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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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