internlm/Lean-Github · Datasets at Hugging Face

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/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	apply h1	case h.e'_1.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ σ v = σ' v	case h.e'_1.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi ∨ isFreeIn v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	left	case h.e'_1.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi ∨ isFreeIn v psi	case h.e'_1.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	exact a1	case h.e'_1.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	apply psi_ih	case h.e'_2 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi	case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	intro v a1	case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v	case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ σ v = σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	apply h1	case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ σ v = σ' v	case h.e'_2.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v phi ∨ isFreeIn v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	right	case h.e'_2.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v phi ∨ isFreeIn v psi	case h.e'_2.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	exact a1	case h.e'_2.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	congr! 1	x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) = exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)	case h.e'_2 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	apply phi_ih	case h.e'_2 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	intro v a1	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	simp only [Function.updateITE]	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ (if v = x then x else σ v) = if v = x then x else σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	split_ifs	case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ (if v = x then x else σ v) = if v = x then x else σ' v	case pos x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi h✝ : v = x ⊢ x = x case neg x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi h✝ : ¬v = x ⊢ σ v = σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	case _ c1 => rfl	x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : v = x ⊢ x = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	case _ c1 => apply h1 tauto	x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ σ v = σ' v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	rfl	x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : v = x ⊢ x = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	apply h1	x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ σ v = σ' v	case a x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ ¬v = x ∧ isFreeIn v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free	[171, 1]	[223, 12]	tauto	case a x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ ¬v = x ∧ isFreeIn v phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	induction F generalizing binders	F : Formula σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders F = replaceFreeAux σ' binders F	case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝) case true_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_ case false_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_ case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	all_goals simp only [replaceFreeAux]	case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝) case true_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_ case false_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_ case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)	case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ eq_ (if a✝¹ ∉ binders then σ a✝¹ else a✝¹) (if a✝ ∉ binders then σ a✝ else a✝) = eq_ (if a✝¹ ∉ binders then σ' a✝¹ else a✝¹) (if a✝ ∉ binders then σ' a✝ else a✝) case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝).not_ = (replaceFreeAux σ' binders a✝).not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).imp_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).imp_ (replaceFreeAux σ' binders a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).and_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).and_ (replaceFreeAux σ' binders a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).or_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).or_ (replaceFreeAux σ' binders a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).iff_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).iff_ (replaceFreeAux σ' binders a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ forall_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = forall_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = exists_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => congr! 1 simp only [List.map_eq_map_iff] intro x _ split_ifs <;> tauto	σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) = def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	case not_ phi phi_ih => simp tauto	σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	case imp_ phi psi phi_ih psi_ih \| and_ phi psi phi_ih psi_ih \| or_ phi psi phi_ih psi_ih \| iff_ phi psi phi_ih psi_ih => simp tauto	σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) = (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => congr! 1 apply phi_ih simp tauto	σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	simp only [replaceFreeAux]	case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)	case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	congr! 1	σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) = def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	simp only [List.map_eq_map_iff]	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	intro x _	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v x : VarName a✝ : x ∈ xs ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	split_ifs <;> tauto	case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v x : VarName a✝ : x ∈ xs ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	congr! 1	σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ eq_ (if x ∉ binders then σ x else x) (if y ∉ binders then σ y else y) = eq_ (if x ∉ binders then σ' x else x) (if y ∉ binders then σ' y else y)	case h.e'_1 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x case h.e'_2 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	split_ifs <;> tauto	case h.e'_1 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	split_ifs <;> tauto	case h.e'_2 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	simp	σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_	σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	tauto	σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	simp	σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) = (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi)	σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧ replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	tauto	σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧ replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	congr! 1	σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)	case h.e'_2 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	apply phi_ih	case h.e'_2 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi	case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	simp	case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v	case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free	[226, 1]	[260, 10]	tauto	case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	induction F generalizing σ	σ : VarName → VarName c : Char F : Formula ⊢ (sub σ c F).freeVarSet = Finset.image σ F.freeVarSet	case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet case true_ c : Char σ : VarName → VarName ⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet case false_ c : Char σ : VarName → VarName ⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	all_goals simp only [sub] simp only [freeVarSet]	case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet case true_ c : Char σ : VarName → VarName ⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet case false_ c : Char σ : VarName → VarName ⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet	case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ {σ a✝¹, σ a✝} = Finset.image σ {a✝¹, a✝} case true_ c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅ case false_ c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅ case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ a✝¹ (if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹)) c a✝).freeVarSet \ {if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹} = Finset.image σ (a✝.freeVarSet \ {a✝¹}) case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ a✝¹ (if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹)) c a✝).freeVarSet \ {if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹} = Finset.image σ (a✝.freeVarSet \ {a✝¹}) case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case pred_const_ X xs \| pred_var_ X xs \| eq_ x y \| def_ X xs => apply Finset.ext intro a simp	c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case true_ \| false_ => simp	c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case not_ phi phi_ih => exact phi_ih σ	c : Char phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp only [phi_ih] simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ] split_ifs case _ c1 => obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at * have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi) apply Finset.image_subset_image simp have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) apply Finset.not_mem_mono s2 s1 calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3 case _ c1 => simp at c1 have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x} apply Finset.image_sdiff_singleton simp only [Function.updateITE] simp simp only [s1] clear s1 have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) simp only [Finset.mem_image] simp simp only [Function.updateITE] simp tauto simp only [Finset.sdiff_singleton_eq_self s2]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)) c phi).freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} = Finset.image σ (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [sub]	case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet	case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [freeVarSet]	case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet	case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	apply Finset.ext	c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset	case a c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	intro a	case a c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset	case a c : Char X : DefName xs : List VarName σ : VarName → VarName a : VarName ⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	case a c : Char X : DefName xs : List VarName σ : VarName → VarName a : VarName ⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	exact phi_ih σ	c : Char phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Finset.image_union]	c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ (phi.freeVarSet ∪ psi.freeVarSet)	c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	congr!	c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet	case h.e'_3 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet case h.e'_4 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	exact phi_ih σ	case h.e'_3 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	exact psi_ih σ	case h.e'_4 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [phi_ih]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)) c phi).freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} = Finset.image σ (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image σ (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image σ (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	split_ifs	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case pos c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName h✝ : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) case neg c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName h✝ : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case _ c1 => obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at * have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi) apply Finset.image_subset_image simp have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) apply Finset.not_mem_mono s2 s1 calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	case _ c1 => simp at c1 have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x} apply Finset.image_sdiff_singleton simp only [Function.updateITE] simp simp only [s1] clear s1 have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) simp only [Finset.mem_image] simp simp only [Function.updateITE] simp tauto simp only [Finset.sdiff_singleton_eq_self s2]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi))	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x s1 : fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at *	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x s1 : fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi)	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	apply Finset.image_subset_image	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2.h c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	case s2.h c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s3 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	apply Finset.not_mem_mono s2 s1	case s3 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x')	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'}	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Function.updateITE σ x x' x = x'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Function.updateITE]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Function.updateITE σ x x' x = x'	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ (if True then x' else σ x) = x'
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ (if True then x' else σ x) = x'	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	exact Finset.sdiff_singleton_eq_self s3	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp at c1	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x}	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	apply Finset.image_sdiff_singleton	case s1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Function.updateITE σ x x x = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Function.updateITE]	case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Function.updateITE σ x x x = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ (if True then x else σ x) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ (if True then x else σ x) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [s1]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	clear s1	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Finset.mem_image]	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Function.updateITE]	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	tauto	case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image	[48, 1]	[125, 52]	simp only [Finset.sdiff_singleton_eq_self s2]	c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	induction F generalizing σ V	D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName c : Char F : Formula ⊢ Holds D I V E (sub σ c F) ↔ Holds D I (V ∘ σ) E F	case pred_const_ D : Type I : Interpretation D E : Env c : Char a✝¹ : PredName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (pred_const_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_const_ a✝¹ a✝) case pred_var_ D : Type I : Interpretation D E : Env c : Char a✝¹ : PredName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (pred_var_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_var_ a✝¹ a✝) case eq_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (eq_ a✝¹ a✝) case true_ D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c true_) ↔ Holds D I (V ∘ σ) E true_ case false_ D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_ case not_ D : Type I : Interpretation D E : Env c : Char a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c a✝.not_) ↔ Holds D I (V ∘ σ) E a✝.not_ case imp_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.imp_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.imp_ a✝) case and_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.and_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.and_ a✝) case or_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.or_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.or_ a✝) case iff_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.iff_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.iff_ a✝) case forall_ D : Type I : Interpretation D E : Env c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (forall_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (forall_ a✝¹ a✝) case exists_ D : Type I : Interpretation D E : Env c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (exists_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (exists_ a✝¹ a✝) case def_ D : Type I : Interpretation D E : Env c : Char a✝¹ : DefName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (def_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	case pred_const_ X xs \| pred_var_ X xs \| eq_ x y => simp only [sub] simp only [Holds] simp	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	case true_ \| false_ => simp only [sub] simp only [Holds]	D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	case not_ phi phi_ih => simp only [sub] simp only [Holds] congr! 1 exact phi_ih V σ	D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [sub]	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [Holds]	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp	D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [sub]	D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_	D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [Holds]	D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [sub]	D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_	D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean	FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem	[128, 1]	[245, 19]	simp only [Holds]	D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_	D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ ¬Holds D I V E (sub σ c phi) ↔ ¬Holds D I (V ∘ σ) E phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

apply h1

case h.e'_1.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ σ v = σ' v

case h.e'_1.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi ∨ isFreeIn v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

left

case h.e'_1.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi ∨ isFreeIn v psi

case h.e'_1.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

exact a1

case h.e'_1.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ isFreeIn v phi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

apply psi_ih

case h.e'_2 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ fastReplaceFree σ psi = fastReplaceFree σ' psi

case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

intro v a1

case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v psi → σ v = σ' v

case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ σ v = σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

apply h1

case h.e'_2.h1 phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ σ v = σ' v

case h.e'_2.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v phi ∨ isFreeIn v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

right

case h.e'_2.h1.a phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v phi ∨ isFreeIn v psi

case h.e'_2.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

exact a1

case h.e'_2.h1.a.h phi psi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi psi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v psi → σ v = σ' v) → fastReplaceFree σ psi = fastReplaceFree σ' psi σ σ' : VarName → VarName h1 : ∀ (v : VarName), isFreeIn v phi ∨ isFreeIn v psi → σ v = σ' v v : VarName a1 : isFreeIn v psi ⊢ isFreeIn v psi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

congr! 1

x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ exists_ x (fastReplaceFree (Function.updateITE σ x x) phi) = exists_ x (fastReplaceFree (Function.updateITE σ' x x) phi)

case h.e'_2 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

apply phi_ih

case h.e'_2 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ fastReplaceFree (Function.updateITE σ x x) phi = fastReplaceFree (Function.updateITE σ' x x) phi

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

intro v a1

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v ⊢ ∀ (v : VarName), isFreeIn v phi → Function.updateITE σ x x v = Function.updateITE σ' x x v

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

simp only [Function.updateITE]

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ Function.updateITE σ x x v = Function.updateITE σ' x x v

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ (if v = x then x else σ v) = if v = x then x else σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

split_ifs

case h.e'_2.h1 x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi ⊢ (if v = x then x else σ v) = if v = x then x else σ' v

case pos x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi h✝ : v = x ⊢ x = x case neg x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi h✝ : ¬v = x ⊢ σ v = σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

case _ c1 => rfl

x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : v = x ⊢ x = x

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

case _ c1 => apply h1 tauto

x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ σ v = σ' v

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

rfl

x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : v = x ⊢ x = x

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

apply h1

x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ σ v = σ' v

case a x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ ¬v = x ∧ isFreeIn v phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.fastReplaceFree_same_on_free

[171, 1]

[223, 12]

tauto

case a x : VarName phi : Formula phi_ih : ∀ (σ σ' : VarName → VarName), (∀ (v : VarName), isFreeIn v phi → σ v = σ' v) → fastReplaceFree σ phi = fastReplaceFree σ' phi σ σ' : VarName → VarName h1 : ∀ (v : VarName), ¬v = x ∧ isFreeIn v phi → σ v = σ' v v : VarName a1 : isFreeIn v phi c1 : ¬v = x ⊢ ¬v = x ∧ isFreeIn v phi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

induction F generalizing binders

F : Formula σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders F = replaceFreeAux σ' binders F

case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝) case true_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_ case false_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_ case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

all_goals simp only [replaceFreeAux]

case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_const_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_const_ a✝¹ a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (pred_var_ a✝¹ a✝) = replaceFreeAux σ' binders (pred_var_ a✝¹ a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (eq_ a✝¹ a✝) = replaceFreeAux σ' binders (eq_ a✝¹ a✝) case true_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders true_ = replaceFreeAux σ' binders true_ case false_ σ σ' : VarName → VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders false_ = replaceFreeAux σ' binders false_ case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders a✝.not_ = replaceFreeAux σ' binders a✝.not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.imp_ a✝) = replaceFreeAux σ' binders (a✝¹.imp_ a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.and_ a✝) = replaceFreeAux σ' binders (a✝¹.and_ a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.or_ a✝) = replaceFreeAux σ' binders (a✝¹.or_ a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (a✝¹.iff_ a✝) = replaceFreeAux σ' binders (a✝¹.iff_ a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (forall_ a✝¹ a✝) = replaceFreeAux σ' binders (forall_ a✝¹ a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (exists_ a✝¹ a✝) = replaceFreeAux σ' binders (exists_ a✝¹ a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)

case pred_const_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = pred_const_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) case pred_var_ σ σ' : VarName → VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = pred_var_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝) case eq_ σ σ' : VarName → VarName a✝¹ a✝ : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ eq_ (if a✝¹ ∉ binders then σ a✝¹ else a✝¹) (if a✝ ∉ binders then σ a✝ else a✝) = eq_ (if a✝¹ ∉ binders then σ' a✝¹ else a✝¹) (if a✝ ∉ binders then σ' a✝ else a✝) case not_ σ σ' : VarName → VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝).not_ = (replaceFreeAux σ' binders a✝).not_ case imp_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).imp_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).imp_ (replaceFreeAux σ' binders a✝) case and_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).and_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).and_ (replaceFreeAux σ' binders a✝) case or_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).or_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).or_ (replaceFreeAux σ' binders a✝) case iff_ σ σ' : VarName → VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝¹ = replaceFreeAux σ' binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders a✝¹).iff_ (replaceFreeAux σ binders a✝) = (replaceFreeAux σ' binders a✝¹).iff_ (replaceFreeAux σ' binders a✝) case forall_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ forall_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = forall_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝) case exists_ σ σ' : VarName → VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders a✝ = replaceFreeAux σ' binders a✝ binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ a✝¹ (replaceFreeAux σ (binders ∪ {a✝¹}) a✝) = exists_ a✝¹ (replaceFreeAux σ' (binders ∪ {a✝¹}) a✝) case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

case pred_const_ X xs | pred_var_ X xs | def_ X xs => congr! 1 simp only [List.map_eq_map_iff] intro x _ split_ifs <;> tauto

σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) = def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

case not_ phi phi_ih => simp tauto

σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

case imp_ phi psi phi_ih psi_ih | and_ phi psi phi_ih psi_ih | or_ phi psi phi_ih psi_ih | iff_ phi psi phi_ih psi_ih => simp tauto

σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) = (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

case forall_ x phi phi_ih | exists_ x phi phi_ih => congr! 1 apply phi_ih simp tauto

σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

simp only [replaceFreeAux]

case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders (def_ a✝¹ a✝) = replaceFreeAux σ' binders (def_ a✝¹ a✝)

case def_ σ σ' : VarName → VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ a✝¹ (List.map (fun x => if x ∉ binders then σ x else x) a✝) = def_ a✝¹ (List.map (fun x => if x ∉ binders then σ' x else x) a✝)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

congr! 1

σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ def_ X (List.map (fun x => if x ∉ binders then σ x else x) xs) = def_ X (List.map (fun x => if x ∉ binders then σ' x else x) xs)

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

simp only [List.map_eq_map_iff]

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ List.map (fun x => if x ∉ binders then σ x else x) xs = List.map (fun x => if x ∉ binders then σ' x else x) xs

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

intro x _

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ x ∈ xs, (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v x : VarName a✝ : x ∈ xs ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

split_ifs <;> tauto

case h.e'_2 σ σ' : VarName → VarName X : DefName xs : List VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v x : VarName a✝ : x ∈ xs ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

congr! 1

σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ eq_ (if x ∉ binders then σ x else x) (if y ∉ binders then σ y else y) = eq_ (if x ∉ binders then σ' x else x) (if y ∉ binders then σ' y else y)

case h.e'_1 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x case h.e'_2 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

split_ifs <;> tauto

case h.e'_1 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if x ∉ binders then σ x else x) = if x ∉ binders then σ' x else x

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

split_ifs <;> tauto

case h.e'_2 σ σ' : VarName → VarName x y : VarName binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (if y ∉ binders then σ y else y) = if y ∉ binders then σ' y else y

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

simp

σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).not_ = (replaceFreeAux σ' binders phi).not_

σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

tauto

σ σ' : VarName → VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

simp

σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ (replaceFreeAux σ binders phi).iff_ (replaceFreeAux σ binders psi) = (replaceFreeAux σ' binders phi).iff_ (replaceFreeAux σ' binders psi)

σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧ replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

tauto

σ σ' : VarName → VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi psi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi ∧ replaceFreeAux σ binders psi = replaceFreeAux σ' binders psi

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

congr! 1

σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ exists_ x (replaceFreeAux σ (binders ∪ {x}) phi) = exists_ x (replaceFreeAux σ' (binders ∪ {x}) phi)

case h.e'_2 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

apply phi_ih

case h.e'_2 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ replaceFreeAux σ (binders ∪ {x}) phi = replaceFreeAux σ' (binders ∪ {x}) phi

case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

simp

case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders ∪ {x}, σ v = σ' v

case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/ReplaceFree.lean

FOL.NV.Sub.Var.All.Rec.replaceFreeAux_same_on_free

[226, 1]

[260, 10]

tauto

case h.e'_2.h1 σ σ' : VarName → VarName x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), (∀ v ∉ binders, σ v = σ' v) → replaceFreeAux σ binders phi = replaceFreeAux σ' binders phi binders : Finset VarName h1 : ∀ v ∉ binders, σ v = σ' v ⊢ ∀ v ∉ binders, ¬v = x → σ v = σ' v

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

induction F generalizing σ

σ : VarName → VarName c : Char F : Formula ⊢ (sub σ c F).freeVarSet = Finset.image σ F.freeVarSet

case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet case true_ c : Char σ : VarName → VarName ⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet case false_ c : Char σ : VarName → VarName ⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

all_goals simp only [sub] simp only [freeVarSet]

case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_const_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_const_ a✝¹ a✝).freeVarSet case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (pred_var_ a✝¹ a✝)).freeVarSet = Finset.image σ (pred_var_ a✝¹ a✝).freeVarSet case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ (sub σ c (eq_ a✝¹ a✝)).freeVarSet = Finset.image σ (eq_ a✝¹ a✝).freeVarSet case true_ c : Char σ : VarName → VarName ⊢ (sub σ c true_).freeVarSet = Finset.image σ true_.freeVarSet case false_ c : Char σ : VarName → VarName ⊢ (sub σ c false_).freeVarSet = Finset.image σ false_.freeVarSet case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝.not_).freeVarSet = Finset.image σ a✝.not_.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.imp_ a✝)).freeVarSet = Finset.image σ (a✝¹.imp_ a✝).freeVarSet case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.and_ a✝)).freeVarSet = Finset.image σ (a✝¹.and_ a✝).freeVarSet case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.or_ a✝)).freeVarSet = Finset.image σ (a✝¹.or_ a✝).freeVarSet case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (a✝¹.iff_ a✝)).freeVarSet = Finset.image σ (a✝¹.iff_ a✝).freeVarSet case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (forall_ a✝¹ a✝)).freeVarSet = Finset.image σ (forall_ a✝¹ a✝).freeVarSet case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c (exists_ a✝¹ a✝)).freeVarSet = Finset.image σ (exists_ a✝¹ a✝).freeVarSet case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet

case pred_const_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset case pred_var_ c : Char a✝¹ : PredName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset case eq_ c : Char a✝¹ a✝ : VarName σ : VarName → VarName ⊢ {σ a✝¹, σ a✝} = Finset.image σ {a✝¹, a✝} case true_ c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅ case false_ c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅ case not_ c : Char a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet case imp_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case and_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case or_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case iff_ c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (σ : VarName → VarName), (sub σ c a✝¹).freeVarSet = Finset.image σ a✝¹.freeVarSet a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub σ c a✝¹).freeVarSet ∪ (sub σ c a✝).freeVarSet = Finset.image σ (a✝¹.freeVarSet ∪ a✝.freeVarSet) case forall_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ a✝¹ (if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹)) c a✝).freeVarSet \ {if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹} = Finset.image σ (a✝.freeVarSet \ {a✝¹}) case exists_ c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (σ : VarName → VarName), (sub σ c a✝).freeVarSet = Finset.image σ a✝.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ a✝¹ (if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹)) c a✝).freeVarSet \ {if ∃ y ∈ a✝.freeVarSet \ {a✝¹}, σ y = a✝¹ then fresh a✝¹ c (sub (Function.updateITE σ a✝¹ a✝¹) c a✝).freeVarSet else a✝¹} = Finset.image σ (a✝.freeVarSet \ {a✝¹}) case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs => apply Finset.ext intro a simp

c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case true_ | false_ => simp

c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case not_ phi phi_ih => exact phi_ih σ

c : Char phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case forall_ x phi phi_ih | exists_ x phi phi_ih => simp only [phi_ih] simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ] split_ifs case _ c1 => obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at * have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi) apply Finset.image_subset_image simp have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) apply Finset.not_mem_mono s2 s1 calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3 case _ c1 => simp at c1 have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x} apply Finset.image_sdiff_singleton simp only [Function.updateITE] simp simp only [s1] clear s1 have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) simp only [Finset.mem_image] simp simp only [Function.updateITE] simp tauto simp only [Finset.sdiff_singleton_eq_self s2]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)) c phi).freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} = Finset.image σ (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [sub]

case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (sub σ c (def_ a✝¹ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet

case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [freeVarSet]

case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (def_ a✝¹ (List.map σ a✝)).freeVarSet = Finset.image σ (def_ a✝¹ a✝).freeVarSet

case def_ c : Char a✝¹ : DefName a✝ : List VarName σ : VarName → VarName ⊢ (List.map σ a✝).toFinset = Finset.image σ a✝.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

apply Finset.ext

c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ (List.map σ xs).toFinset = Finset.image σ xs.toFinset

case a c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

intro a

case a c : Char X : DefName xs : List VarName σ : VarName → VarName ⊢ ∀ (a : VarName), a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset

case a c : Char X : DefName xs : List VarName σ : VarName → VarName a : VarName ⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

case a c : Char X : DefName xs : List VarName σ : VarName → VarName a : VarName ⊢ a ∈ (List.map σ xs).toFinset ↔ a ∈ Finset.image σ xs.toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

c : Char σ : VarName → VarName ⊢ ∅ = Finset.image σ ∅

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

exact phi_ih σ

c : Char phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Finset.image_union]

c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ (phi.freeVarSet ∪ psi.freeVarSet)

c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

congr!

c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet ∪ (sub σ c psi).freeVarSet = Finset.image σ phi.freeVarSet ∪ Finset.image σ psi.freeVarSet

case h.e'_3 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet case h.e'_4 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

exact phi_ih σ

case h.e'_3 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

exact psi_ih σ

case h.e'_4 c : Char phi psi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet psi_ih : ∀ (σ : VarName → VarName), (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet σ : VarName → VarName ⊢ (sub σ c psi).freeVarSet = Finset.image σ psi.freeVarSet

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [phi_ih]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ (sub (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x)) c phi).freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (sub (Function.updateITE σ x x) c phi).freeVarSet else x} = Finset.image σ (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image σ (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [<- Finset.image_sdiff_singleton_updateITE phi.freeVarSet x x σ]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image σ (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

split_ifs

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName ⊢ Finset.image (Function.updateITE σ x (if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x)) phi.freeVarSet \ {if ∃ y ∈ phi.freeVarSet \ {x}, σ y = x then fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) else x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case pos c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName h✝ : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) case neg c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName h✝ : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case _ c1 => obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at * have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi) apply Finset.image_subset_image simp have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) apply Finset.not_mem_mono s2 s1 calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

case _ c1 => simp at c1 have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x} apply Finset.image_sdiff_singleton simp only [Function.updateITE] simp simp only [s1] clear s1 have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) simp only [Finset.mem_image] simp simp only [Function.updateITE] simp tauto simp only [Finset.sdiff_singleton_eq_self s2]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

obtain s1 := fresh_not_mem x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi))

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x s1 : fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

generalize ( fresh x c (Finset.image (Function.updateITE σ x x) (freeVarSet phi)) ) = x' at *

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x s1 : fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet) ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x (fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet))) phi.freeVarSet \ {fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet)} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

have s2 : Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) ⊆ Finset.image (Function.updateITE σ x x) (freeVarSet phi)

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

apply Finset.image_subset_image

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2.h c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

case s2.h c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ phi.freeVarSet \ {x} ⊆ phi.freeVarSet c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

have s3 : x' ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s3 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

apply Finset.not_mem_mono s2 s1

case s3 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet ⊢ x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

calc Finset.image (Function.updateITE σ x x') (freeVarSet phi) \ {x'} = Finset.image (Function.updateITE σ x x') (freeVarSet phi \ {x}) \ {x'} := by { apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x') simp only [Function.updateITE] simp } _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x'} := by simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x] _ = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) := by exact Finset.sdiff_singleton_eq_self s3

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

apply Finset.image_sdiff_singleton phi.freeVarSet x x' (Function.updateITE σ x x')

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') phi.freeVarSet \ {x'} = Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'}

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Function.updateITE σ x x' x = x'

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Function.updateITE]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Function.updateITE σ x x' x = x'

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ (if True then x' else σ x) = x'

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ (if True then x' else σ x) = x'

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Finset.image_congr_update_ite phi.freeVarSet x x' x]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x') (phi.freeVarSet \ {x}) \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

exact Finset.sdiff_singleton_eq_self s3

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∃ y ∈ phi.freeVarSet \ {x}, σ y = x x' : VarName s1 : x' ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet s2 : Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊆ Finset.image (Function.updateITE σ x x) phi.freeVarSet s3 : x' ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x'} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp at c1

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ¬∃ y ∈ phi.freeVarSet \ {x}, σ y = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

have s1 : Finset.image (Function.updateITE σ x x) (freeVarSet phi) \ {x} = Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x}) \ {x}

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

apply Finset.image_sdiff_singleton

case s1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Function.updateITE σ x x x = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Function.updateITE]

case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Function.updateITE σ x x x = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ (if True then x else σ x) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

case s1.h1 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ (if True then x else σ x) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [s1]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

clear s1

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s1 : Finset.image (Function.updateITE σ x x) phi.freeVarSet \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

have s2 : x ∉ Finset.image (Function.updateITE σ x x) (freeVarSet phi \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Finset.mem_image]

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ¬∃ a ∈ phi.freeVarSet \ {x}, Function.updateITE σ x x a = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Function.updateITE]

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬Function.updateITE σ x x x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬(if x_1 = x then x else σ x_1) = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

tauto

case s2 c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x ⊢ ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬x_1 = x ∧ ¬σ x_1 = x c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.freeVarSet_sub_eq_freeVarSet_image

[48, 1]

[125, 52]

simp only [Finset.sdiff_singleton_eq_self s2]

c : Char x : VarName phi : Formula phi_ih : ∀ (σ : VarName → VarName), (sub σ c phi).freeVarSet = Finset.image σ phi.freeVarSet σ : VarName → VarName c1 : ∀ x_1 ∈ phi.freeVarSet, ¬x_1 = x → ¬σ x_1 = x s2 : x ∉ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) ⊢ Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x}) \ {x} = Finset.image (Function.updateITE σ x x) (phi.freeVarSet \ {x})

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

induction F generalizing σ V

D : Type I : Interpretation D V : VarAssignment D E : Env σ : VarName → VarName c : Char F : Formula ⊢ Holds D I V E (sub σ c F) ↔ Holds D I (V ∘ σ) E F

case pred_const_ D : Type I : Interpretation D E : Env c : Char a✝¹ : PredName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (pred_const_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_const_ a✝¹ a✝) case pred_var_ D : Type I : Interpretation D E : Env c : Char a✝¹ : PredName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (pred_var_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (pred_var_ a✝¹ a✝) case eq_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (eq_ a✝¹ a✝) case true_ D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c true_) ↔ Holds D I (V ∘ σ) E true_ case false_ D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_ case not_ D : Type I : Interpretation D E : Env c : Char a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c a✝.not_) ↔ Holds D I (V ∘ σ) E a✝.not_ case imp_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.imp_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.imp_ a✝) case and_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.and_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.and_ a✝) case or_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.or_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.or_ a✝) case iff_ D : Type I : Interpretation D E : Env c : Char a✝¹ a✝ : Formula a_ih✝¹ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝¹) ↔ Holds D I (V ∘ σ) E a✝¹ a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (a✝¹.iff_ a✝)) ↔ Holds D I (V ∘ σ) E (a✝¹.iff_ a✝) case forall_ D : Type I : Interpretation D E : Env c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (forall_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (forall_ a✝¹ a✝) case exists_ D : Type I : Interpretation D E : Env c : Char a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c a✝) ↔ Holds D I (V ∘ σ) E a✝ V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (exists_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (exists_ a✝¹ a✝) case def_ D : Type I : Interpretation D E : Env c : Char a✝¹ : DefName a✝ : List VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (def_ a✝¹ a✝)) ↔ Holds D I (V ∘ σ) E (def_ a✝¹ a✝)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

case pred_const_ X xs | pred_var_ X xs | eq_ x y => simp only [sub] simp only [Holds] simp

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

case true_ | false_ => simp only [sub] simp only [Holds]

D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

case not_ phi phi_ih => simp only [sub] simp only [Holds] congr! 1 exact phi_ih V σ

D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [sub]

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c (eq_ x y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [Holds]

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (eq_ (σ x) (σ y)) ↔ Holds D I (V ∘ σ) E (eq_ x y)

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp

D : Type I : Interpretation D E : Env c : Char x y : VarName V : VarAssignment D σ : VarName → VarName ⊢ V (σ x) = V (σ y) ↔ (V ∘ σ) x = (V ∘ σ) y

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [sub]

D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c false_) ↔ Holds D I (V ∘ σ) E false_

D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [Holds]

D : Type I : Interpretation D E : Env c : Char V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E false_ ↔ Holds D I (V ∘ σ) E false_

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [sub]

D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi.not_) ↔ Holds D I (V ∘ σ) E phi.not_

D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/NV/Sub/Var/All/Rec/Fresh/Sub.lean

FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem

[128, 1]

[245, 19]

simp only [Holds]

D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ Holds D I V E (sub σ c phi).not_ ↔ Holds D I (V ∘ σ) E phi.not_

D : Type I : Interpretation D E : Env c : Char phi : Formula phi_ih : ∀ (V : VarAssignment D) (σ : VarName → VarName), Holds D I V E (sub σ c phi) ↔ Holds D I (V ∘ σ) E phi V : VarAssignment D σ : VarName → VarName ⊢ ¬Holds D I V E (sub σ c phi) ↔ ¬Holds D I (V ∘ σ) E phi