url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
case eq_3 h1_Γ h1_Δ h1_n h1_name h1_xs h1_ys =>
sorry
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_n : ℕ
h1_name : PredName
h1_xs h1_ys : Fin h1_n → VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (eqSubPred h1_name h1_n h1_xs h1_ys)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
case conv h1_Γ h1_Δ h1_phi h1_phi' h1_1 h1_2 h1_3 h1_ih =>
intro V
have s1 : Holds D I V M E h1_phi := h1_ih M nf hyp V
simp only [← holds_conv D I V M E h1_phi h1_phi' h2 h1_3]
exact s1
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi'
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : h1_phi ∈ h1_Δ
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : h1_phi ∈ h1_Δ
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact hyp h1_phi V h1_2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : h1_phi ∈ h1_Δ
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Holds D I V M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds] at h1_ih_2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_2 : IsProof E h1_Γ h1_Δ (h1_phi.imp_ h1_psi)
h1_ih_1 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ h1_psi)
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_psi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_2 : IsProof E h1_Γ h1_Δ (h1_phi.imp_ h1_psi)
h1_ih_1 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi
⊢ ∀ (V : Valuation D), Holds D I V M E h1_psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_2 : IsProof E h1_Γ h1_Δ (h1_phi.imp_ h1_psi)
h1_ih_1 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi
⊢ ∀ (V : Valuation D), Holds D I V M E h1_psi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_2 : IsProof E h1_Γ h1_Δ (h1_phi.imp_ h1_psi)
h1_ih_1 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi
V : Valuation D
⊢ Holds D I V M E h1_psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact h1_ih_2 M nf hyp V (h1_ih_1 M nf hyp V)
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_2 : IsProof E h1_Γ h1_Δ (h1_phi.imp_ h1_psi)
h1_ih_1 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi
V : Valuation D
⊢ Holds D I V M E h1_psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (h1_psi.imp_ h1_phi))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi
a2 : Holds D I V M E h1_psi
⊢ Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a1
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi
a2 : Holds D I V M E h1_psi
⊢ Holds D I V M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
Holds D I V M E ((h1_phi.imp_ (h1_psi.imp_ h1_chi)).imp_ ((h1_phi.imp_ h1_psi).imp_ (h1_phi.imp_ h1_chi)))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi) →
(Holds D I V M E h1_phi → Holds D I V M E h1_psi) → Holds D I V M E h1_phi → Holds D I V M E h1_chi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a2 a3
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi) →
(Holds D I V M E h1_phi → Holds D I V M E h1_psi) → Holds D I V M E h1_phi → Holds D I V M E h1_chi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi
a2 : Holds D I V M E h1_phi → Holds D I V M E h1_psi
a3 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_chi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply a1 a3
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi
a2 : Holds D I V M E h1_phi → Holds D I V M E h1_psi
a3 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_chi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi
a2 : Holds D I V M E h1_phi → Holds D I V M E h1_psi
a3 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a2 a3
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi h1_chi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
h1_3 : IsMetaVarOrAllDefInEnv E h1_chi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : Holds D I V M E h1_phi → Holds D I V M E h1_psi → Holds D I V M E h1_chi
a2 : Holds D I V M E h1_phi → Holds D I V M E h1_psi
a3 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E ((h1_phi.not_.imp_ h1_psi.not_).imp_ (h1_psi.imp_ h1_phi))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi) → Holds D I V M E h1_psi → Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi) → Holds D I V M E h1_psi → Holds D I V M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi
a2 : Holds D I V M E h1_psi
⊢ Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
by_contra contra
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi
a2 : Holds D I V M E h1_psi
⊢ Holds D I V M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi
a2 : Holds D I V M E h1_psi
contra : ¬Holds D I V M E h1_phi
⊢ False
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a1 contra a2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ¬Holds D I V M E h1_phi → ¬Holds D I V M E h1_psi
a2 : Holds D I V M E h1_psi
contra : ¬Holds D I V M E h1_phi
⊢ False
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (forall_ h1_x h1_phi)
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D) (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V d
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D) (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact h1_ih M nf hyp (Function.updateITE V h1_x d)
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
Holds D I V M E ((forall_ h1_x (h1_phi.imp_ h1_psi)).imp_ ((forall_ h1_x h1_phi).imp_ (forall_ h1_x h1_psi)))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(∀ (d : D),
Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi) →
(∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi) →
∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a2 d
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D),
(∀ (d : D),
Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi) →
(∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi) →
∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_psi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi
a2 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply a1 d
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi
a2 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_psi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi
a2 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a2 d
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_psi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : IsMetaVarOrAllDefInEnv E h1_psi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi → Holds D I (Function.updateITE V h1_x d) M E h1_psi
a2 : ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
d : D
⊢ Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
have s1 : IsNotFree D I M E h1_phi h1_x
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
case s1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ IsNotFree D I M E h1_phi h1_x
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply not_free_imp_is_not_free D I M E h1_phi h1_Γ h1_x h1_2
|
case s1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ IsNotFree D I M E h1_phi h1_x
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
case s1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (X : MetaVarName), (h1_x, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) h1_x
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact nf h1_x
|
case s1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (X : MetaVarName), (h1_x, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) h1_x
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E (h1_phi.imp_ (forall_ h1_x h1_phi))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi → ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi → ∀ (d : D), Holds D I (Function.updateITE V h1_x d) M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I (Function.updateITE V h1_x a) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
unfold IsNotFree at s1
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : IsNotFree D I M E h1_phi h1_x
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I (Function.updateITE V h1_x a) M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : ∀ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I (Function.updateITE V h1_x a) M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [← s1 V a]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : ∀ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I (Function.updateITE V h1_x a) M E h1_phi
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : ∀ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a1
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi : Formula
h1_x : VarName
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi
h1_2 : NotFree h1_Γ h1_x h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
s1 : ∀ (V : Valuation D) (d : D), Holds D I V M E h1_phi ↔ Holds D I (Function.updateITE V h1_x d) M E h1_phi
V : Valuation D
a1 : Holds D I V M E h1_phi
a : D
⊢ Holds D I V M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
unfold exists_
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (exists_ h1_x (eq_ h1_x h1_y))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (forall_ h1_x (eq_ h1_x h1_y).not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (forall_ h1_x (eq_ h1_x h1_y).not_).not_
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), ¬∀ (d : D), ¬Function.updateITE V h1_x d h1_x = Function.updateITE V h1_x d h1_y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), ¬∀ (d : D), ¬Function.updateITE V h1_x d h1_x = Function.updateITE V h1_x d h1_y
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), ∃ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), ∃ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ ∃ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply Exists.intro (V h1_y)
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ ∃ x, Function.updateITE V h1_x x h1_x = Function.updateITE V h1_x x h1_y
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Function.updateITE V h1_x (V h1_y) h1_x = Function.updateITE V h1_x (V h1_y) h1_y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
unfold Function.updateITE
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Function.updateITE V h1_x (V h1_y) h1_x = Function.updateITE V h1_x (V h1_y) h1_y
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ (if h1_x = h1_x then V h1_y else V h1_x) = if h1_y = h1_x then V h1_y else V h1_y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y : VarName
h1_1 : ¬h1_y = h1_x
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ (if h1_x = h1_x then V h1_y else V h1_x) = if h1_y = h1_x then V h1_y else V h1_y
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E ((eq_ h1_x h1_y).imp_ ((eq_ h1_x h1_z).imp_ (eq_ h1_y h1_z)))
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), V h1_x = V h1_y → V h1_x = V h1_z → V h1_y = V h1_z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V a1 a2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), V h1_x = V h1_y → V h1_x = V h1_z → V h1_y = V h1_z
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_y = V h1_z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
trans V h1_x
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_y = V h1_z
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_y = V h1_x
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_x = V h1_z
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [a1]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_y = V h1_x
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact a2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_x h1_y h1_z : VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
a1 : V h1_x = V h1_y
a2 : V h1_x = V h1_z
⊢ V h1_x = V h1_z
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
sorry
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_n : ℕ
h1_name : PredName
h1_xs h1_ys : Fin h1_n → VarName
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (eqSubPred h1_name h1_n h1_xs h1_ys)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
obtain ⟨σ', a1⟩ := h1_σ.2
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
⊢ ∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
have s1 : IsMetaVarOrAllDefInEnv E h1_phi := is_proof_imp_is_meta_var_or_all_def_in_env E h1_Γ h1_Δ h1_phi h1_4
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
⊢ ∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
⊢ ∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
⊢ ∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [← holds_sub D I V M E h1_σ σ' h1_τ h1_phi s1 a1]
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ Holds D I V M E (sub h1_σ h1_τ h1_phi)
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ Holds D I (V ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply h1_ih_2
|
case intro
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ Holds D I (V ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E h1_phi
|
case intro.nf
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ ∀ (v : VarName) (X : MetaVarName),
(v, X) ∈ h1_Γ → IsNotFree D I (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E (meta_var_ X) v
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro v X a2
|
case intro.nf
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ ∀ (v : VarName) (X : MetaVarName),
(v, X) ∈ h1_Γ → IsNotFree D I (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E (meta_var_ X) v
|
case intro.nf
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
v : VarName
X : MetaVarName
a2 : (v, X) ∈ h1_Γ
⊢ IsNotFree D I (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E (meta_var_ X) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact lem_1 D I M E h1_Γ h1_Γ' h1_σ σ' h1_τ a1 nf h1_2 v X a2
|
case intro.nf
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
v : VarName
X : MetaVarName
a2 : (v, X) ∈ h1_Γ
⊢ IsNotFree D I (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E (meta_var_ X) v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro psi V' a2
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
⊢ ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E F
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
have s2 : IsMetaVarOrAllDefInEnv E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s2
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsMetaVarOrAllDefInEnv E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply lem_2_b E h1_σ h1_τ
|
case s2
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsMetaVarOrAllDefInEnv E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s2.h1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsMetaVarOrAllDefInEnv E (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
apply is_proof_imp_is_meta_var_or_all_def_in_env E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
|
case s2.h1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsMetaVarOrAllDefInEnv E (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s2.h1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact h1_3 psi a2
|
case s2.h1
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
⊢ IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
have s3 : ∀ V'' : Valuation D, Holds D I (V'' ∘ h1_σ.val) (fun (X' : MetaVarName) (V' : Valuation D) => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V''
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
⊢ ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
V'' : Valuation D
⊢ Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [holds_sub D I V'' M E h1_σ σ' h1_τ psi s2 a1]
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
V'' : Valuation D
⊢ Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
V'' : Valuation D
⊢ Holds D I V'' M E (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact h1_ih_1 psi a2 M nf hyp V''
|
case s3
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
V'' : Valuation D
⊢ Holds D I V'' M E (sub h1_σ h1_τ psi)
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
specialize s3 (V' ∘ σ')
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : ∀ (V'' : Valuation D), Holds D I (V'' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I ((V' ∘ σ') ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Function.comp.assoc] at s3
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I ((V' ∘ σ') ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I (V' ∘ σ' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [a1.right] at s3
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I (V' ∘ σ' ∘ ↑h1_σ) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I (V' ∘ id) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [Function.comp_id] at s3
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I (V' ∘ id) (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact s3
|
case intro.hyp
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ h1_Γ' : List (VarName × MetaVarName)
h1_Δ h1_Δ' : List Formula
h1_phi : Formula
h1_σ : Instantiation
h1_τ : MetaInstantiation
h1_1 : ∀ X ∈ h1_phi.metaVarSet, IsMetaVarOrAllDefInEnv E (h1_τ X)
h1_2 : ∀ (x : VarName) (X : MetaVarName), (x, X) ∈ h1_Γ → NotFree h1_Γ' (↑h1_σ x) (h1_τ X)
h1_3 : ∀ psi ∈ h1_Δ, IsProof E h1_Γ' h1_Δ' (sub h1_σ h1_τ psi)
h1_4 : IsProof E h1_Γ h1_Δ h1_phi
h1_ih_1 :
∀ psi ∈ h1_Δ,
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F) →
∀ (V : Valuation D), Holds D I V M E (sub h1_σ h1_τ psi)
h1_ih_2 :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ' → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ' → Holds D I V M E F
σ' : VarName → VarName
a1 : ↑h1_σ ∘ σ' = id ∧ σ' ∘ ↑h1_σ = id
s1 : IsMetaVarOrAllDefInEnv E h1_phi
V : Valuation D
psi : Formula
V' : Valuation D
a2 : psi ∈ h1_Δ
s2 : IsMetaVarOrAllDefInEnv E psi
s3 : Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
⊢ Holds D I V' (fun X' V' => Holds D I (V' ∘ σ') M E (h1_τ X')) E psi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
intro V
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
⊢ ∀ (V : Valuation D), Holds D I V M E h1_phi'
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Holds D I V M E h1_phi'
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
have s1 : Holds D I V M E h1_phi := h1_ih M nf hyp V
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
⊢ Holds D I V M E h1_phi'
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
s1 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_phi'
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
simp only [← holds_conv D I V M E h1_phi h1_phi' h2 h1_3]
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
s1 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_phi'
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
s1 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_is_proof
|
[2060, 1]
|
[2162, 13]
|
exact s1
|
D : Type
I : Interpretation D
E : Env
Γ : List (VarName × MetaVarName)
Δ : List Formula
F : Formula
h2 : E.WellFormed
h1_Γ : List (VarName × MetaVarName)
h1_Δ : List Formula
h1_phi h1_phi' : Formula
h1_1 : IsMetaVarOrAllDefInEnv E h1_phi'
h1_2 : IsProof E h1_Γ h1_Δ h1_phi
h1_3 : IsConv E h1_phi h1_phi'
h1_ih :
∀ (M : MetaValuation D),
(∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v) →
(∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F) → ∀ (V : Valuation D), Holds D I V M E h1_phi
M : MetaValuation D
nf : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ h1_Γ → IsNotFree D I M E (meta_var_ X) v
hyp : ∀ (F : Formula) (V : Valuation D), F ∈ h1_Δ → Holds D I V M E F
V : Valuation D
s1 : Holds D I V M E h1_phi
⊢ Holds D I V M E h1_phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
DA.memAccepts
|
[145, 1]
|
[151, 67]
|
rfl
|
α σ : Type
D : DA α σ
input : List α
⊢ D.accepts input ↔ D.evalFrom D.startingState input ∈ D.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NA.memAccepts
|
[154, 1]
|
[161, 38]
|
rfl
|
α σ : Type
N : NA α σ
input : List α
⊢ N.accepts input ↔ ∃ s ∈ N.evalFrom N.startingStates input, s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
ext cs
|
α σ : Type
N : NA α σ
⊢ N.toDA.accepts = N.accepts
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.accepts cs ↔ N.accepts cs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [DA.memAccepts]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.accepts cs ↔ N.accepts cs
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔ N.accepts cs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [NA.memAccepts]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔ N.accepts cs
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp only [NA.toDA]
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ N.toDA.evalFrom N.toDA.startingState cs ∈ N.toDA.acceptingStates ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ { step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∈
{S | ∃ s ∈ S, s ∈ N.acceptingStates} ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ { step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∈
{S | ∃ s ∈ S, s ∈ N.acceptingStates} ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
constructor
|
case h.a
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) ↔
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
|
case h.a.mp
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) →
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
all_goals
simp
intro s a1 a2
apply Exists.intro s
tauto
|
case h.a.mp
α σ : Type
N : NA α σ
cs : List α
⊢ (∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates) →
∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
simp
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ (∃ s ∈ N.evalFrom N.startingStates cs, s ∈ N.acceptingStates) →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ ∀ x ∈ N.evalFrom N.startingStates cs,
x ∈ N.acceptingStates →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
intro s a1 a2
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
⊢ ∀ x ∈ N.evalFrom N.startingStates cs,
x ∈ N.acceptingStates →
∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ ∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
apply Exists.intro s
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ ∃
s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs,
s ∈ N.acceptingStates
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∧
s ∈ N.acceptingStates
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/Compute.lean
|
NAtoDAisEquiv
|
[167, 1]
|
[183, 10]
|
tauto
|
case h.a.mpr
α σ : Type
N : NA α σ
cs : List α
s : σ
a1 : s ∈ N.evalFrom N.startingStates cs
a2 : s ∈ N.acceptingStates
⊢ s ∈
{ step := N.stepSet, startingState := N.startingStates,
acceptingStates := {S | ∃ s ∈ S, s ∈ N.acceptingStates} }.evalFrom
N.startingStates cs ∧
s ∈ N.acceptingStates
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/NFA.lean
|
NFA.mem_accepts
|
[106, 1]
|
[115, 43]
|
rfl
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : NFA α σ
input : List α
⊢ e.accepts input ↔ ∃ s ∈ e.eval_from e.starting_state_list input, s ∈ e.accepting_state_list
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_def
|
[111, 1]
|
[124, 9]
|
simp only [eval_one_no_eps]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈ e.eval_one_no_eps starting_state_list symbol ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_def
|
[111, 1]
|
[124, 9]
|
simp
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, stop_state ∈ symbol_arrow_list_to_fun e.symbol_arrow_list state symbol
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [EpsilonNFA.eval_one_no_eps]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈ e.eval_one_no_eps starting_state_list symbol ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [EpsilonNFA.toAbstract]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list, e.toAbstract.symbol state symbol stop_state
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp only [symbol_arrow_list_to_fun]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map (fun state => symbol_arrow_list_to_fun e.symbol_arrow_list state symbol)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state ∧ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
simp
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ stop_state ∈
(List.map
(fun state =>
(List.filterMap
(fun arrow =>
if arrow.start_state = state ∧ arrow.symbol = symbol then some arrow.stop_state_list else none)
e.symbol_arrow_list).join.dedup)
starting_state_list).join.dedup ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
constructor
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) ↔
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
case mp
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) →
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
case mpr
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list) →
∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
rintro ⟨_, h1, _, ⟨⟨⟩, h2, ⟨rfl, rfl⟩, rfl⟩, h3⟩
|
case mp
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l) →
∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
stop_state start_state✝ : σ
symbol✝ : α
stop_state_list✝ : List σ
h2 : { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ } ∈ e.symbol_arrow_list
h1 :
{ start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.start_state ∈
starting_state_list
h3 :
stop_state ∈ { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.stop_state_list
⊢ ∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state,
symbol := { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.symbol,
stop_state_list := stop_state_list } ∈
e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
exact ⟨_, h1, _, h2, h3⟩
|
case mp.intro.intro.intro.intro.intro.mk.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
stop_state start_state✝ : σ
symbol✝ : α
stop_state_list✝ : List σ
h2 : { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ } ∈ e.symbol_arrow_list
h1 :
{ start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.start_state ∈
starting_state_list
h3 :
stop_state ∈ { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.stop_state_list
⊢ ∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state,
symbol := { start_state := start_state✝, symbol := symbol✝, stop_state_list := stop_state_list✝ }.symbol,
stop_state_list := stop_state_list } ∈
e.symbol_arrow_list ∧
stop_state ∈ stop_state_list
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
rintro ⟨_, h1, _, h2, h3⟩
|
case mpr
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state : σ
⊢ (∃ state ∈ starting_state_list,
∃ stop_state_list,
{ start_state := state, symbol := symbol, stop_state_list := stop_state_list } ∈ e.symbol_arrow_list ∧
stop_state ∈ stop_state_list) →
∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l
|
case mpr.intro.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state w✝¹ : σ
h1 : w✝¹ ∈ starting_state_list
w✝ : List σ
h2 : { start_state := w✝¹, symbol := symbol, stop_state_list := w✝ } ∈ e.symbol_arrow_list
h3 : stop_state ∈ w✝
⊢ ∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.eval_one_no_eps_iff
|
[354, 1]
|
[374, 53]
|
exact ⟨_, h1, _, ⟨_, h2, ⟨rfl, rfl⟩, rfl⟩, h3⟩
|
case mpr.intro.intro.intro.intro
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
symbol : α
stop_state w✝¹ : σ
h1 : w✝¹ ∈ starting_state_list
w✝ : List σ
h2 : { start_state := w✝¹, symbol := symbol, stop_state_list := w✝ } ∈ e.symbol_arrow_list
h3 : stop_state ∈ w✝
⊢ ∃ a ∈ starting_state_list,
∃ l,
(∃ a_1 ∈ e.symbol_arrow_list, (a_1.start_state = a ∧ a_1.symbol = symbol) ∧ a_1.stop_state_list = l) ∧
stop_state ∈ l
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp only [epsilon_closure]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ e.epsilon_closure starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp only [dft_iff]
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ state ∈ dft (epsilon_arrow_list_to_graph e.epsilon_arrow_list) starting_state_list ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ (∃ s ∈ starting_state_list,
Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ b ∈ l) s
state) ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
congr! with a b c
|
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state : σ
⊢ (∃ s ∈ starting_state_list,
Relation.ReflTransGen (fun a b => ∃ l, (a, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ b ∈ l) s
state) ↔
∃ start_state ∈ starting_state_list, e.toAbstract.EpsilonClosure start_state state
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔ e.toAbstract.epsilon b c
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Parsing/EpsilonNFA.lean
|
EpsilonNFA.epsilon_closure_iff
|
[385, 1]
|
[402, 83]
|
simp [toAbstract]
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔ e.toAbstract.epsilon b c
|
case a.h.e'_2.h.h.e'_2.h.e.h.e'_2.h.h.a
α : Type
inst✝¹ : DecidableEq α
σ : Type
inst✝ : DecidableEq σ
e : EpsilonNFA α σ
starting_state_list : List σ
state a b c : σ
⊢ (∃ l, (b, l) ∈ epsilon_arrow_list_to_graph e.epsilon_arrow_list ∧ c ∈ l) ↔
∃ stop_state_list,
{ start_state := b, stop_state_list := stop_state_list } ∈ e.epsilon_arrow_list ∧ c ∈ stop_state_list
|
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