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Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' x x v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' x x v) case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' x x v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' v) = x then d else V' (if v = x then x else σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' v) = x then d else V' (if v = x then x else σ' v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' x x v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' v) = x then d else V' (if v = x then x else σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ' v) = x then d else V' (if v = x then x else σ' v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : ¬v = x ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : ¬v = x ⊢ V v = if σ' v = x then d else V' (σ' v)	case neg.inl D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x h✝ : v ∈ binders' ⊢ V v = if σ' v = x then d else V' (σ' v) case neg.inr D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x h✝ : v = x ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v = x ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = if σ' v = x then d else V' (σ' v)	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = if False then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = if False then d else V' (σ' v)	D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ V v = V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v = x ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h3 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v) case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : ¬v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬(if v = x then x else σ' v) = x → (if v = x then d else V v) = if (if v = x then x else σ' v) = x then d else V' (if v = x then x else σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬(if v = x then x else σ' v) = x → (if v = x then d else V v) = if (if v = x then x else σ' v) = x then d else V' (if v = x then x else σ' v)	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬True → 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : v = x ⊢ ¬True → 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : Function.updateITE σ' x x v ∉ binders' c1 : ¬v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : (if v = x then x else σ' v) ∉ binders' c1 : ¬v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : (if v = x then x else σ' v) ∉ binders' c1 : ¬v = x ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬Function.updateITE σ' x x v = x → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬(if v = x then x else σ' v) = x → (if v = x then d else V v) = if (if v = x then x else σ' v) = x then d else V' (if v = x then x else σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬(if v = x then x else σ' v) = x → (if v = x then d else V v) = if (if v = x then x else σ' v) = x then d else V' (if v = x then x else σ' v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬σ' v = x → V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' ⊢ ¬σ' v = x → V v = if σ' v = x then d else V' (σ' v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' a2 : ¬σ' v = x ⊢ V v = if σ' v = x then d else V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' a2 : ¬σ' v = x ⊢ V v = if σ' v = x then d else V' (σ' v)	case neg D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' a2 : ¬σ' v = x ⊢ V v = V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x a1 : σ' v ∉ binders' a2 : ¬σ' v = x ⊢ V v = V' (σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D ⊢ ∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v	case h.h4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊢ v = Function.updateITE σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊢ v = Function.updateITE σ' x x v	case h.h4 D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊢ v = if v = x then x else σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x ⊢ v = if v = x then x else σ' v	case pos D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x h✝ : ¬v = x ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : ¬v = x ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName a1 : v ∈ binders' ∨ v = x c1 : ¬v = x ⊢ v = σ' v	case inl D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x h✝ : v ∈ binders' ⊢ v = σ' v case inr D : Type I : Interpretation D E : Env σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x h✝ : v = x ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v = x ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v ∈ binders' ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName x : VarName phi phi' : Formula a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ih_2 : ∀ (V V' : VarAssignment D), (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), Function.updateITE σ' x x v ∉ binders' → ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) → (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) → (Holds D I V E phi ↔ Holds D I V' E phi') s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x d : D v : VarName c1 : ¬v = x c2 : v = x ⊢ v = σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V E (def_ X' xs') ↔ Holds D I V' E (def_ X' (List.map σ' xs'))	case nil D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map σ' xs')) case cons D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' 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 σ' xs')) ⊢ Holds D I V (head✝ :: tail✝) (def_ X' xs') ↔ Holds D I V' (head✝ :: tail✝) (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v ⊢ Holds D I V [] (def_ X' xs') ↔ Holds D I V' [] (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) ⊢ Holds D I V (hd :: tl) (def_ X' xs') ↔ Holds D I V' (hd :: tl) (def_ X' (List.map σ' xs'))	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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 σ' xs').length = hd.args.length then Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map σ' xs'))) tl hd.q else Holds D I V' tl (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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 σ' xs').length = hd.args.length then Holds D I (Function.updateListITE V' hd.args (List.map V' (List.map σ' xs'))) tl hd.q else Holds D I V' tl (def_ X' (List.map σ' xs'))	case pos D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) h✝¹ : X' = hd.name ∧ xs'.length = hd.args.length h✝ : X' = hd.name ∧ (List.map σ' 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 σ' xs'))) tl hd.q case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) h✝¹ : X' = hd.name ∧ xs'.length = hd.args.length h✝ : ¬(X' = hd.name ∧ (List.map σ' 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 σ' xs')) case pos D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) h✝¹ : ¬(X' = hd.name ∧ xs'.length = hd.args.length) h✝ : X' = hd.name ∧ (List.map σ' 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 σ' xs'))) tl hd.q case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) h✝¹ : ¬(X' = hd.name ∧ xs'.length = hd.args.length) h✝ : ¬(X' = hd.name ∧ (List.map σ' xs').length = hd.args.length) ⊢ Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : ¬(X' = hd.name ∧ (List.map σ' 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 σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : X' = hd.name ∧ (List.map σ' 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 σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : ¬(X' = hd.name ∧ (List.map σ' xs').length = hd.args.length) ⊢ Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' 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 σ' xs'))) tl hd.q	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' 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' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' 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' ∘ σ') xs')) tl hd.q	case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length ⊢ ∀ (v : VarName), isFreeIn v hd.q → Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length ⊢ ∀ (v : VarName), isFreeIn v hd.q → Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' 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' ∘ σ') 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' ∘ σ') xs'	case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' 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' ∘ σ') xs') v	case s1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊢ List.map V xs' = List.map (V' ∘ σ') xs' case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊢ List.map V xs' = List.map (V' ∘ σ') xs' case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case s1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q ⊢ ∀ x ∈ xs', V x = (V' ∘ σ') x case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case s1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' ⊢ V x = (V' ∘ σ') x case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' ⊢ V x = (V' ∘ σ') x case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case pos D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∈ binders' ⊢ V x = (V' ∘ σ') x case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∉ binders' ⊢ V x = (V' ∘ σ') x case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map V xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map (V' ∘ σ') xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ Function.updateListITE V hd.args (List.map (V' ∘ σ') xs') v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs') v	case h1.h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args.toFinset ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args.toFinset ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h1 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName s1 : List.map V xs' = List.map (V' ∘ σ') xs' a1 : v ∈ hd.q.freeVarSet s2 : v ∈ hd.args ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q s1 : List.map V xs' = List.map (V' ∘ σ') xs' ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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' ∘ σ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊢ hd.args.length = (List.map (V' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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' ∘ σ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊢ hd.args.length = (List.map (V' ∘ σ') xs').length	case h1.h2 D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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' ∘ σ') 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 σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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' ∘ σ') xs' c2 : X' = hd.name ∧ xs'.length = hd.args.length ⊢ hd.args.length = xs'.length	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 h2 x c3	case pos D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∈ binders' ⊢ V x = (V' ∘ σ') 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 x	case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∉ binders' ⊢ V x = (V' ∘ σ') x	case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∉ binders' ⊢ σ' x ∉ 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 x a2 c3	case neg D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : X' = hd.name ∧ (List.map σ' xs').length = hd.args.length v : VarName a1 : isFreeIn v hd.q x : VarName a2 : x ∈ xs' c3 : x ∉ binders' ⊢ σ' x ∉ 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 [List.length_map] at c2	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : ¬(X' = hd.name ∧ (List.map σ' 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 σ' xs'))	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : ¬(X' = hd.name ∧ 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 σ' 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]	contradiction	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : X' = hd.name ∧ xs'.length = hd.args.length c2 : ¬(X' = hd.name ∧ 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 σ' 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 at c2	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : X' = hd.name ∧ (List.map σ' 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 σ' xs'))) tl hd.q	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : X' = hd.name ∧ 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 σ' 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]	contradiction	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : X' = hd.name ∧ 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 σ' 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]	exact ih	D : Type I : Interpretation D σ : VarName → VarName binders : Finset VarName F F' : Formula σ' : VarName → VarName binders' : Finset VarName X' : DefName xs' : List VarName ih_1 : ∀ v ∈ xs', v ∉ binders' → σ' v ∉ binders' V V' : VarAssignment D h2 : ∀ v ∈ binders', V v = V' (σ' v) h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v) h4 : ∀ v ∈ binders', v = σ' v hd : Definition tl : List Definition ih : Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' xs')) c1 : ¬(X' = hd.name ∧ xs'.length = hd.args.length) c2 : ¬(X' = hd.name ∧ (List.map σ' xs').length = hd.args.length) ⊢ Holds D I V tl (def_ X' xs') ↔ Holds D I V' tl (def_ X' (List.map σ' 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	[276, 1]	[289, 9]	apply substitution_theorem_aux D I (V ∘ σ) V E σ ∅ F F' h1	D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ Holds D I (V ∘ σ) E F ↔ Holds D I V E F'	case h2 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ v ∈ ∅, (V ∘ σ) v = V (σ v) case h3 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ (v : VarName), σ v ∉ ∅ → (V ∘ σ) v = V (σ v) case h4 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ v ∈ ∅, v = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_theorem	[276, 1]	[289, 9]	simp	case h2 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ v ∈ ∅, (V ∘ σ) v = V (σ 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	[276, 1]	[289, 9]	simp	case h3 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ (v : VarName), σ v ∉ ∅ → (V ∘ σ) v = V (σ 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	[276, 1]	[289, 9]	simp	case h4 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' ⊢ ∀ v ∈ ∅, v = σ 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_is_valid	[292, 1]	[304, 11]	simp only [IsValid] at h2	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : F.IsValid ⊢ F'.IsValid	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ F'.IsValid
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_is_valid	[292, 1]	[304, 11]	simp only [IsValid]	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ F'.IsValid	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_is_valid	[292, 1]	[304, 11]	intro D I V E	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F'	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E F'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_is_valid	[292, 1]	[304, 11]	simp only [← substitution_theorem D I V E σ F F' h1]	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E F'	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I (V ∘ σ) E F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Ind/Sub.lean	FOL.NV.Sub.Var.All.Ind.substitution_is_valid	[292, 1]	[304, 11]	apply h2	σ : VarName → VarName F F' : Formula h1 : IsSub σ F F' h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I (V ∘ σ) E F	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	induction h1	Δ : List Formula F : Formula h1 : IsDeduct Δ F ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ, Holds D I V E H) → Holds D I V E F	case struct_1_ Δ : List Formula F : Formula Δ✝ : List Formula H✝ phi✝ : Formula a✝ : IsDeduct Δ✝ phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ H✝ :: Δ✝, Holds D I V E H) → Holds D I V E phi✝ case struct_2_ Δ : List Formula F : Formula Δ✝ : List Formula H✝ phi✝ : Formula a✝ : IsDeduct (H✝ :: H✝ :: Δ✝) phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ H✝ :: H✝ :: Δ✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ H✝ :: Δ✝, Holds D I V E H) → Holds D I V E phi✝ case struct_3_ Δ : List Formula F : Formula Δ_1✝ Δ_2✝ : List Formula H_1✝ H_2✝ phi✝ : Formula a✝ : IsDeduct (Δ_1✝ ++ [H_1✝] ++ [H_2✝] ++ Δ_2✝) phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1✝ ++ [H_1✝] ++ [H_2✝] ++ Δ_2✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1✝ ++ [H_2✝] ++ [H_1✝] ++ Δ_2✝, Holds D I V E H) → Holds D I V E phi✝ case assume_ Δ : List Formula F phi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [phi✝], Holds D I V E H) → Holds D I V E phi✝ case prop_0_ Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E true_ case prop_1_ Δ : List Formula F phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi✝.imp_ (psi✝.imp_ phi✝)) case prop_2_ Δ : List Formula F phi✝ psi✝ chi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi✝.imp_ (psi✝.imp_ chi✝)).imp_ ((phi✝.imp_ psi✝).imp_ (phi✝.imp_ chi✝))) case prop_3_ Δ : List Formula F phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi✝.not_.imp_ psi✝.not_).imp_ (psi✝.imp_ phi✝)) case mp_ Δ : List Formula F : Formula Δ✝ : List Formula phi✝ psi✝ : Formula a✝¹ : IsDeduct Δ✝ (phi✝.imp_ psi✝) a✝ : IsDeduct Δ✝ phi✝ a_ih✝¹ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E (phi✝.imp_ psi✝) a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E psi✝ case dt_ Δ : List Formula F : Formula Δ✝ : List Formula H✝ phi✝ : Formula a✝ : IsDeduct (H✝ :: Δ✝) phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ H✝ :: Δ✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E (H✝.imp_ phi✝) case pred_1_ Δ : List Formula F : Formula v✝ : VarName phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v✝ (phi✝.imp_ psi✝)).imp_ ((forall_ v✝ phi✝).imp_ (forall_ v✝ psi✝))) case pred_2_ Δ : List Formula F : Formula v✝ t✝ : VarName phi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v✝ phi✝).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v✝ t✝) freshChar phi✝)) case pred_3_ Δ : List Formula F : Formula v✝ : VarName phi✝ : Formula a✝ : ¬isFreeIn v✝ phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi✝.imp_ (forall_ v✝ phi✝)) case gen_ Δ : List Formula F : Formula v✝ : VarName phi✝ : Formula a✝ : IsDeduct [] phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (forall_ v✝ phi✝) case eq_1_ Δ : List Formula F : Formula v✝ : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (eq_ v✝ v✝) case eq_2_pred_var_ Δ : List Formula F : Formula name✝ : PredName xs✝ ys✝ : List VarName a✝ : xs✝.length = ys✝.length ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((List.foldr and_ true_ (List.zipWith eq_ xs✝ ys✝)).imp_ ((pred_var_ name✝ xs✝).iff_ (pred_var_ name✝ ys✝))) case eq_2_eq_ Δ : List Formula F : Formula x_0✝ x_1✝ y_0✝ y_1✝ : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (((eq_ x_0✝ y_0✝).and_ (eq_ x_1✝ y_1✝)).imp_ ((eq_ x_0✝ x_1✝).iff_ (eq_ y_0✝ y_1✝))) case def_false_ Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (false_.iff_ true_.not_) case def_and_ Δ : List Formula F phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi✝.and_ psi✝).iff_ (phi✝.imp_ psi✝.not_).not_) case def_or_ Δ : List Formula F phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi✝.or_ psi✝).iff_ (phi✝.not_.imp_ psi✝)) case def_iff_ Δ : List Formula F phi✝ psi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (((phi✝.iff_ psi✝).imp_ ((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_).imp_ (((phi✝.imp_ psi✝).imp_ (psi✝.imp_ phi✝).not_).not_.imp_ (phi✝.iff_ psi✝)).not_).not_ case def_exists_ Δ : List Formula F : Formula v✝ : VarName phi✝ : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((exists_ v✝ phi✝).iff_ (forall_ v✝ phi✝.not_).not_) case sub_ Δ : List Formula F : Formula Δ✝ : List Formula phi✝ : Formula τ✝ : PredName → ℕ → Option (List VarName × Formula) a✝ : IsDeduct Δ✝ phi✝ a_ih✝ : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ✝, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar τ✝) Δ✝, Holds D I V E H) → Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar τ✝ phi✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case struct_1_ Δ' H _ _ ih_2 => intro D I V E a1 apply ih_2 intro H' a2 simp at a1 cases a1 case _ a1_left a1_right => exact a1_right H' a2	Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct Δ' phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case struct_2_ Δ' H _ _ ih_2 => intro D I V E a1 apply ih_2 intro H' a2 simp at a1 cases a1 case _ a1_left a1_right => simp at a2 cases a2 case _ a2 => simp only [a2] exact a1_left case _ a2 => exact a1_right H' a2	Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct (H :: H :: Δ') phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case struct_3_ Δ_1 Δ_2 H_1 H_2 _ _ ih_2 => intro D I V E a1 apply ih_2 intro H' a2 simp at a1 apply a1 simp at a2 tauto	Δ : List Formula F : Formula Δ_1 Δ_2 : List Formula H_1 H_2 phi✝ : Formula a✝ : IsDeduct (Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2) phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_1] ++ [H_2] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ_1 ++ [H_2] ++ [H_1] ++ Δ_2, Holds D I V E H) → Holds D I V E phi✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case assume_ phi => intro D I V E a1 simp at a1 exact a1	Δ : List Formula F phi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [phi], Holds D I V E H) → Holds D I V E phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case prop_0_ => intro D I V E _ simp only [Holds]	Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E true_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case prop_1_ phi psi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (psi.imp_ phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case prop_2_ phi psi chi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi chi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi)))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case prop_3_ phi psi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case pred_1_ v phi psi => intro D I V E _ simp only [Holds] intro a2 a3 d apply a2 d exact a3 d	Δ : List Formula F : Formula v : VarName phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi)))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case pred_2_ v t phi => intro D I V E _ obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi simp only [Holds] intro a2 simp only [s1] simp only [Function.updateITE_comp_left] simp exact a2 (V t)	Δ : List Formula F : Formula v t : VarName phi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case pred_3_ v phi ih => intro D I V E _ simp only [Holds] intro a2 d have s1 : Holds D I (Function.updateITE V v d) E phi ↔ Holds D I V E phi { apply Holds_coincide_Var intro v' a1 simp only [Function.updateITE] split_ifs case _ c1 => subst c1 contradiction case _ c1 => rfl } simp only [s1] exact a2	Δ : List Formula F : Formula v : VarName phi : Formula ih : ¬isFreeIn v phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (phi.imp_ (forall_ v phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case gen_ v phi _ ih_2 => intro D I V E _ simp only [Holds] intro d apply ih_2 simp	Δ : List Formula F : Formula v : VarName phi : Formula a✝ : IsDeduct [] phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (forall_ v phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case eq_1_ v => intro D I V E _ simp only [Holds]	Δ : List Formula F : Formula v : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (eq_ v v)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case eq_2_eq_ x_0 x_1 y_0 y_1 => intro D I V E _ simp only [Holds] intro a2 cases a2 case _ a2_left a2_right => simp only [a2_left] simp only [a2_right]	Δ : List Formula F : Formula x_0 x_1 y_0 y_1 : VarName ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1)))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case def_false_ => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (false_.iff_ true_.not_)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case def_and_ phi psi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case def_or_ phi psi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case def_iff_ phi psi => intro D I V E _ simp only [Holds] tauto	Δ : List Formula F phi psi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E (((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_ (((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case def_exists_ v phi => intro D I V E _ simp only [Holds] simp	Δ : List Formula F : Formula v : VarName phi : Formula ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ [], Holds D I V E H) → Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	case sub_ Δ' phi τ _ ih_2 => intro D I V E a1 simp at a1 obtain s1 := Sub.Pred.All.Rec.Option.Fresh.substitution_theorem D I V E freshChar τ simp only [← s1] at a1 simp only [← s1] apply ih_2 exact a1	Δ : List Formula F : Formula Δ' : List Formula phi : Formula τ : PredName → ℕ → Option (List VarName × Formula) a✝ : IsDeduct Δ' phi ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar τ) Δ', Holds D I V E H) → Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar τ phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	intro D I V E a1	Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct Δ' phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝ ⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H_1 ∈ H :: Δ', Holds D I V E H_1) → Holds D I V E phi✝	Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct Δ' phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1 ⊢ Holds D I V E phi✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Program/Backend.lean	FOL.NV.Program.Backend.soundness	[793, 1]	[979, 13]	apply ih_2	Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct Δ' phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1 ⊢ Holds D I V E phi✝	case a Δ : List Formula F : Formula Δ' : List Formula H phi✝ : Formula a✝ : IsDeduct Δ' phi✝ ih_2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (∀ H ∈ Δ', Holds D I V E H) → Holds D I V E phi✝ D : Type I : Interpretation D V : VarAssignment D E : Env a1 : ∀ H_1 ∈ H :: Δ', Holds D I V E H_1 ⊢ ∀ H ∈ Δ', Holds D I V E H

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