Datasets:

internlm
/

Lean-Github

Dataset card Data Studio Files Files and versions Community

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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	by_cases c1 : y ∈ binders	case a.h.e'_3.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders ⊢ y ∈ binders ∨ σ' y ∉ binders	case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∈ binders ⊢ y ∈ binders ∨ σ' y ∉ binders case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ y ∈ binders ∨ σ' y ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	left	case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∈ binders ⊢ y ∈ binders ∨ σ' y ∉ binders	case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∈ binders ⊢ y ∈ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact c1	case pos.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∈ binders ⊢ y ∈ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	right	case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ y ∈ binders ∨ σ' y ∉ binders	case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ σ' y ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [h3 y c1]	case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ σ' y ∉ binders	case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ σ y ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact h1_right c1	case neg.h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x y : VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : x ∉ binders → σ x ∉ binders h1_right : y ∉ binders → σ y ∉ binders c1 : y ∉ binders ⊢ σ y ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	congr! 1	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ binders phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ binders phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact phi_ih V V' σ σ' binders h1 h2 h2' h3	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ binders phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	cases h1	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ binders phi ∧ admitsAux σ binders psi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔ (Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi))	case intro D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v left✝ : admitsAux σ binders phi right✝ : admitsAux σ binders psi ⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔ (Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	congr! 1	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : admitsAux σ binders phi h1_right : admitsAux σ binders psi ⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔ (Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi) ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi))	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : admitsAux σ binders phi h1_right : admitsAux σ binders psi ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi) case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : admitsAux σ binders phi h1_right : admitsAux σ binders psi ⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact phi_ih V V' σ σ' binders h1_left h2 h2' h3	case a.h.e'_1.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : admitsAux σ binders phi h1_right : admitsAux σ binders psi ⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact psi_ih V V' σ σ' binders h1_right h2 h2' h3	case a.h.e'_2.a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi psi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) psi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders psi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v h1_left : admitsAux σ binders phi h1_right : admitsAux σ binders psi ⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' psi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	first \| apply forall_congr' \| apply exists_congr	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro d	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply phi_ih (Function.updateITE V x d) (Function.updateITE V' x d) σ (Function.updateITE σ' x x) (binders ∪ {x}) h1	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ (v : VarName), v ∈ binders ∪ {x} ∨ Function.updateITE σ' x x v ∉ binders ∪ {x} → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v) case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ v ∈ binders ∪ {x}, v = Function.updateITE σ' x x v case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ v ∉ binders ∪ {x}, Function.updateITE σ' x x v = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply forall_congr'	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply exists_congr	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔ ∃ d, Holds D I (Function.updateITE V' x d) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)	case h D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔ Holds D I (Function.updateITE V' x a) (head✝ :: tail✝) (fastReplaceFree (Function.updateITE σ' x x) phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro v a1	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ (v : VarName), v ∈ binders ∪ {x} ∨ Function.updateITE σ' x x v ∉ binders ∪ {x} → Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ∨ Function.updateITE σ' x x v ∉ binders ∪ {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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [Function.updateITE] at a1	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ∨ Function.updateITE σ' x x v ∉ binders ∪ {x} ⊢ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ∨ (if v = x then x else σ' v) ∉ binders ∪ {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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp at a1	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ∨ (if v = x then x else σ' v) ∉ binders ∪ {x} ⊢ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [Function.updateITE]	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x ⊢ Function.updateITE V x d v = Function.updateITE V' x d (Function.updateITE σ' x x v)	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	split_ifs	case h.h2 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬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 head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x h✝¹ : v = x h✝ : x = x ⊢ d = d case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x h✝¹ : v = x h✝ : ¬x = x ⊢ d = V' x case pos D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x h✝¹ : ¬v = x h✝ : σ' v = x ⊢ V v = d case neg D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x h✝¹ : ¬v = x h✝ : ¬σ' v = x ⊢ V v = V' (σ' v)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => rfl	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : v = x c2 : x = x ⊢ d = d	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => contradiction	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : v = x c2 : ¬x = x ⊢ d = V' x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => subst c2 tauto	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : ¬v = x c2 : σ' v = x ⊢ V v = d	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => simp only [if_neg c1] at a1 apply h2 tauto	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : ¬v = x c2 : ¬σ' v = x ⊢ V v = V' (σ' v)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	rfl	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : v = x c2 : x = x ⊢ d = d	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	contradiction	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : v = x c2 : ¬x = x ⊢ d = V' x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	subst c2	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : ¬v = x c2 : σ' v = x ⊢ V v = d	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName h1 : admitsAux σ (binders ∪ {σ' v}) phi a1 : (v ∈ binders ∨ v = σ' v) ∨ (if v = σ' v then σ' v else σ' v) ∉ binders ∧ ¬v = σ' v ∧ ¬σ' v = σ' v c1 : ¬v = σ' v ⊢ V v = d
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	tauto	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName h1 : admitsAux σ (binders ∪ {σ' v}) phi a1 : (v ∈ binders ∨ v = σ' v) ∨ (if v = σ' v then σ' v else σ' v) ∉ binders ∧ ¬v = σ' v ∧ ¬σ' v = σ' v c1 : ¬v = σ' v ⊢ V v = d	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [if_neg c1] at a1	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : (v ∈ binders ∨ v = x) ∨ (if v = x then x else σ' v) ∉ binders ∧ ¬v = x ∧ ¬σ' v = x c1 : ¬v = x c2 : ¬σ' v = x ⊢ V v = V' (σ' v)	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName c1 : ¬v = x c2 : ¬σ' v = x a1 : (v ∈ binders ∨ v = x) ∨ σ' v ∉ binders ∧ ¬v = x ∧ ¬σ' v = x ⊢ V v = V' (σ' v)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply h2	D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName c1 : ¬v = x c2 : ¬σ' v = x a1 : (v ∈ binders ∨ v = x) ∨ σ' v ∉ binders ∧ ¬v = x ∧ ¬σ' v = x ⊢ V v = V' (σ' v)	case a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName c1 : ¬v = x c2 : ¬σ' v = x a1 : (v ∈ binders ∨ v = x) ∨ σ' v ∉ binders ∧ ¬v = x ∧ ¬σ' v = x ⊢ v ∈ binders ∨ σ' v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	tauto	case a D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName c1 : ¬v = x c2 : ¬σ' v = x a1 : (v ∈ binders ∨ v = x) ∨ σ' v ∉ binders ∧ ¬v = x ∧ ¬σ' v = x ⊢ v ∈ binders ∨ σ' v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro v a1	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ v ∈ binders ∪ {x}, v = Function.updateITE σ' x x v	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ⊢ v = Function.updateITE σ' x x v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp at a1	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∪ {x} ⊢ v = Function.updateITE σ' x x v	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [Function.updateITE]	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∨ v = x ⊢ v = Function.updateITE σ' x x v	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	split_ifs <;> tauto	case h.h2' D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∈ binders ∨ v = x ⊢ v = if v = x then x else σ' v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro v a1	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D ⊢ ∀ v ∉ binders ∪ {x}, Function.updateITE σ' x x v = σ v	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∪ {x} ⊢ Function.updateITE σ' x x v = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp at a1	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∪ {x} ⊢ Function.updateITE σ' x x v = σ v	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∧ ¬v = x ⊢ Function.updateITE σ' x x v = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [Function.updateITE]	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∧ ¬v = x ⊢ Function.updateITE σ' x x v = σ v	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∧ ¬v = x ⊢ (if v = x then x else σ' v) = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	split_ifs <;> tauto	case h.h3 D : Type I : Interpretation D head✝ : Definition tail✝ : List Definition tail_ih✝ : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ (fastReplaceFree σ' F)) x : VarName phi : Formula phi_ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName), admitsAux σ binders phi → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) (fastReplaceFree σ' phi)) V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : admitsAux σ (binders ∪ {x}) phi h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v d : D v : VarName a1 : v ∉ binders ∧ ¬v = x ⊢ (if v = x then x else σ' v) = σ v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	split_ifs	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v ⊢ (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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => simp only [List.length_map] at c2 contradiction	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => simp at c2 contradiction	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	case _ c1 c2 => specialize ih V V' σ σ' binders (def_ X xs) simp only [fastReplaceFree] at ih apply ih h1 h2 h2' h3	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	have s1 : List.map V xs = List.map (V' ∘ σ') xs	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length ⊢ List.map V xs = List.map (V' ∘ σ') xs D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [s1]	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ 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	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ Holds D I (Function.updateListITE V hd.args (List.map (V' ∘ σ') 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply Holds_coincide_Var	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ Holds D I (Function.updateListITE V hd.args (List.map (V' ∘ σ') 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ ∀ (v : VarName), isFreeIn v hd.q → 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro v a1	case h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs ⊢ ∀ (v : VarName), isFreeIn v hd.q → Function.updateListITE V hd.args (List.map (V' ∘ σ') xs) v = Function.updateListITE V' hd.args (List.map (V' ∘ σ') xs) v	case h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply Function.updateListITE_mem_eq_len	case h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ v ∈ hd.args case h1.h2 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ hd.args.length = (List.map (V' ∘ σ') xs).length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [List.map_eq_map_iff]	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length ⊢ List.map V xs = List.map (V' ∘ σ') xs	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length ⊢ ∀ x ∈ xs, V x = (V' ∘ σ') x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	intro x a1	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length ⊢ ∀ x ∈ xs, V x = (V' ∘ σ') x	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ V x = (V' ∘ σ') x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ V x = (V' ∘ σ') x	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ V x = V' (σ' x)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply h2	case s1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ V x = V' (σ' x)	case s1.a D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ x ∈ binders ∨ σ' x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	by_cases c3 : x ∈ binders	case s1.a D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs ⊢ x ∈ binders ∨ σ' x ∉ binders	case pos D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∈ binders ⊢ x ∈ binders ∨ σ' x ∉ binders case neg D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ x ∈ binders ∨ σ' x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	left	case pos D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∈ binders ⊢ x ∈ binders ∨ σ' x ∉ binders	case pos.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∈ binders ⊢ x ∈ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact c3	case pos.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∈ binders ⊢ x ∈ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	right	case neg D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ x ∈ binders ∨ σ' x ∉ binders	case neg.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ σ' x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [h3 x c3]	case neg.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ σ' x ∉ binders	case neg.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ σ x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	exact h1 x a1 c3	case neg.h D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length x : VarName a1 : x ∈ xs c3 : x ∉ binders ⊢ σ x ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [isFreeIn_iff_mem_freeVarSet] at a1	case h1.h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ v ∈ hd.args	case h1.h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [← List.mem_toFinset]	case h1.h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args	case h1.h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply Finset.mem_of_subset hd.h1 a1	case h1.h1 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : v ∈ hd.q.freeVarSet ⊢ v ∈ hd.args.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp	case h1.h2 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ hd.args.length = (List.map (V' ∘ σ') xs).length	case h1.h2 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ hd.args.length = xs.length
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	tauto	case h1.h2 D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : X = hd.name ∧ xs.length = hd.args.length c2 : X = hd.name ∧ (List.map σ' xs).length = hd.args.length s1 : List.map V xs = List.map (V' ∘ σ') xs v : VarName a1 : isFreeIn v hd.q ⊢ hd.args.length = xs.length	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [List.length_map] at c2	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	contradiction	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp at c2	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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 hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	contradiction	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	specialize ih V V' σ σ' binders (def_ X xs)	D : Type I : Interpretation D hd : Definition tl : List Definition ih : ∀ (V V' : VarAssignment D) (σ σ' : VarName → VarName) (binders : Finset VarName) (F : Formula), admitsAux σ binders F → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl F ↔ Holds D I V' tl (fastReplaceFree σ' F)) X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v 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))	D : Type I : Interpretation D hd : Definition tl : List Definition X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) c2 : ¬(X = hd.name ∧ (List.map σ' xs).length = hd.args.length) ih : admitsAux σ binders (def_ X xs) → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (fastReplaceFree σ' (def_ X xs))) ⊢ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	simp only [fastReplaceFree] at ih	D : Type I : Interpretation D hd : Definition tl : List Definition X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) c2 : ¬(X = hd.name ∧ (List.map σ' xs).length = hd.args.length) ih : admitsAux σ binders (def_ X xs) → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (fastReplaceFree σ' (def_ X xs))) ⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X (List.map σ' xs))	D : Type I : Interpretation D hd : Definition tl : List Definition X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) c2 : ¬(X = hd.name ∧ (List.map σ' xs).length = hd.args.length) ih : admitsAux σ binders (def_ X xs) → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X (List.map σ' xs))) ⊢ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem_aux	[74, 1]	[207, 28]	apply ih h1 h2 h2' h3	D : Type I : Interpretation D hd : Definition tl : List Definition X : DefName xs : List VarName V V' : VarAssignment D σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∈ xs, v ∉ binders → σ v ∉ binders h2 : ∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v) h2' : ∀ v ∈ binders, v = σ' v h3 : ∀ v ∉ binders, σ' v = σ v c1 : ¬(X = hd.name ∧ xs.length = hd.args.length) c2 : ¬(X = hd.name ∧ (List.map σ' xs).length = hd.args.length) ih : admitsAux σ binders (def_ X xs) → (∀ (v : VarName), v ∈ binders ∨ σ' v ∉ binders → V v = V' (σ' v)) → (∀ v ∈ binders, v = σ' v) → (∀ v ∉ binders, σ' v = σ v) → (Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ X (List.map σ' xs))) ⊢ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem	[210, 1]	[224, 9]	apply substitution_theorem_aux D I (V ∘ σ) V E σ σ ∅ F h1	D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ Holds D I (V ∘ σ) E F ↔ Holds D I V E (fastReplaceFree σ F)	case h2 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ (v : VarName), v ∈ ∅ ∨ σ v ∉ ∅ → (V ∘ σ) v = V (σ v) case h2' D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ v ∈ ∅, v = σ v case h3 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ v ∉ ∅, σ v = σ v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem	[210, 1]	[224, 9]	simp	case h2 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ (v : VarName), v ∈ ∅ ∨ σ v ∉ ∅ → (V ∘ σ) v = V (σ v)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem	[210, 1]	[224, 9]	simp	case h2' D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ v ∈ ∅, v = σ v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_theorem	[210, 1]	[224, 9]	simp	case h3 D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName F : Formula h1 : admits σ F ⊢ ∀ v ∉ ∅, σ v = σ v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_is_valid	[227, 1]	[239, 25]	simp only [IsValid] at h2	σ : VarName → VarName F : Formula h1 : admits σ F h2 : F.IsValid ⊢ (fastReplaceFree σ F).IsValid	σ : VarName → VarName F : Formula h1 : admits σ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ (fastReplaceFree σ F).IsValid
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_is_valid	[227, 1]	[239, 25]	simp only [IsValid]	σ : VarName → VarName F : Formula h1 : admits σ F h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F ⊢ (fastReplaceFree σ F).IsValid	σ : VarName → VarName F : Formula h1 : admits σ 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 (fastReplaceFree σ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_is_valid	[227, 1]	[239, 25]	intro D I V E	σ : VarName → VarName F : Formula h1 : admits σ 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 (fastReplaceFree σ F)	σ : VarName → VarName F : Formula h1 : admits σ 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 (fastReplaceFree σ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_is_valid	[227, 1]	[239, 25]	simp only [← substitution_theorem D I V E σ F h1]	σ : VarName → VarName F : Formula h1 : admits σ 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 (fastReplaceFree σ F)	σ : VarName → VarName F : Formula h1 : admits σ 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/Rec/Admits.lean	FOL.NV.Sub.Var.All.Rec.substitution_is_valid	[227, 1]	[239, 25]	exact h2 D I (V ∘ σ) E	σ : VarName → VarName F : Formula h1 : admits σ 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/Parsing/DFT.lean	list_cons_to_set_union	[9, 1]	[17, 9]	ext a	α : Type inst : DecidableEq α ys : List α x : α ⊢ ↑(x :: ys).toFinset = {x} ∪ ↑ys.toFinset	case h α : Type inst : DecidableEq α ys : List α x a : α ⊢ a ∈ ↑(x :: ys).toFinset ↔ a ∈ {x} ∪ ↑ys.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_cons_to_set_union	[9, 1]	[17, 9]	simp	case h α : Type inst : DecidableEq α ys : List α x a : α ⊢ a ∈ ↑(x :: ys).toFinset ↔ a ∈ {x} ∪ ↑ys.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_append_to_set_union	[19, 1]	[26, 9]	ext a	α : Type inst : DecidableEq α xs ys : List α ⊢ ↑(xs ++ ys).toFinset = ↑xs.toFinset ∪ ↑ys.toFinset	case h α : Type inst : DecidableEq α xs ys : List α a : α ⊢ a ∈ ↑(xs ++ ys).toFinset ↔ a ∈ ↑xs.toFinset ∪ ↑ys.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	list_append_to_set_union	[19, 1]	[26, 9]	simp	case h α : Type inst : DecidableEq α xs ys : List α a : α ⊢ a ∈ ↑(xs ++ ys).toFinset ↔ a ∈ ↑xs.toFinset ∪ ↑ys.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	induction g	Node : Type inst✝ : DecidableEq Node g : Graph Node ⊢ ∀ (x y : Node), y ∈ direct_succ_list g x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ g	case nil Node : Type inst✝ : DecidableEq Node ⊢ ∀ (x y : Node), y ∈ direct_succ_list [] x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ [] case cons Node : Type inst✝ : DecidableEq Node head✝ : Node × List Node tail✝ : List (Node × List Node) tail_ih✝ : ∀ (x y : Node), y ∈ direct_succ_list tail✝ x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tail✝ ⊢ ∀ (x y : Node), y ∈ direct_succ_list (head✝ :: tail✝) x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ head✝ :: tail✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	case nil => simp only [direct_succ_list] simp	Node : Type inst✝ : DecidableEq Node ⊢ ∀ (x y : Node), y ∈ direct_succ_list [] x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	simp only [direct_succ_list]	Node : Type inst✝ : DecidableEq Node ⊢ ∀ (x y : Node), y ∈ direct_succ_list [] x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ []	Node : Type inst✝ : DecidableEq Node ⊢ ∀ (x y : Node), y ∈ [] ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ []
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	simp	Node : Type inst✝ : DecidableEq Node ⊢ ∀ (x y : Node), y ∈ [] ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	simp only [direct_succ_list]	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl ⊢ ∀ (x y : Node), y ∈ direct_succ_list (hd :: tl) x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl ⊢ ∀ (x y : Node), (y ∈ if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	intro x y	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl ⊢ ∀ (x y : Node), (y ∈ if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl x y : Node ⊢ (y ∈ if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	split_ifs	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl x y : Node ⊢ (y ∈ if hd.1 = x then hd.2 ++ direct_succ_list tl x else direct_succ_list tl x) ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl	case pos Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl x y : Node h✝ : hd.1 = x ⊢ y ∈ hd.2 ++ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl case neg Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl x y : Node h✝ : ¬hd.1 = x ⊢ y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	subst c1	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl x y : Node c1 : hd.1 = x ⊢ y ∈ hd.2 ++ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ hd :: tl	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ y ∈ hd.2 ++ direct_succ_list tl hd.1 ↔ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ hd :: tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	simp	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ y ∈ hd.2 ++ direct_succ_list tl hd.1 ↔ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ hd :: tl	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ y ∈ hd.2 ∨ y ∈ direct_succ_list tl hd.1 ↔ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	simp only [ih]	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ y ∈ hd.2 ∨ y ∈ direct_succ_list tl hd.1 ↔ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ (y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl) ↔ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	constructor	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ (y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl) ↔ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)	case mp Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ (y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl) → ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl) case mpr Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ (∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)) → y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	intro a1	case mp Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node ⊢ (y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl) → ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)	case mp Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node a1 : y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl ⊢ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	cases a1	case mp Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node a1 : y ∈ hd.2 ∨ ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl ⊢ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)	case mp.inl Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node h✝ : y ∈ hd.2 ⊢ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl) case mp.inr Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node h✝ : ∃ ys, y ∈ ys ∧ (hd.1, ys) ∈ tl ⊢ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/DFT.lean	mem_direct_succ_list_iff	[65, 1]	[122, 16]	case _ left => apply Exists.intro hd.snd tauto	Node : Type inst✝ : DecidableEq Node hd : Node × List Node tl : List (Node × List Node) ih : ∀ (x y : Node), y ∈ direct_succ_list tl x ↔ ∃ ys, y ∈ ys ∧ (x, ys) ∈ tl y : Node left : y ∈ hd.2 ⊢ ∃ ys, y ∈ ys ∧ ((hd.1, ys) = hd ∨ (hd.1, ys) ∈ tl)	no goals

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.