset-struct : ∀ {j} (cov : Cover A j)
→ GsetStructure π1A (Fiber cov a₁) (Fiber-is-set cov a₁)
cover-to-gset-struct cov = record
{ act = cover-trace cov
; unit-r = cover-trace-idp₀ cov
; assoc = cover-paste cov
}
cover-to-gset : ∀ {j} → Cover A j → Gset π1A j
cover-to-gset cov = record
{ El = Fiber cov a₁
; El-level = Fiber-level cov a₁
; gset-struct = cover-to-gset-struct cov
}
-- This is derivable from connectedness condition.
module _ where
abstract
[base-path] : ∀ {a₂ : A} → Trunc -1 (a₁ == a₂)
[base-path] {a₂} =
–> (Trunc=-equiv [ a₁ ] [ a₂ ]) (contr-has-all-paths A-conn [ a₁ ] [ a₂ ])
-- Part 1: Show that the synthesized cover (ribbon) is fiberwisely
-- equivalent to the original fiber.
private
module _ {j} (cov : Cover A j) where
-- Suppose that we get the path, we can compute the ribbon easily.
fiber+path-to-ribbon : ∀ {a₂} (a↑ : Fiber cov a₂) (p : a₁ == a₂)
→ Ribbon A∙ (cover-to-gset cov) a₂
fiber+path-to-ribbon {a₂} a↑ p =
trace (cover-trace cov a↑ [ ! p ]) [ p ]
abstract
-- Our construction is "constant" with respect to paths.
fiber+path-to-ribbon-is-path-irrelevant :
∀ {a₂} (a↑ : Fiber cov a₂) (p₁ p₂ : a₁ == a₂)
→ fiber+path-to-ribbon a↑ p₁ == fiber+path-to-ribbon a↑ p₂
fiber+path-to-ribbon-is-path-irrelevant a↑ p idp =
trace (cover-trace cov a↑ [ ! p ]) [ p ]
=⟨ paste a↑ [ ! p ] [ p ] ⟩
trace a↑ [ ! p ∙ p ]
=⟨ ap (trace a↑) $ !₀-inv-l [ p ] ⟩
trace a↑ idp₀
∎
open import homotopy.ConstantToSetFactorization
fiber+path₋₁-to-ribbon : ∀ {a₂} (a↑ : Cover.Fiber cov a₂)
→ Trunc -1 (a₁ == a₂) → Ribbon A∙ (cover-to-gset cov) a₂
fiber+path₋₁-to-ribbon a↑ = cst-extend
Ribbon-is-set
(fiber+path-to-ribbon a↑)
(fiber+path-to-ribbon-is-path-irrelevant a↑)
-- So the conversion from fiber to ribbon is done.
fiber-to-ribbon : ∀ {j} (cov : Cover A j)
→ {a₂ : A} (a↑ : Cover.Fiber cov a₂)
→ Ribbon A∙ (cover-to-gset cov) a₂
fiber-to-ribbon cov a↑ = fiber+path₋₁-to-ribbon cov a↑ [base-path]
-- The other direction is easy.
ribbon-to-fiber : ∀ {j} (cov : Cover A j) {a₂}
→ Ribbon A∙ (cover-to-gset cov) a₂ → Cover.Fiber cov a₂
ribbon-to-fiber cov {a₂} r =
Ribbon-rec (Fiber-is-set cov a₂) (cover-trace cov) (cover-paste cov) r
private
-- Some routine computations.
abstract
ribbon-to-fiber-to-ribbon : ∀ {j} (cov : Cover A j) {a₂}
→ (r : Ribbon A∙ (cover-to-gset cov) a₂)
→ fiber-to-ribbon cov (ribbon-to-fiber cov r) == r
ribbon-to-fiber-to-ribbon cov {a₂} = Ribbon-elim
{P = λ r → fiber-to-ribbon cov (ribbon-to-fiber cov r) == r}
(λ _ → =-preserves-set Ribbon-is-set)
(λ a↑ p → Trunc-elim
-- All ugly things will go away when bp = proj bp′
(λ bp → Ribbon-is-set
(fiber+path₋₁-to-ribbon cov (cover-trace cov a↑ p) bp)
(trace a↑ p))
(lemma a↑ p)
[base-path])
(λ _ _ _ → prop-has-all-paths-↓ (Ribbon-is-set _ _))
where
abstract
lemma : ∀ {a₂} (a↑ : Cover.Fiber cov a₁) (p : a₁ =₀ a₂) (bp : a₁ == a₂)
→ trace {A∙ = A∙} {gs = cover-to-gset cov}
(cover-trace cov (cover-trace cov a↑ p) [ ! bp ]) [ bp ]
== trace {A∙ = A∙} {gs = cover-to-gset cov} a↑ p
lemma a↑ p idp =
trace (cover-trace cov a↑ p) idp₀
=⟨ paste a↑ p idp₀ ⟩
trace a↑ (p ∙₀ idp₀)
=⟨ ap (trace a↑) $ ∙₀-unit-r p ⟩
trace a↑ p
∎
fiber-to-ribbon-to-fiber : ∀ {j} (cov : Cover A j) {a₂}
→ (a↑ : Cover.Fiber cov a₂)
→ ribbon-to-fiber cov (fiber-to-ribbon cov {a₂} a↑) == a↑
fiber-to-ribbon-to-fiber cov {a₂} a↑ = Trunc-elim
-- All ugly things will go away when bp = proj bp′
(λ bp → Cover.Fiber-is-set cov a₂
(ribbon-to-fiber cov
(fiber+path₋₁-to-ribbon cov a↑ bp))
a↑)
(lemma a↑)
[base-path]
where
abstract
lemma : ∀ {a₂} (a↑ : Cover.Fiber cov a₂) (bp : a₁ == a₂)
→ cover-trace cov (cover-trace cov a↑ [ ! bp ]) [ bp ]
== a↑
lemma a↑ idp = idp
cover-to-gset-to-cover : ∀ {j} (cov : Cover A (lmax i j))
→ gset-to-cover (cover-to-gset cov) == cov
cover-to-gset-to-cover cov = cover= λ _ →
ribbon-to-fiber cov , is-eq
(ribbon-to-fiber cov)
(fiber-to-ribbon cov)
(fiber-to-ribbon-to-fiber cov)
(ribbon-to-fiber-to-ribbon cov)
-- The second direction : gset -> covering -> gset
-- Part 2.1: The fiber over the point a is the carrier.
ribbon-a₁-to-El : ∀ {j} {gs : Gset π1A j}
→ Ribbon A∙ gs a₁ → Gset.El gs
ribbon-a₁-to-El {j} {gs} = let open Gset gs in
Ribbon-rec El-level act assoc
ribbon-a₁-to-El-equiv : ∀ {j} {gs : Gset π1A j}
→ Ribbon A∙ gs a₁ ≃ Gset.El gs
ribbon-a₁-to-El-equiv {j} {gs} = let open Gset gs in
ribbon-a₁-to-El , is-eq _
(λ r → trace r idp₀)
(λ a↑ → unit-r a↑)
(Ribbon-elim
{P = λ r → trace (ribbon-a₁-to-El r) idp₀ == r}
(λ _ → =-preserves-set Ribbon-is-set)
(λ y p →
trace (act y p) idp₀
=⟨ paste y p idp₀ ⟩
trace y (p ∙₀ idp₀)
=⟨ ap (trace y) $ ∙₀-unit-r p ⟩
trace y p
∎)
(λ _ _ _ → prop-has-all-paths-↓ (Ribbon-is-set _ _)))
gset-to-cover-to-gset : ∀ {j} (gs : Gset π1A (lmax i j))
→ cover-to-gset (gset-to-cover gs) == gs
gset-to-cover-to-gset gs =
gset=
ribbon-a₁-to-El-equiv
(λ {x₁}{x₂} x= → Trunc-elim (λ _ → =-preserves-set $ Gset.El-is-set gs) λ g →
ribbon-a₁-to-El (transport (Ribbon A∙ gs) g x₁)
=⟨ ap (λ x → ribbon-a₁-to-El (transport (Ribbon A∙ gs) g x))
$ ! $ <–-inv-l ribbon-a₁-to-El-equiv x₁ ⟩
ribbon-a₁-to-El (transport (Ribbon A∙ gs) g (trace (ribbon-a₁-to-El x₁) idp₀))
=⟨ ap (λ x → ribbon-a₁-to-El (transport (Ribbon A∙ gs) g (trace x idp₀))) x= ⟩
ribbon-a₁-to-El (transport (Ribbon A∙ gs) g (trace x₂ idp₀))
=⟨ ap ribbon-a₁-to-El $ trans-trace g x₂ idp₀ ⟩
Gset.act gs x₂ [ g ]
∎)
-- Finally...
gset-to-cover-equiv : ∀ {j}
→ Gset π1A (lmax i j) ≃ Cover A (lmax i j)
gset-to-cover-equiv {j} =
gset-to-cover , is-eq
_
(λ c → cover-to-gset c)
(λ c → cover-to-gset-to-cover {lmax i j} c)
(gset-to-cover-to-gset {lmax i j})
|
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module Agda.Builtin.Coinduction where
infix 1000 ♯_
postulate
∞ : ∀ {a} (A : Set a) → Set a
♯_ : ∀ {a} {A : Set a} → A → ∞ A
♭ : ∀ {a} {A : Set a} → ∞ A → A
{-# BUILTIN INFINITY ∞ #-}
{-# BUILTIN SHARP ♯_ #-}
{-# BUILTIN FLAT ♭ #-}
|
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-- Andreas, 2013-02-27
module Issue653 where
postulate
P : {A : Set} → (A → Set) → Set
mutual
A : Set
A = P B -- note A = P {A} B is non-terminating
data B : A → Set where
c : (a : A) → B a
-- This caused a stack overflow due to infinite reduction
-- in the positivity checker.
-- Now functions that do not pass the termination checker are not unfolded
-- any more.
-- So, it should report positivity violation now.
|
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{-# OPTIONS --cubical --no-import-sorts --safe #-}
module Cubical.HITs.Ints.QuoInt.Properties where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.HLevels
open import Cubical.Foundations.Equiv
open import Cubical.Relation.Nullary
open import Cubical.Data.Nat as ℕ using (ℕ; zero; suc)
open import Cubical.Data.Bool as Bool using (Bool; not; notnot)
open import Cubical.Data.Empty
open import Cubical.Data.Unit renaming (Unit to ⊤)
open import Cubical.HITs.Ints.QuoInt.Base
·S-comm : ∀ x y → x ·S y ≡ y ·S x
·S-comm = Bool.⊕-comm
·S-assoc : ∀ x y z → x ·S (y ·S z) ≡ (x ·S y) ·S z
·S-assoc = Bool.⊕-assoc
not-·Sˡ : ∀ x y → not (x ·S y) ≡ not x ·S y
not-·Sˡ = Bool.not-⊕ˡ
snotz : ∀ s n s' → ¬ (signed s (suc n) ≡ signed s' zero)
snotz s n s' eq = subst (λ { (signed s (suc n)) → ⊤
; _ → ⊥ }) eq tt
+-zeroʳ : ∀ s m → m + signed s zero ≡ m
+-zeroʳ s (signed s' zero) = signed-zero s s'
+-zeroʳ s (pos (suc n)) = cong sucℤ (+-zeroʳ s (pos n))
+-zeroʳ s (neg (suc n)) = cong predℤ (+-zeroʳ s (neg n))
+-zeroʳ spos (posneg i) j = posneg (i ∧ j)
+-zeroʳ sneg (posneg i) j = posneg (i ∨ ~ j)
+-identityʳ : ∀ m → m + pos zero ≡ m
+-identityʳ = +-zeroʳ spos
sucℤ-+ʳ : ∀ m n → sucℤ (m + n) ≡ m + sucℤ n
sucℤ-+ʳ (signed _ zero) _ = refl
sucℤ-+ʳ (posneg _) _ = refl
sucℤ-+ʳ (pos (suc m)) n = cong sucℤ (sucℤ-+ʳ (pos m) n)
sucℤ-+ʳ (neg (suc m)) n =
sucPredℤ (neg m + n) ∙∙
sym (predSucℤ (neg m + n)) ∙∙
cong predℤ (sucℤ-+ʳ (neg m) n)
-- I wonder if we could somehow use negEq to get this for free from the above...
predℤ-+ʳ : ∀ m n → predℤ (m + n) ≡ m + predℤ n
predℤ-+ʳ (signed _ zero) _ = refl
predℤ-+ʳ (posneg _) _ = refl
predℤ-+ʳ (neg (suc m)) n = cong predℤ (predℤ-+ʳ (neg m) n)
predℤ-+ʳ (pos (suc m)) n =
predSucℤ (pos m + n) ∙∙
sym (sucPredℤ (pos m + n)) ∙∙
cong sucℤ (predℤ-+ʳ (pos m) n)
+-comm : ∀ m n → m + n ≡ n + m
+-comm (signed s zero) n = sym (+-zeroʳ s n)
+-comm (pos (suc m)) n = cong sucℤ (+-comm (pos m) n) ∙ sucℤ-+ʳ n (pos m)
+-comm (neg (suc m)) n = cong predℤ (+-comm (neg m) n) ∙ predℤ-+ʳ n (neg m)
+-comm (posneg i) n = lemma n i
where
-- get some help from:
-- https://www.codewars.com/kata/reviews/5c878e3ef1abb10001e32eb1/groups/5cca3f9e840f4b0001d8ac98
lemma : ∀ n → (λ i → (posneg i + n) ≡ (n + posneg i))
[ sym (+-zeroʳ spos n) ≡ sym (+-zeroʳ sneg n) ]
lemma (pos zero) i j = posneg (i ∧ j)
lemma (pos (suc n)) i = cong sucℤ (lemma (pos n) i)
lemma (neg zero) i j = posneg (i ∨ ~ j)
lemma (neg (suc n)) i = cong predℤ (lemma (neg n) i)
lemma (posneg i) j k = posneg ((i ∧ ~ k) ∨ (i ∧ j) ∨ (j ∧ k))
sucℤ-+ˡ : ∀ m n → sucℤ (m + n) ≡ sucℤ m + n
sucℤ-+ˡ (pos _) n = refl
sucℤ-+ˡ (neg zero) n = refl
sucℤ-+ˡ (neg (suc m)) n = sucPredℤ _
sucℤ-+ˡ (posneg _) n = refl
-- I wonder if we could somehow use negEq to get this for free from the above...
predℤ-+ˡ : ∀ m n → predℤ (m + n) ≡ predℤ m + n
predℤ-+ˡ (pos zero) n = refl
predℤ-+ˡ (pos (suc m)) n = predSucℤ _
predℤ-+ˡ (neg _) n = refl
predℤ-+ˡ (posneg _) n = refl
+-assoc : ∀ m n o → m + (n + o) ≡ m + n + o
+-assoc (signed s zero) n o = refl
+-assoc (posneg i) n o = refl
+-assoc (pos (suc m)) n o = cong sucℤ (+-assoc (pos m) n o) ∙ sucℤ-+ˡ (pos m + n) o
+-assoc (neg (suc m)) n o = cong predℤ (+-assoc (neg m) n o) ∙ predℤ-+ˡ (neg m + n) o
sucℤ-inj : ∀ m n → sucℤ m ≡ sucℤ n → m ≡ n
sucℤ-inj m n p = sym (predSucℤ m) ∙ cong predℤ p ∙ predSucℤ n
predℤ-inj : ∀ m n → predℤ m ≡ predℤ n → m ≡ n
predℤ-inj m n p = sym (sucPredℤ m) ∙ cong sucℤ p ∙ sucPredℤ n
+-injˡ : ∀ o m n → o + m ≡ o + n → m ≡ n
+-injˡ (signed _ zero) _ _ p = p
+-injˡ (posneg _) _ _ p = p
+-injˡ (pos (suc o)) m n p = +-injˡ (pos o) m n (sucℤ-inj _ _ p)
+-injˡ (neg (suc o)) m n p = +-injˡ (neg o) m n (predℤ-inj _ _ p)
+-injʳ : ∀ m n o → m + o ≡ n + o → m ≡ n
+-injʳ m n o p = +-injˡ o m n (+-comm o m ∙ p ∙ +-comm n o)
·-comm : ∀ m n → m · n ≡ n · m
·-comm m n i = signed (·S-comm (sign m) (sign n) i) (ℕ.·-comm (abs m) (abs n) i)
·-identityˡ : ∀ n → pos 1 · n ≡ n
·-identityˡ n = cong (signed (sign n)) (ℕ.+-zero (abs n)) ∙ signed-inv n
·-identityʳ : ∀ n → n · pos 1 ≡ n
·-identityʳ n = ·-comm n (pos 1) ∙ ·-identityˡ n
·-zeroˡ : ∀ {s} n → signed s zero · n ≡ signed s zero
·-zeroˡ _ = signed-zero _ _
·-zeroʳ : ∀ {s} n → n · signed s zero ≡ signed s zero
·-zeroʳ n = cong (signed _) (sym (ℕ.0≡m·0 (abs n))) ∙ signed-zero _ _
·-signed-pos : ∀ {s} m n → signed s m · pos n ≡ signed s (m ℕ.· n)
·-signed-pos {s} zero n = ·-zeroˡ {s} (pos n)
·-signed-pos {s} (suc m) n i = signed (·S-comm s (sign-pos n i) i) (suc m ℕ.· n)
-- this proof is why we defined ℤ using `signed` instead of `pos` and `neg`
-- based on that in: https://github.com/danr/Agda-Numerics
·-assoc : ∀ m n o → m · (n · o) ≡ m · n · o
·-assoc (signed s zero) n o =
·-zeroˡ (n · o)
·-assoc m@(signed _ (suc _)) (signed s zero) o =
·-zeroʳ {sign o} m ∙ signed-zero _ _ ∙ cong (_· o) (sym (·-zeroʳ {s} m))
·-assoc m@(signed _ (suc _)) n@(signed _ (suc _)) (signed s zero) =
cong (m ·_) (·-zeroʳ {s} n) ∙ ·-zeroʳ {s} m ∙ sym (·-zeroʳ {s} (m · n))
·-assoc (signed sm (suc m)) (signed sn (suc n)) (signed so (suc o)) i =
signed (·S-assoc sm sn so i) (ℕ.·-assoc (suc m) (suc n) (suc o) i)
·-assoc (posneg i) n o j =
isSet→isSet' isSetℤ (·-assoc (pos zero) n o) (·-assoc (neg zero) n o)
(λ i → posneg i · (n · o)) (λ i → posneg i · n · o) i j
·-assoc m@(signed _ (suc _)) (posneg i) o j =
isSet→isSet' isSetℤ (·-assoc m (pos zero) o) (·-assoc m (neg zero) o)
(λ i → m · (posneg i · o)) (λ i → m · posneg i · o) i j
·-assoc m@(signed _ (suc _)) n@(signed _ (suc _)) (posneg i) j =
isSet→isSet' isSetℤ (·-assoc m n (pos zero)) (·-assoc m n (neg zero))
(λ i → m · (n · posneg i)) (λ i → m · n · posneg i) i j
negateSuc : ∀ n → - sucℤ n ≡ predℤ (- n)
negateSuc n i = - sucℤ (negate-invol n (~ i))
negatePred : ∀ n → - predℤ n ≡ sucℤ (- n)
negatePred n i = negate-invol (sucℤ (- n)) i
negate-+ : ∀ m n → - (m + n) ≡ (- m) + (- n)
negate-+ (signed _ zero) n = refl
negate-+ (posneg _) n = refl
negate-+ (pos (suc m)) n = negateSuc (pos m + n) ∙ cong predℤ (negate-+ (pos m) n)
negate-+ (neg (suc m)) n = negatePred (neg m + n) ∙ cong sucℤ (negate-+ (neg m) n)
negate-·ˡ : ∀ m n → - (m · n) ≡ (- m) · n
negate-·ˡ (signed _ zero) n = signed-zero (not (sign n)) (sign n)
negate-·ˡ (signed ss (suc m)) n i = signed (not-·Sˡ ss (sign n) i) (suc m ℕ.· abs n)
negate-·ˡ (posneg i) n j =
isSet→isSet' isSetℤ (signed-zero (not (sign n)) _) (signed-zero _ _)
refl (λ i → posneg (~ i) · n) i j
signed-distrib : ∀ s m n → signed s (m ℕ.+ n) ≡ signed s m + signed s n
signed-distrib s zero n = refl
signed-distrib spos (suc m) n = cong sucℤ (signed-distrib spos m n)
signed-distrib sneg (suc m) n = cong predℤ (signed-distrib sneg m n)
·-pos-suc : ∀ m n → pos (suc m) · n ≡ n + pos m · n
·-pos-suc m n = signed-distrib (sign n) (abs n) (m ℕ.· abs n)
∙ (λ i → signed-inv n i + signed (sign-pos m (~ i) ·S sign n) (m ℕ.· abs n))
-- the below is based on that in: https://github.com/danr/Agda-Numerics
·-distribˡ-pos : ∀ o m n → (pos o · m) + (pos o · n) ≡ pos o · (m + n)
·-distribˡ-pos zero m n = signed-zero (sign n) (sign (m + n))
·-distribˡ-pos (suc o) m n =
pos (suc o) · m + pos (suc o) · n ≡[ i ]⟨ ·-pos-suc o m i + ·-pos-suc o n i ⟩
m + pos o · m + (n + pos o · n) ≡⟨ +-assoc (m + pos o · m) n (pos o · n) ⟩
m + pos o · m + n + pos o · n ≡[ i ]⟨ +-assoc m (pos o · m) n (~ i) + pos o · n ⟩
m + (pos o · m + n) + pos o · n ≡[ i ]⟨ m + +-comm (pos o · m) n i + pos o · n ⟩
m + (n + pos o · m) + pos o · n ≡[ i ]⟨ +-assoc m n (pos o · m) i + pos o · n ⟩
m + n + pos o · m + pos o · n ≡⟨ sym (+-assoc (m + n) (pos o · m) (pos o · n)) ⟩
m + n + (pos o · m + pos o · n) ≡⟨ cong ((m + n) +_) (·-distribˡ-pos o m n) ⟩
m + n + pos o · (m + n) ≡⟨ sym (·-pos-suc o (m + n)) ⟩
pos (suc o) · (m + n) ∎
·-distribˡ-neg : ∀ o m n → (neg o · m) + (neg o · n) ≡ neg o · (m + n)
·-distribˡ-neg o m n =
neg o · m + neg o · n ≡[ i ]⟨ negate-·ˡ (pos o) m (~ i) + negate-·ˡ (pos o) n (~ i) ⟩
- (pos o · m) + - (pos o · n) ≡⟨ sym (negate-+ (pos o · m) (pos o · n)) ⟩
- (pos o · m + pos o · n) ≡⟨ cong -_ (·-distribˡ-pos o m n) ⟩
- (pos o · (m + n)) ≡⟨ negate-·ˡ (pos o) (m + n) ⟩
neg o · (m + n) ∎
·-distribˡ : ∀ o m n → (o · m) + (o · n) ≡ o · (m + n)
·-distribˡ (pos o) m n = ·-distribˡ-pos o m n
·-distribˡ (neg o) m n = ·-distribˡ-neg o m n
·-distribˡ (posneg i) m n j =
isSet→isSet' isSetℤ (·-distribˡ-pos zero m n) (·-distribˡ-neg zero m n)
(λ i → posneg i · n) (λ i → posneg i · (m + n)) i j
·-distribʳ : ∀ m n o → (m · o) + (n · o) ≡ (m + n) · o
·-distribʳ m n o = (λ i → ·-comm m o i + ·-comm n o i) ∙ ·-distribˡ o m n ∙ ·-comm o (m + n)
sign-pos-suc-· : ∀ m n → sign (pos (suc m) · n) ≡ sign n
sign-pos-suc-· m (signed s zero) = sign-pos (suc m ℕ.· zero)
sign-pos-suc-· m (posneg i) = sign-pos (suc m ℕ.· zero)
sign-pos-suc-· m (signed s (suc n)) = refl
·-injˡ : ∀ o m n → pos (suc o) · m ≡ pos (suc o) · n → m ≡ n
·-injˡ o m n p = sym (signed-inv m) ∙ (λ i → signed (sign-eq i) (abs-eq i)) ∙ signed-inv n
where sign-eq = sym (sign-pos-suc-· o m) ∙ cong sign p ∙ sign-pos-suc-· o n
abs-eq = ℕ.inj-sm· {o} (cong abs p)
·-injʳ : ∀ m n o → m · pos (suc o) ≡ n · pos (suc o) → m ≡ n
·-injʳ m n o p = ·-injˡ o m n (·-comm (pos (suc o)) m ∙ p ∙ ·-comm n (pos (suc o)))
|
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{-# OPTIONS --without-K #-}
module Model.Type where
open import Model.Exponential public
open import Model.Nat public
open import Model.Product public
open import Model.Terminal public
open import Model.Type.Core public
open import Model.Stream public
open import Model.Quantification public
open import Model.Size as MS using
( ≤-IsProp ; ⟦_⟧Δ ; ⟦_⟧n ; ⟦_⟧σ )
open import Source.Size.Substitution.Theory
open import Source.Size.Substitution.Universe as SS using (Sub⊢ᵤ ; ⟨_⟩)
open import Util.Prelude hiding (id ; _∘_ ; _×_ ; ⊤)
import Source.Size as SS
import Source.Type as ST
open MS._≤_
⟦_⟧T : ∀ {Δ} (T : ST.Type Δ) → ⟦Type⟧ ⟦ Δ ⟧Δ
⟦ ST.Nat n ⟧T = subT ⟦ n ⟧n Nat
⟦ ST.Stream n ⟧T = subT ⟦ n ⟧n Stream
⟦ T ST.⇒ U ⟧T = ⟦ T ⟧T ↝ ⟦ U ⟧T
⟦ ST.Π n , T ⟧T = ⟦∀⟧ n ⟦ T ⟧T
⟦_⟧Γ : ∀ {Δ} (Γ : ST.Ctx Δ) → ⟦Type⟧ ⟦ Δ ⟧Δ
⟦ ST.[] ⟧Γ = ⊤
⟦ Γ ST.∙ T ⟧Γ = ⟦ Γ ⟧Γ × ⟦ T ⟧T
⇒-resp-≈⟦Type⟧ : ∀ {Γ} {T T′ U U′ : ⟦Type⟧ Γ}
→ T ≈⟦Type⟧ T′
→ U ≈⟦Type⟧ U′
→ T ⇒ U
→ T′ ⇒ U′
⇒-resp-≈⟦Type⟧ T≈T′ U≈U′ f = U≈U′ .forth ∘ f ∘ T≈T′ .back
⟦subT⟧ : ∀ {Δ Ω σ} (⊢σ : σ ∶ Δ ⇒ᵤ Ω) T
→ ⟦ T [ σ ]ᵤ ⟧T ≈⟦Type⟧ subT ⟦ ⊢σ ⟧σ ⟦ T ⟧T
⟦subT⟧ ⊢σ (ST.Nat n) = record
{ forth = record
{ fobj = castℕ≤ (reflexive (MS.⟦sub⟧ ⊢σ n))
; feq = λ γ≈γ′ m≡m → m≡m
}
; back = record
{ fobj = castℕ≤ (reflexive (sym (MS.⟦sub⟧ ⊢σ n)))
; feq = λ γ≈γ′ m≡m → m≡m
}
; back-forth = ≈⁺ λ γ x → ℕ≤-≡⁺ _ _ refl
; forth-back = ≈⁺ λ γ x → ℕ≤-≡⁺ _ _ refl
}
⟦subT⟧ ⊢σ (ST.Stream n) = record
{ forth = record
{ fobj = castColist (reflexive (sym (MS.⟦sub⟧ ⊢σ n)))
; feq = λ γ≈γ′ xs≈ys a a₁ a₂ → xs≈ys _ _ _
}
; back = record
{ fobj = castColist (reflexive (MS.⟦sub⟧ ⊢σ n))
; feq = λ γ≈γ′ xs≈ys a a₁ a₂ → xs≈ys _ _ _
}
; back-forth = ≈⁺ λ γ xs → Colist-≡⁺ λ m m≤n → cong (xs m) (≤-IsProp _ _)
; forth-back = ≈⁺ λ γ xs → Colist-≡⁺ λ m m≤n → cong (xs m) (≤-IsProp _ _)
}
⟦subT⟧ {Δ} {Ω} {σ} ⊢σ (T ST.⇒ U)
= ≈⟦Type⟧-trans (↝-resp-≈⟦Type⟧ _ _ _ _ (⟦subT⟧ ⊢σ T) (⟦subT⟧ ⊢σ U))
(subT-↝ ⟦ ⊢σ ⟧σ ⟦ T ⟧T ⟦ U ⟧T)
⟦subT⟧ {Δ} {Ω} {σ} ⊢σ (ST.Π n , T)
= ≈⟦Type⟧-trans (⟦∀⟧-resp-≈⟦Type⟧ (n [ σ ]ᵤ) (⟦subT⟧ (SS.Lift ⊢σ refl) T))
(subT-⟦∀⟧ ⊢σ ⟦ T ⟧T)
⟦subΓ⟧ : ∀ {Δ Ω σ} (⊢σ : σ ∶ Δ ⇒ᵤ Ω) Γ
→ ⟦ Γ [ σ ]ᵤ ⟧Γ ≈⟦Type⟧ subT ⟦ ⊢σ ⟧σ ⟦ Γ ⟧Γ
⟦subΓ⟧ σ ST.[] = record
{ forth = record { fobj = λ x → x ; feq = λ γ≈γ′ x → x }
; back = record { fobj = λ x → x ; feq = λ γ≈γ′ x → x }
; back-forth = ≈⁺ λ γ x → refl
; forth-back = ≈⁺ λ γ x → refl
}
⟦subΓ⟧ σ (Γ ST.∙ T) = ×-resp-≈⟦Type⟧ (⟦subΓ⟧ σ Γ) (⟦subT⟧ σ T)
subₛ
: ∀ {Δ Ω σ Γ T}
→ (⊢σ : σ ∶ Δ ⇒ᵤ Ω)
→ ⟦ Γ ⟧Γ ⇒ ⟦ T ⟧T
→ ⟦ Γ [ σ ]ᵤ ⟧Γ ⇒ ⟦ T [ σ ]ᵤ ⟧T
subₛ {Γ = Γ} {T} ⊢σ f
= ⟦subT⟧ ⊢σ T .back ∘ subt ⟦ ⊢σ ⟧σ f ∘ ⟦subΓ⟧ ⊢σ Γ .forth
≡→≈⟦Type⟧Γ : ∀ {Δ} {Γ Ψ : ST.Ctx Δ}
→ Γ ≡ Ψ
→ ⟦ Γ ⟧Γ ≈⟦Type⟧ ⟦ Ψ ⟧Γ
≡→≈⟦Type⟧Γ p = ≡→≈⟦Type⟧ (cong ⟦_⟧Γ p)
≡→≈⟦Type⟧T : ∀ {Δ} {T U : ST.Type Δ}
→ T ≡ U
→ ⟦ T ⟧T ≈⟦Type⟧ ⟦ U ⟧T
≡→≈⟦Type⟧T p = ≡→≈⟦Type⟧ (cong ⟦_⟧T p)
|
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open import Function using ( _∘_ )
open import Data.Product using ( ∃ ; _×_ ; _,_ )
open import Data.Sum using ( _⊎_ ; inj₁ ; inj₂ )
open import Data.Empty using ( ⊥ ; ⊥-elim )
open import Data.Nat using ( ℕ ; zero ; suc ) renaming ( _+_ to _+ℕ_ ; _≤_ to _≤ℕ_ )
open import Relation.Binary.PropositionalEquality using
( _≡_ ; _≢_ ; refl ; sym ; cong ; cong₂ ; subst₂ ; inspect ; [_] )
open import Relation.Nullary using ( ¬_ ; Dec ; yes ; no )
open import FRP.LTL.Util using ( _trans_ ; _∋_ ; m+n≡0-impl-m≡0 ; ≤0-impl-≡0 ; 1+n≰n )
renaming ( +-comm to +ℕ-comm ; +-assoc to +ℕ-assoc )
open Relation.Binary.PropositionalEquality.≡-Reasoning using ( begin_ ; _≡⟨_⟩_ ; _∎ )
module FRP.LTL.Time where
infix 2 _≤_ _≥_ _≰_ _≱_ _<_ _>_
infixr 4 _,_
infixr 5 _≤-trans_ _<-transˡ_ _<-transʳ_ _≤-asym_ _≤-total_
infixl 6 _+_ _∸_
-- Time has a cancellative action _+_ which respects the monoid structure of ℕ
postulate
Time : Set
_+_ : Time → ℕ → Time
+-unit : ∀ t → (t + 0 ≡ t)
+-assoc : ∀ t m n → ((t + m) + n ≡ t + (m +ℕ n))
+-cancelˡ : ∀ t {m n} → (t + m ≡ t + n) → (m ≡ n)
+-cancelʳ : ∀ {s t} n → (s + n ≡ t + n) → (s ≡ t)
-- The order on time is derived from +
data _≤_ (t u : Time) : Set where
_,_ : ∀ n → (t + n ≡ u) → (t ≤ u)
-- Floored subtraction t ∸ u is the smallest n such that t ≤ u + n
postulate
_∸_ : Time → Time → ℕ
t≤u+t∸u : ∀ {t u} → (t ≤ u + (t ∸ u))
∸-min : ∀ {t u n} → (t ≤ u + n) → (t ∸ u ≤ℕ n)
-- End of postulates.
suc-cancelʳ : ∀ {t u m n} → (t + suc m ≡ u + suc n) → (t + m ≡ u + n)
suc-cancelʳ {t} {u} {m} {n} t+1+m≡u+1+n =
+-cancelʳ 1
(+-assoc t m 1 trans
cong₂ _+_ refl (+ℕ-comm m 1) trans
t+1+m≡u+1+n trans
cong₂ _+_ refl (+ℕ-comm 1 n) trans
sym (+-assoc u n 1))
-- Syntax sugar for ≤
_≥_ : Time → Time → Set
t ≥ u = u ≤ t
_≰_ : Time → Time → Set
t ≰ u = ¬(t ≤ u)
_≱_ : Time → Time → Set
t ≱ u = u ≰ t
_<_ : Time → Time → Set
t < u = (t ≤ u) × (u ≰ t)
_>_ : Time → Time → Set
t > u = u < t
-- ≤ is a decidable total order
≤-refl : ∀ {t} → (t ≤ t)
≤-refl {t} = (0 , +-unit t)
_≤-trans_ : ∀ {t u v} → (t ≤ u) → (u ≤ v) → (t ≤ v)
_≤-trans_ {t} {u} {v} (m , t+m≡u) (n , u+n≡v) =
(m +ℕ n , (sym (+-assoc t m n)) trans (cong₂ _+_ t+m≡u refl) trans u+n≡v)
≡-impl-≤ : ∀ {t u} → (t ≡ u) → (t ≤ u)
≡-impl-≤ refl = ≤-refl
≡-impl-≥ : ∀ {t u} → (t ≡ u) → (t ≥ u)
≡-impl-≥ refl = ≤-refl
_≤-asym_ : ∀ {t u} → (t ≤ u) → (u ≤ t) → (t ≡ u)
(m , t+m≡u) ≤-asym (n , u+n≡t) =
sym (+-unit _) trans cong₂ _+_ refl (sym m≡0) trans t+m≡u where
m≡0 : m ≡ 0
m≡0 = m+n≡0-impl-m≡0 m n (+-cancelˡ _
(sym (+-assoc _ m n) trans
cong₂ _+_ t+m≡u refl trans
u+n≡t trans sym (+-unit _)))
≤-impl-∸≡0 : ∀ {t u} → (t ≤ u) → (t ∸ u ≡ 0)
≤-impl-∸≡0 t≤u with (∸-min (t≤u ≤-trans ≡-impl-≤ (sym (+-unit _))))
≤-impl-∸≡0 t≤u | t∸u≤0 = ≤0-impl-≡0 t∸u≤0
∸≡0-impl-≤ : ∀ {t u} → (t ∸ u ≡ 0) → (t ≤ u)
∸≡0-impl-≤ t∸u≡0 = t≤u+t∸u ≤-trans ≡-impl-≤ (cong₂ _+_ refl t∸u≡0 trans +-unit _)
∸≢0-impl-≰ : ∀ {t u n} → (t ∸ u ≡ suc n) → (t ≰ u)
∸≢0-impl-≰ t∸u≡1+n t≤u
with sym t∸u≡1+n trans ≤0-impl-≡0 (∸-min (t≤u ≤-trans ≡-impl-≤ (sym (+-unit _))))
∸≢0-impl-≰ t∸u≡1+n t≤u
| ()
t∸u≢0-impl-u∸t≡0 : ∀ t u {n} → (t ∸ u ≡ suc n) → (u ∸ t ≡ 0)
t∸u≢0-impl-u∸t≡0 t u {n} t∸u≡1+n with t≤u+t∸u {t} {u}
t∸u≢0-impl-u∸t≡0 t u {n} t∸u≡1+n | (zero , t+0≡u+t∸u) =
≤-impl-∸≡0 (t ∸ u , sym t+0≡u+t∸u trans +-unit t)
t∸u≢0-impl-u∸t≡0 t u {n} t∸u≡1+n | (suc m , t+1+m≡u+t∸u) =
⊥-elim (1+n≰n n (subst₂ _≤ℕ_ t∸u≡1+n refl
(∸-min (m , suc-cancelʳ (t+1+m≡u+t∸u trans cong₂ _+_ refl t∸u≡1+n)))))
_≤-total_ : ∀ t u → (t ≤ u) ⊎ (u < t)
t ≤-total u with t ∸ u | inspect (_∸_ t) u
t ≤-total u | zero | [ t∸u≡0 ] = inj₁ (∸≡0-impl-≤ t∸u≡0)
t ≤-total u | suc n | [ t∸u≡1+n ] with t∸u≢0-impl-u∸t≡0 t u t∸u≡1+n
t ≤-total u | suc n | [ t∸u≡1+n ] | u∸t≡0 = inj₂ (∸≡0-impl-≤ u∸t≡0 , ∸≢0-impl-≰ t∸u≡1+n)
-- Case analysis on ≤
data _≤-Case_ (t u : Time) : Set where
lt : .(t < u) → (t ≤-Case u)
eq : .(t ≡ u) → (t ≤-Case u)
gt : .(u < t) → (t ≤-Case u)
_≤-case_ : ∀ t u → (t ≤-Case u)
t ≤-case u with (t ∸ u) | inspect (_∸_ t) u | u ∸ t | inspect (_∸_ u) t
t ≤-case u | zero | [ t∸u≡0 ] | zero | [ u∸t≡0 ] = eq (∸≡0-impl-≤ t∸u≡0 ≤-asym ∸≡0-impl-≤ u∸t≡0)
t ≤-case u | suc n | [ t∸u≡1+n ] | zero | [ u∸t≡0 ] = gt (∸≡0-impl-≤ u∸t≡0 , ∸≢0-impl-≰ t∸u≡1+n)
t ≤-case u | zero | [ t∸u≡0 ] | suc w₁ | [ u∸t≡1+n ] = lt (∸≡0-impl-≤ t∸u≡0 , ∸≢0-impl-≰ u∸t≡1+n)
t ≤-case u | suc m | [ t∸u≡1+m ] | suc n | [ u∸t≡1+n ] with sym u∸t≡1+n trans t∸u≢0-impl-u∸t≡0 t u t∸u≡1+m
t ≤-case u | suc m | [ t∸u≡1+m ] | suc n | [ u∸t≡1+n ] | ()
-- + is monotone
+-resp-≤ : ∀ {t u} → (t ≤ u) → ∀ n → (t + n ≤ u + n)
+-resp-≤ (m , t+m≡u) n =
( m
, +-assoc _ n m trans
cong₂ _+_ refl (+ℕ-comm n m) trans
sym (+-assoc _ m n) trans
cong₂ _+_ t+m≡u refl )
+-refl-≤ : ∀ {t u} n → (t + n ≤ u + n) → (t ≤ u)
+-refl-≤ n (m , t+n+m≡u+n) =
( m
, +-cancelʳ n
(+-assoc _ m n trans
cong₂ _+_ refl (+ℕ-comm m n) trans
sym (+-assoc _ n m) trans
t+n+m≡u+n) )
-- Lemmas about <
<-impl-≤ : ∀ {t u} → (t < u) → (t ≤ u)
<-impl-≤ (t≤u , u≰t) = t≤u
<-impl-≱ : ∀ {t u} → (t < u) → (u ≰ t)
<-impl-≱ (t≤u , u≰t) = u≰t
_<-transˡ_ : ∀ {t u v} → (t < u) → (u ≤ v) → (t < v)
_<-transˡ_ (t≤u , u≰t) u≤v = (t≤u ≤-trans u≤v , λ v≤t → u≰t (u≤v ≤-trans v≤t))
_<-transʳ_ : ∀ {t u v} → (t ≤ u) → (u < v) → (t < v)
_<-transʳ_ t≤u (u≤v , v≰u) = (t≤u ≤-trans u≤v , λ v≤t → v≰u (v≤t ≤-trans t≤u))
≤-proof-irrel′ : ∀ {t u m n} → (m ≡ n) → (t+m≡u : t + m ≡ u) → (t+n≡u : t + n ≡ u) →
(t ≤ u) ∋ (m , t+m≡u) ≡ (n , t+n≡u)
≤-proof-irrel′ refl refl refl = refl
t≤t+1 : ∀ {t} → (t ≤ t + 1)
t≤t+1 = (1 , refl)
t≱t+1 : ∀ {t} → (t ≱ t + 1)
t≱t+1 {t} (m , t+1+m≡t) with +-cancelˡ t (sym (+-assoc t 1 m) trans t+1+m≡t trans sym (+-unit t))
t≱t+1 (m , t+1+m≡t) | ()
t<t+1 : ∀ {t} → (t < t + 1)
t<t+1 = (t≤t+1 , t≱t+1)
<-impl-+1≤ : ∀ {t u} → (t < u) → (t + 1 ≤ u)
<-impl-+1≤ {t} ((zero , t+0≡u) , u≰t) = ⊥-elim (u≰t (≡-impl-≥ (sym (+-unit t) trans t+0≡u)))
<-impl-+1≤ {t} ((suc n , t+1+n≡u) , u≰t) = (n , +-assoc t 1 n trans t+1+n≡u)
+-resp-< : ∀ {t u} → (t < u) → ∀ n → (t + n < u + n)
+-resp-< (t≤u , t≱u) n = (+-resp-≤ t≤u n , λ u+n≤t+n → t≱u (+-refl-≤ n u+n≤t+n))
-- Proof irrelevance for ≤
≤-proof-irrel : ∀ {t u} → (p q : t ≤ u) → (p ≡ q)
≤-proof-irrel {t} (m , t+m≡u) (n , t+n≡u) =
≤-proof-irrel′ (+-cancelˡ t (t+m≡u trans (sym t+n≡u))) t+m≡u t+n≡u
-- Well ordering of < on an interval
_≮[_]_ : Time → ℕ → Time → Set
s ≮[ zero ] u = ⊥
s ≮[ suc n ] u = ∀ {t} → (s ≤ t) → (t < u) → (s ≮[ n ] t)
<-wo′ : ∀ n {s u} → (s ≤ u) → (u ≤ s + n) → (s ≮[ suc n ] u)
<-wo′ zero {s} s≤u u≤s+0 s≤t t<u =
<-impl-≱ t<u (u≤s+0 ≤-trans ≡-impl-≤ (+-unit s) ≤-trans s≤t)
<-wo′ (suc n) s≤u u≤s+1+n {t} s≤t ((zero , t+0≡u) , t≱u) =
⊥-elim (t≱u (≡-impl-≤ ((sym t+0≡u) trans (+-unit t))))
<-wo′ (suc n) {s} {u} s≤u (l , u+l≡s+1+n) {t} s≤t ((suc m , t+1+m≡u) , t≱u) =
<-wo′ n s≤t (l +ℕ m , suc-cancelʳ t+1+l+m≡s+1+n) where
t+1+l+m≡s+1+n : t + suc (l +ℕ m) ≡ s + suc n
t+1+l+m≡s+1+n =
cong₂ _+_ refl (cong suc (+ℕ-comm l m)) trans
sym (+-assoc t (1 +ℕ m) l) trans
cong₂ _+_ t+1+m≡u refl trans
u+l≡s+1+n
<-wo : ∀ {s u} → (s ≤ u) → ∃ λ n → (s ≮[ n ] u)
<-wo (n , s+n≡u) = (suc n , λ {t} → <-wo′ n (n , s+n≡u) (≡-impl-≤ (sym s+n≡u)) {t})
|
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open import MLib.Algebra.PropertyCode
open import MLib.Algebra.PropertyCode.Structures
module MLib.Matrix.Equality {c ℓ} (struct : Struct bimonoidCode c ℓ) where
open import MLib.Prelude
open import MLib.Matrix.Core
import Relation.Binary.Indexed as I
module S = Struct struct renaming (Carrier to S; _≈_ to _≈′_)
open S hiding (isEquivalence; setoid; refl; sym; trans) public
module _ {m n} where
-- Pointwise equality --
infix 4 _≈_
_≈_ : Rel (Matrix S m n) _
A ≈ B = ∀ i j → A i j ≈′ B i j
isEquivalence : IsEquivalence _≈_
isEquivalence = record { Proofs } where
module Proofs where
refl : Reflexive _≈_
refl _ _ = S.refl
sym : Symmetric _≈_
sym p = λ i j → S.sym (p i j)
trans : Transitive _≈_
trans p q = λ i j → S.trans (p i j) (q i j)
setoid : Setoid _ _
setoid = record { isEquivalence = isEquivalence }
open Setoid setoid public using (refl; sym; trans)
module FunctionProperties = Algebra.FunctionProperties _≈_
-- Size-heterogeneous pointwise equality
infix 4 _≃_
record _≃_ {m n p q} (A : Matrix S m n) (B : Matrix S p q) : Set (c ⊔ˡ ℓ) where
field
m≡p : m ≡ p
n≡q : n ≡ q
equal : ∀ {i i′ j j′} → i ≅ i′ → j ≅ j′ → A i j ≈′ B i′ j′
open _≃_
≃-refl : ∀ {m n} {A : Matrix S m n} → A ≃ A
≃-refl .m≡p = ≡.refl
≃-refl .n≡q = ≡.refl
≃-refl .equal ≅.refl ≅.refl = S.refl
≃-trans :
∀ {m n p q r s} {A : Matrix S m n} {B : Matrix S p q} {C : Matrix S r s} →
A ≃ B → B ≃ C → A ≃ C
≃-trans x y .m≡p = ≡.trans (x .m≡p) (y .m≡p)
≃-trans x y .n≡q = ≡.trans (x .n≡q) (y .n≡q)
≃-trans {m} {n} {p} {q} {r} {s} x y .equal i≅i′′ j≅j′′ =
let i≅i′ = ≅.sym (≅.≡-subst-removable Fin (x .m≡p) _)
i′≅i′′ = ≅.trans (≅.sym i≅i′) i≅i′′
j≅j′ = ≅.sym (≅.≡-subst-removable Fin (x .n≡q) _)
j′≅j′′ = ≅.trans (≅.sym j≅j′) j≅j′′
in S.trans (x .equal i≅i′ j≅j′) (y .equal i′≅i′′ j′≅j′′)
≃-sym :
∀ {m n p q} {A : Matrix S m n} {B : Matrix S p q} →
A ≃ B → B ≃ A
≃-sym A≃B .m≡p = ≡.sym (A≃B .m≡p)
≃-sym A≃B .n≡q = ≡.sym (A≃B .n≡q)
≃-sym A≃B .equal i≅i′ j≅j′ = S.sym (A≃B .equal (≅.sym i≅i′) (≅.sym j≅j′))
≃-setoid : I.Setoid (ℕ × ℕ) _ _
≃-setoid = record
{ Carrier = uncurry (Matrix S)
; _≈_ = _≃_
; isEquivalence = record
{ refl = ≃-refl
; sym = ≃-sym
; trans = ≃-trans
}
}
≡-subst-≃₁ : ∀ {m n p} {A : Matrix S m n} (m≡p : m ≡ p) → ≡.subst (λ h → Matrix S h n) m≡p A ≃ A
≡-subst-≃₁ ≡.refl = ≃-refl
≡-subst-≃₂ : ∀ {m n q} {A : Matrix S m n} (n≡q : n ≡ q) → ≡.subst (Matrix S m) n≡q A ≃ A
≡-subst-≃₂ ≡.refl = ≃-refl
|
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module Type.Identity.Heterogenous where
import Lvl
open import Type
data HId {ℓ} : ∀{A : Type{ℓ}}{B : Type{ℓ}} → A → B → Type{Lvl.𝐒(ℓ)} where
instance intro : ∀{T : Type{ℓ}}{x : T} → HId x x
|
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{-# OPTIONS --cubical --safe #-}
module Data.Sigma.Properties where
open import Prelude hiding (B; C)
open import Cubical.Foundations.HLevels using (isOfHLevelΣ) public
open import Cubical.Data.Sigma.Properties using (Σ≡Prop) public
private
variable
B : A → Type b
C : Σ A B → Type c
reassoc : Σ (Σ A B) C ⇔ Σ[ x ⦂ A ] × Σ[ y ⦂ B x ] × C (x , y)
reassoc .fun ((x , y) , z) = x , (y , z)
reassoc .inv (x , (y , z)) = (x , y) , z
reassoc .leftInv ((x , y) , z) i = ((x , y) , z)
reassoc .rightInv (x , (y , z)) i = (x , (y , z))
≃ΣProp≡ : ∀ {A : Type a} {u} {U : A → Type u} → ((x : A) → isProp (U x)) → {p q : Σ A U} → (p ≡ q) ≃ (fst p ≡ fst q)
≃ΣProp≡ {A = A} {U = U} pA {p} {q} = isoToEquiv (iso to fro (λ _ → refl) (J Jt Jp))
where
to : {p q : Σ A U} → p ≡ q → fst p ≡ fst q
to = cong fst
fro : ∀ {p q} → fst p ≡ fst q → p ≡ q
fro = Σ≡Prop pA
Jt : (q : Σ A U) → p ≡ q → Type _
Jt q q≡ = fro (to q≡) ≡ q≡
Jp : Jt p refl
Jp i j .fst = p .fst
Jp i j .snd = isProp→isSet (pA (p .fst)) (p .snd) (p .snd) (λ k → fro {p} {p} (to (refl {x = p})) k .snd) refl i j
|
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