Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
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2.09M
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case a.h.e'_4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' ⊒ V x = (V' ∘ Οƒ') x
case a.h.e'_4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' ⊒ V x = V' (Οƒ' x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c1 : x ∈ binders'
case a.h.e'_4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' ⊒ V x = V' (Οƒ' x)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x ∈ binders' ⊒ V x = V' (Οƒ' x) case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x βˆ‰ binders' ⊒ V x = V' (Οƒ' x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h2 x c1
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x ∈ binders' ⊒ V x = V' (Οƒ' x)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply h3
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x βˆ‰ binders' ⊒ V x = V' (Οƒ' x)
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x βˆ‰ binders' ⊒ Οƒ' x βˆ‰ binders'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact ih_1 x a1 c1
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : PredName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v x : VarName a1 : x ∈ xs' c1 : x βˆ‰ binders' ⊒ Οƒ' x βˆ‰ binders'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E (eq_ x y) ↔ Holds D I V' E (eq_ (Οƒ' x) (Οƒ' y))
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V x = V y ↔ V' (Οƒ' x) = V' (Οƒ' y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
congr! 1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V x = V y ↔ V' (Οƒ' x) = V' (Οƒ' y)
case a.h.e'_2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V x = V' (Οƒ' x) case a.h.e'_3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V y = V' (Οƒ' y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c1 : x ∈ binders'
case a.h.e'_2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V x = V' (Οƒ' x)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x ∈ binders' ⊒ V x = V' (Οƒ' x) case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ V x = V' (Οƒ' x)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h2 x c1
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x ∈ binders' ⊒ V x = V' (Οƒ' x)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply h3
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ V x = V' (Οƒ' x)
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ Οƒ' x βˆ‰ binders'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply ih_1
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ Οƒ' x βˆ‰ binders'
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ x = x ∨ x = y case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ x βˆ‰ binders'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ x = x ∨ x = y
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact c1
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : x βˆ‰ binders' ⊒ x βˆ‰ binders'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c1 : y ∈ binders'
case a.h.e'_3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ V y = V' (Οƒ' y)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y ∈ binders' ⊒ V y = V' (Οƒ' y) case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ V y = V' (Οƒ' y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h2 y c1
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y ∈ binders' ⊒ V y = V' (Οƒ' y)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply h3
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ V y = V' (Οƒ' y)
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ Οƒ' y βˆ‰ binders'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply ih_1
case neg.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ Οƒ' y βˆ‰ binders'
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ y = x ∨ y = y case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ y βˆ‰ binders'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ y = x ∨ y = y
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact c1
case neg.a.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x y : VarName ih_1 : βˆ€ (v : VarName), v = x ∨ v = y β†’ v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v c1 : y βˆ‰ binders' ⊒ y βˆ‰ binders'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒβœ : VarName β†’ VarName binders✝ : Finset VarName V V' : VarAssignment D h2 : βˆ€ v ∈ binders✝, V v = V' (Οƒβœ v) h3 : βˆ€ (v : VarName), Οƒβœ v βˆ‰ binders✝ β†’ V v = V' (Οƒβœ v) h4 : βˆ€ v ∈ binders✝, v = Οƒβœ v ⊒ Holds D I V E false_ ↔ Holds D I V' E false_
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux Οƒ' binders' phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E phi.not_ ↔ Holds D I V' E phi'.not_
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux Οƒ' binders' phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Β¬Holds D I V E phi ↔ Β¬Holds D I V' E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
congr! 1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux Οƒ' binders' phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Β¬Holds D I V E phi ↔ Β¬Holds D I V' E phi'
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux Οƒ' binders' phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E phi ↔ Holds D I V' E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact ih_2 V V' h2 h3 h4
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi phi' : Formula a✝ : IsSubAux Οƒ' binders' phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E phi ↔ Holds D I V' E phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E (phi.iff_ psi) ↔ Holds D I V' E (phi'.iff_ psi')
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ (Holds D I V E phi ↔ Holds D I V E psi) ↔ (Holds D I V' E phi' ↔ Holds D I V' E psi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
congr! 1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ (Holds D I V E phi ↔ Holds D I V E psi) ↔ (Holds D I V' E phi' ↔ Holds D I V' E psi')
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E phi ↔ Holds D I V' E phi' case a.h.e'_2.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E psi ↔ Holds D I V' E psi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply phi_ih_2 V V' h2 h3 h4
case a.h.e'_1.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E phi ↔ Holds D I V' E phi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply psi_ih_2 V V' h2 h3 h4
case a.h.e'_2.a D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName phi psi phi' psi' : Formula a✝¹ : IsSubAux Οƒ' binders' phi phi' a✝ : IsSubAux Οƒ' binders' psi psi' phi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') psi_ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders', V v = V' (Οƒ' v)) β†’ (βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v)) β†’ (βˆ€ v ∈ binders', v = Οƒ' v) β†’ (Holds D I V E psi ↔ Holds D I V' E psi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E psi ↔ Holds D I V' E psi'
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp at ih_2
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ v ∈ binders' βˆͺ {x}, V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' βˆͺ {x} β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ v ∈ binders' βˆͺ {x}, v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
have s1 : βˆ€ (v : VarName), Β¬ v = x β†’ v ∈ binders' β†’ Β¬ Οƒ' v = x
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') ⊒ βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro v a1 a2 contra
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') ⊒ βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ False D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply a1
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ False D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ v = x D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [← contra]
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ v = x D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ v = Οƒ' v D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h4 v a2
case s1 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') v : VarName a1 : Β¬v = x a2 : v ∈ binders' contra : Οƒ' v = x ⊒ v = Οƒ' v D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ (βˆƒ d, Holds D I (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V' x d) E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
first | apply forall_congr'| apply exists_congr
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ (βˆƒ d, Holds D I (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V' x d) E phi'
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro d
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ Holds D I (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V' x d) E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply ih_2
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ Holds D I (Function.updateITE V x d) E phi ↔ Holds D I (Function.updateITE V' x d) E phi'
case h.h2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v) case h.h3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v) case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply forall_congr'
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ (βˆ€ (d : D), Holds D I (Function.updateITE V x d) E phi) ↔ βˆ€ (d : D), Holds D I (Function.updateITE V' x d) E phi'
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply exists_congr
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ (βˆƒ d, Holds D I (Function.updateITE V x d) E phi) ↔ βˆƒ d, Holds D I (Function.updateITE V' x d) E phi'
case h D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x ⊒ βˆ€ (a : D), Holds D I (Function.updateITE V x a) E phi ↔ Holds D I (Function.updateITE V' x a) E phi'
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro v a1
case h.h2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case h.h2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c1 : v = x
case h.h2 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v) case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE]
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_pos c1]
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ d = if True then d else V' x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ d = if True then d else V' x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE]
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_neg c1]
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
cases a1
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
case neg.inl D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x h✝ : v ∈ binders' ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v) case neg.inr D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x h✝ : v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c2 => simp only [s1 v c1 c2] simp exact h2 v c2
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c2 => contradiction
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [s1 v c1 c2]
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = if False then d else V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = if False then d else V' (Οƒ' v)
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h2 v c2
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ V v = V' (Οƒ' v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
contradiction
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro v a1
case h.h3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case h.h3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c1 : v = x
case h.h3 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v) case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : Β¬v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE]
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬(if v = x then x else Οƒ' v) = x β†’ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_pos c1]
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬(if v = x then x else Οƒ' v) = x β†’ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬True β†’ d = if True then d else V' x
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : v = x ⊒ Β¬True β†’ d = if True then d else V' x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE] at a1
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : Function.updateITE Οƒ' x x v βˆ‰ binders' c1 : Β¬v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : (if v = x then x else Οƒ' v) βˆ‰ binders' c1 : Β¬v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_neg c1] at a1
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : (if v = x then x else Οƒ' v) βˆ‰ binders' c1 : Β¬v = x ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE]
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ Β¬Function.updateITE Οƒ' x x v = x β†’ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE Οƒ' x x v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ Β¬(if v = x then x else Οƒ' v) = x β†’ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_neg c1]
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ Β¬(if v = x then x else Οƒ' v) = x β†’ (if v = x then d else V v) = if (if v = x then x else Οƒ' v) = x then d else V' (if v = x then x else Οƒ' v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ ¬σ' v = x β†’ V v = if Οƒ' v = x then d else V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro a2
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' ⊒ ¬σ' v = x β†’ V v = if Οƒ' v = x then d else V' (Οƒ' v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' a2 : ¬σ' v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [if_neg a2]
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' a2 : ¬σ' v = x ⊒ V v = if Οƒ' v = x then d else V' (Οƒ' v)
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' a2 : ¬σ' v = x ⊒ V v = V' (Οƒ' v)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply h3 v a1
case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x a1 : Οƒ' v βˆ‰ binders' a2 : ¬σ' v = x ⊒ V v = V' (Οƒ' v)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro v a1
case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D ⊒ βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v
case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ v = Function.updateITE Οƒ' x x v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Function.updateITE]
case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ v = Function.updateITE Οƒ' x x v
case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ v = if v = x then x else Οƒ' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
split_ifs
case h.h4 D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊒ v = if v = x then x else Οƒ' v
case pos D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x h✝ : v = x ⊒ v = x case neg D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x h✝ : Β¬v = x ⊒ v = Οƒ' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c1 => exact c1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c1 => cases a1 case _ c2 => exact h4 v c2 case _ c2 => contradiction
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ v = Οƒ' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact c1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : v = x ⊒ v = x
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
cases a1
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : Β¬v = x ⊒ v = Οƒ' v
case inl D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x h✝ : v ∈ binders' ⊒ v = Οƒ' v case inr D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x h✝ : v = x ⊒ v = Οƒ' v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c2 => exact h4 v c2
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ v = Οƒ' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c2 => contradiction
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v = x ⊒ v = Οƒ' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact h4 v c2
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v ∈ binders' ⊒ v = Οƒ' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
contradiction
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE Οƒ' x x) (binders' βˆͺ {x}) phi phi' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ih_2 : βˆ€ (V V' : VarAssignment D), (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), Function.updateITE Οƒ' x x v βˆ‰ binders' β†’ Β¬Function.updateITE Οƒ' x x v = x β†’ V v = V' (Function.updateITE Οƒ' x x v)) β†’ (βˆ€ (v : VarName), v ∈ binders' ∨ v = x β†’ v = Function.updateITE Οƒ' x x v) β†’ (Holds D I V E phi ↔ Holds D I V' E phi') s1 : βˆ€ (v : VarName), Β¬v = x β†’ v ∈ binders' β†’ ¬σ' v = x d : D v : VarName c1 : Β¬v = x c2 : v = x ⊒ v = Οƒ' v
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
induction E
D : Type I : Interpretation D E : Env Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V E (def_ X' xs') ↔ Holds D I V' E (def_ X' (List.map Οƒ' xs'))
case nil D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map Οƒ' xs')) case cons D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v head✝ : Definition tail✝ : List Definition tail_ih✝ : Holds D I V tail✝ (def_ X' xs') ↔ Holds D I V' tail✝ (def_ X' (List.map Οƒ' xs')) ⊒ Holds D I V (head✝ :: tail✝) (def_ X' xs') ↔ Holds D I V' (head✝ :: tail✝) (def_ X' (List.map Οƒ' xs'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case nil => simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map Οƒ' xs'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v ⊒ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map Οƒ' xs'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [Holds]
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) ⊒ Holds D I V (hd :: tl) (def_ X' xs') ↔ Holds D I V' (hd :: tl) (def_ X' (List.map Οƒ' xs'))
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) ⊒ (if X' = hd.name ∧ xs'.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q else Holds D I V tl (def_ X' xs')) ↔ if X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length then Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q else Holds D I V' tl (def_ X' (List.map Οƒ' xs'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
split_ifs
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) ⊒ (if X' = hd.name ∧ xs'.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q else Holds D I V tl (def_ X' xs')) ↔ if X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length then Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q else Holds D I V' tl (def_ X' (List.map Οƒ' xs'))
case pos D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) h✝¹ : X' = hd.name ∧ xs'.length = hd.args.length h✝ : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q case neg D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) h✝¹ : X' = hd.name ∧ xs'.length = hd.args.length h✝ : Β¬(X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length) ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) case pos D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) h✝¹ : Β¬(X' = hd.name ∧ xs'.length = hd.args.length) h✝ : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I V tl (def_ X' xs') ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q case neg D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) h✝¹ : Β¬(X' = hd.name ∧ xs'.length = hd.args.length) h✝ : Β¬(X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length) ⊒ Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c1 c2 => simp only [List.length_map] at c2 contradiction
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : Β¬(X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length) ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c1 c2 => simp at c2 contradiction
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : Β¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I V tl (def_ X' xs') ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
case _ c1 c2 => exact ih
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : Β¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : Β¬(X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length) ⊒ Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map Οƒ' xs'))) tl hd.q
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I (Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs')) tl hd.q
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply Holds_coincide_Var
D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ Holds D I (Function.updateListITE V hd.args (List.map V xs')) tl hd.q ↔ Holds D I (Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs')) tl hd.q
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v hd.q β†’ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro v a1
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length ⊒ βˆ€ (v : VarName), isFreeIn v hd.q β†’ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
have s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs'
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ List.map V xs' = List.map (V' ∘ Οƒ') xs' case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [List.map_eq_map_iff]
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ List.map V xs' = List.map (V' ∘ Οƒ') xs' case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ βˆ€ x ∈ xs', V x = (V' ∘ Οƒ') x case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
intro x a2
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊒ βˆ€ x ∈ xs', V x = (V' ∘ Οƒ') x case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' ⊒ V x = (V' ∘ Οƒ') x case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
by_cases c3 : x ∈ binders'
case s1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' ⊒ V x = (V' ∘ Οƒ') x case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case pos D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∈ binders' ⊒ V x = (V' ∘ Οƒ') x case neg D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x βˆ‰ binders' ⊒ V x = (V' ∘ Οƒ') x case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [s1]
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map (V' ∘ Οƒ') xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
apply Function.updateListITE_mem_eq_len
case h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ Function.updateListITE V hd.args (List.map (V' ∘ Οƒ') xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ Οƒ') xs') v
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp only [isFreeIn_iff_mem_freeVarSet] at a1
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
obtain s2 := hd.h1 a1
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args.toFinset ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp at s2
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args.toFinset ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
exact s2
case h1.h1 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args ⊒ v ∈ hd.args case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp at c2
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map Οƒ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
simp
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊒ hd.args.length = (List.map (V' ∘ Οƒ') xs').length
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊒ hd.args.length = xs'.length
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Sub/Var/All/Ind/Sub.lean
FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
[129, 1]
[273, 17]
tauto
case h1.h2 D : Type I : Interpretation D Οƒ : VarName β†’ VarName binders : Finset VarName F F' : Formula Οƒ' : VarName β†’ VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : βˆ€ v ∈ xs', v βˆ‰ binders' β†’ Οƒ' v βˆ‰ binders' V V' : VarAssignment D h2 : βˆ€ v ∈ binders', V v = V' (Οƒ' v) h3 : βˆ€ (v : VarName), Οƒ' v βˆ‰ binders' β†’ V v = V' (Οƒ' v) h4 : βˆ€ v ∈ binders', v = Οƒ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map Οƒ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ Οƒ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊒ hd.args.length = xs'.length
no goals