pkuAI4M/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	input stringlengths 73 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [occursIn]	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [varSet]	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	congr!	v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [occursIn]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	simp only [varSet]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.occursIn_iff_mem_varSet	[186, 1]	[208, 17]	exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	induction F	v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet	case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (pred_ a✝¹ a✝).freeVarSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v a✝.not_ ∧ v.isFree ↔ v ∈ a✝.not_.freeVarSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ∧ v.isFree ↔ v ∈ a✝¹.freeVarSet a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v (a✝¹.imp_ a✝) ∧ v.isFree ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v (forall_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case not_ phi phi_ih => simp only [Formula.freeVarSet] simp only [occursIn] exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case imp_ phi psi phi_ih psi_ih => simp only [Formula.freeVarSet] simp only [occursIn] simp tauto	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case forall_ _ phi phi_ih => simp only [Formula.freeVarSet] simp only [occursIn] exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Formula.freeVarSet]	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ (pred_ X vs).freeVarSet	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ (pred_ X vs).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [occursIn]	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	constructor	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet	case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	intro a1	case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	apply Exists.intro v	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	cases v	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet	case mp.free_ X : String vs : List Var a✝ : String a1 : free_ a✝ ∈ vs ∧ (free_ a✝).isFree ⊢ free_ a✝ ∈ vs ∧ free_ a✝ ∈ (free_ a✝).freeVarSet case mp.bound_ X : String vs : List Var a✝ : ℕ a1 : bound_ a✝ ∈ vs ∧ (bound_ a✝).isFree ⊢ bound_ a✝ ∈ vs ∧ bound_ a✝ ∈ (bound_ a✝).freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ x => simp only [Var.freeVarSet] simp cases a1 case _ a1_left a1_right => exact a1_left	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ i => simp only [isFree] at a1 cases a1 case _ a1_left a1_right => contradiction	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Var.freeVarSet]	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x}	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x}	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	cases a1	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs	case intro X : String vs : List Var x : String left✝ : free_ x ∈ vs right✝ : (free_ x).isFree ⊢ free_ x ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ a1_left a1_right => exact a1_left	X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	exact a1_left	X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [isFree] at a1	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	cases a1	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	case intro X : String vs : List Var i : ℕ left✝ : bound_ i ∈ vs right✝ : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ a1_left a1_right => contradiction	X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	contradiction	X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	intro a1	case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	apply Exists.elim a1	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	intro u a2	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	cases u	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree	case mpr.free_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet a✝ : String a2 : free_ a✝ ∈ vs ∧ v ∈ (free_ a✝).freeVarSet ⊢ v ∈ vs ∧ v.isFree case mpr.bound_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet a✝ : ℕ a2 : bound_ a✝ ∈ vs ∧ v ∈ (bound_ a✝).freeVarSet ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ x => simp only [Var.freeVarSet] at a2 simp at a2 cases a2 case _ a2_left a2_right => subst a2_right simp only [isFree] simp exact a2_left	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ i => simp only [Var.freeVarSet] at a2 simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Var.freeVarSet] at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	cases a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree	case intro v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String left✝ : free_ x ∈ vs right✝ : v = free_ x ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	case _ a2_left a2_right => subst a2_right simp only [isFree] simp exact a2_left	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	subst a2_right	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [isFree]	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	exact a2_left	X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Var.freeVarSet] at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Formula.freeVarSet]	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [occursIn]	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Formula.freeVarSet]	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [occursIn]	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	tauto	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [Formula.freeVarSet]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	simp only [occursIn]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet	[211, 1]	[264, 17]	exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	induction F	v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ F.boundVarSet	case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ∧ v.isBound ↔ v ∈ (pred_ a✝¹ a✝).boundVarSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v a✝.not_ ∧ v.isBound ↔ v ∈ a✝.not_.boundVarSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ∧ v.isBound ↔ v ∈ a✝¹.boundVarSet a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v (a✝¹.imp_ a✝) ∧ v.isBound ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v (forall_ a✝¹ a✝) ∧ v.isBound ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ F.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case not_ phi phi_ih => simp only [Formula.boundVarSet] simp only [occursIn] exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case imp_ phi psi phi_ih psi_ih => simp only [Formula.boundVarSet] simp only [occursIn] simp tauto	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case forall_ _ phi phi_ih => simp only [Formula.boundVarSet] simp only [occursIn] exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ (forall_ a✝ phi).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ (forall_ a✝ phi).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Formula.boundVarSet]	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ (pred_ X vs).boundVarSet	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ (pred_ X vs).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [occursIn]	v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	constructor	v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet	case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	intro a1	case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	apply Exists.intro v	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	cases v	case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet	case mp.free_ X : String vs : List Var a✝ : String a1 : free_ a✝ ∈ vs ∧ (free_ a✝).isBound ⊢ free_ a✝ ∈ vs ∧ free_ a✝ ∈ (free_ a✝).boundVarSet case mp.bound_ X : String vs : List Var a✝ : ℕ a1 : bound_ a✝ ∈ vs ∧ (bound_ a✝).isBound ⊢ bound_ a✝ ∈ vs ∧ bound_ a✝ ∈ (bound_ a✝).boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ x => simp only [Var.isBound] at a1 cases a1 case _ a1_left a1_right => contradiction	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ i => simp only [Var.boundVarSet] simp cases a1 case _ a1_left a1_right => exact a1_left	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Var.isBound] at a1	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	cases a1	X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	case intro X : String vs : List Var x : String left✝ : free_ x ∈ vs right✝ : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ a1_left a1_right => contradiction	X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	contradiction	X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Var.boundVarSet]	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i}	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i}	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i} TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	cases a1	X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs	case intro X : String vs : List Var i : ℕ left✝ : bound_ i ∈ vs right✝ : (bound_ i).isBound ⊢ bound_ i ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ a1_left a1_right => exact a1_left	X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	exact a1_left	X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	intro a1	case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	apply Exists.elim a1	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	intro u a2	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	cases u	case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound	case mpr.free_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet a✝ : String a2 : free_ a✝ ∈ vs ∧ v ∈ (free_ a✝).boundVarSet ⊢ v ∈ vs ∧ v.isBound case mpr.bound_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet a✝ : ℕ a2 : bound_ a✝ ∈ vs ∧ v ∈ (bound_ a✝).boundVarSet ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ x => simp only [Var.boundVarSet] at a2 simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ i => simp only [Var.boundVarSet] at a2 simp at a2 cases a2 case _ a2_left a2_right => subst a2_right simp only [isBound] simp exact a2_left	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Var.boundVarSet] at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Var.boundVarSet] at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp at a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	cases a2	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound	case intro v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ left✝ : bound_ i ∈ vs right✝ : v = bound_ i ⊢ v ∈ vs ∧ v.isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	case _ a2_left a2_right => subst a2_right simp only [isBound] simp exact a2_left	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	subst a2_right	v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound	Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [isBound]	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	exact a2_left	X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs	no goals	Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Formula.boundVarSet]	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [occursIn]	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	exact phi_ih	v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet	no goals	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Formula.boundVarSet]	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [occursIn]	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet TACTIC:
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet	Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [occursIn]

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v (phi.imp_ psi) ↔ v ∈ (phi.imp_ psi).varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [varSet]

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ (phi.imp_ psi).varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∪ psi.varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

congr!

v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet psi_ih : occursIn v psi ↔ v ∈ psi.varSet ⊢ occursIn v phi ∨ occursIn v psi ↔ v ∈ phi.varSet ∨ v ∈ psi.varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [occursIn]

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v (forall_ a✝ phi) ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

simp only [varSet]

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ (forall_ a✝ phi).varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.occursIn_iff_mem_varSet

[186, 1]

[208, 17]

exact phi_ih

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ↔ v ∈ phi.varSet ⊢ occursIn v phi ↔ v ∈ phi.varSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

induction F

v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet

case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (pred_ a✝¹ a✝).freeVarSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v a✝.not_ ∧ v.isFree ↔ v ∈ a✝.not_.freeVarSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ∧ v.isFree ↔ v ∈ a✝¹.freeVarSet a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v (a✝¹.imp_ a✝) ∧ v.isFree ↔ v ∈ (a✝¹.imp_ a✝).freeVarSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isFree ↔ v ∈ a✝.freeVarSet ⊢ occursIn v (forall_ a✝¹ a✝) ∧ v.isFree ↔ v ∈ (forall_ a✝¹ a✝).freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case not_ phi phi_ih => simp only [Formula.freeVarSet] simp only [occursIn] exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case imp_ phi psi phi_ih psi_ih => simp only [Formula.freeVarSet] simp only [occursIn] simp tauto

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case forall_ _ phi phi_ih => simp only [Formula.freeVarSet] simp only [occursIn] exact phi_ih

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Formula.freeVarSet]

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ (pred_ X vs).freeVarSet

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ (pred_ X vs).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [occursIn]

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ v ∈ vs.toFinset.biUnion Var.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

constructor

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet

case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree ↔ ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

intro a1

case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isFree → ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

apply Exists.intro v

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ ∃ a ∈ vs, v ∈ a.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

cases v

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet

case mp.free_ X : String vs : List Var a✝ : String a1 : free_ a✝ ∈ vs ∧ (free_ a✝).isFree ⊢ free_ a✝ ∈ vs ∧ free_ a✝ ∈ (free_ a✝).freeVarSet case mp.bound_ X : String vs : List Var a✝ : ℕ a1 : bound_ a✝ ∈ vs ∧ (bound_ a✝).isFree ⊢ bound_ a✝ ∈ vs ∧ bound_ a✝ ∈ (bound_ a✝).freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isFree ⊢ v ∈ vs ∧ v ∈ v.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ x => simp only [Var.freeVarSet] simp cases a1 case _ a1_left a1_right => exact a1_left

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ i => simp only [isFree] at a1 cases a1 case _ a1_left a1_right => contradiction

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Var.freeVarSet]

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x}

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x}

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs ∧ free_ x ∈ {free_ x} TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

cases a1

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs

case intro X : String vs : List Var x : String left✝ : free_ x ∈ vs right✝ : (free_ x).isFree ⊢ free_ x ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ a1_left a1_right => exact a1_left

X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

exact a1_left

X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : (free_ x).isFree ⊢ free_ x ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [isFree] at a1

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isFree ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

cases a1

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

case intro X : String vs : List Var i : ℕ left✝ : bound_ i ∈ vs right✝ : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ a1_left a1_right => contradiction

X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

contradiction

X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : False ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

intro a1

case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.freeVarSet) → v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

apply Exists.elim a1

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

intro u a2

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.freeVarSet → v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

cases u

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree

case mpr.free_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet a✝ : String a2 : free_ a✝ ∈ vs ∧ v ∈ (free_ a✝).freeVarSet ⊢ v ∈ vs ∧ v.isFree case mpr.bound_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet a✝ : ℕ a2 : bound_ a✝ ∈ vs ∧ v ∈ (bound_ a✝).freeVarSet ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ x => simp only [Var.freeVarSet] at a2 simp at a2 cases a2 case _ a2_left a2_right => subst a2_right simp only [isFree] simp exact a2_left

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ i => simp only [Var.freeVarSet] at a2 simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Var.freeVarSet] at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ {free_ x} ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

cases a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree

case intro v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String left✝ : free_ x ∈ vs right✝ : v = free_ x ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2 : free_ x ∈ vs ∧ v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

case _ a2_left a2_right => subst a2_right simp only [isFree] simp exact a2_left

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

subst a2_right

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet x : String a2_left : free_ x ∈ vs a2_right : v = free_ x ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [isFree]

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ (free_ x).isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs ∧ True TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

exact a2_left

X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a2_left : free_ x ∈ vs a1 : ∃ a ∈ vs, free_ x ∈ a.freeVarSet ⊢ free_ x ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Var.freeVarSet] at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).freeVarSet ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.freeVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isFree TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Formula.freeVarSet]

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.not_.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [occursIn]

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi.not_ ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Formula.freeVarSet]

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ (phi.imp_ psi).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [occursIn]

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∪ psi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

tauto

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet psi_ih : occursIn v psi ∧ v.isFree ↔ v ∈ psi.freeVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isFree ↔ v ∈ phi.freeVarSet ∨ v ∈ psi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [Formula.freeVarSet]

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ (forall_ a✝ phi).freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

simp only [occursIn]

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isFreeIn_iff_mem_freeVarSet

[211, 1]

[264, 17]

exact phi_ih

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet ⊢ occursIn v phi ∧ v.isFree ↔ v ∈ phi.freeVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

induction F

v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ F.boundVarSet

case pred_ v : Var a✝¹ : String a✝ : List Var ⊢ occursIn v (pred_ a✝¹ a✝) ∧ v.isBound ↔ v ∈ (pred_ a✝¹ a✝).boundVarSet case not_ v : Var a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v a✝.not_ ∧ v.isBound ↔ v ∈ a✝.not_.boundVarSet case imp_ v : Var a✝¹ a✝ : Formula a_ih✝¹ : occursIn v a✝¹ ∧ v.isBound ↔ v ∈ a✝¹.boundVarSet a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v (a✝¹.imp_ a✝) ∧ v.isBound ↔ v ∈ (a✝¹.imp_ a✝).boundVarSet case forall_ v : Var a✝¹ : String a✝ : Formula a_ih✝ : occursIn v a✝ ∧ v.isBound ↔ v ∈ a✝.boundVarSet ⊢ occursIn v (forall_ a✝¹ a✝) ∧ v.isBound ↔ v ∈ (forall_ a✝¹ a✝).boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ F.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case not_ phi phi_ih => simp only [Formula.boundVarSet] simp only [occursIn] exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case imp_ phi psi phi_ih psi_ih => simp only [Formula.boundVarSet] simp only [occursIn] simp tauto

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case forall_ _ phi phi_ih => simp only [Formula.boundVarSet] simp only [occursIn] exact phi_ih

v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ (forall_ a✝ phi).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ (forall_ a✝ phi).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Formula.boundVarSet]

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ (pred_ X vs).boundVarSet

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ (pred_ X vs).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [occursIn]

v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ occursIn v (pred_ X vs) ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ v ∈ vs.toFinset.biUnion Var.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

constructor

v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet

case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound ↔ ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

intro a1

case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var ⊢ v ∈ vs ∧ v.isBound → ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

apply Exists.intro v

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ ∃ a ∈ vs, v ∈ a.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

cases v

case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet

case mp.free_ X : String vs : List Var a✝ : String a1 : free_ a✝ ∈ vs ∧ (free_ a✝).isBound ⊢ free_ a✝ ∈ vs ∧ free_ a✝ ∈ (free_ a✝).boundVarSet case mp.bound_ X : String vs : List Var a✝ : ℕ a1 : bound_ a✝ ∈ vs ∧ (bound_ a✝).isBound ⊢ bound_ a✝ ∈ vs ∧ bound_ a✝ ∈ (bound_ a✝).boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: case mp v : Var X : String vs : List Var a1 : v ∈ vs ∧ v.isBound ⊢ v ∈ vs ∧ v ∈ v.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ x => simp only [Var.isBound] at a1 cases a1 case _ a1_left a1_right => contradiction

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ i => simp only [Var.boundVarSet] simp cases a1 case _ a1_left a1_right => exact a1_left

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Var.isBound] at a1

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ (free_ x).isBound ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

cases a1

X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

case intro X : String vs : List Var x : String left✝ : free_ x ∈ vs right✝ : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1 : free_ x ∈ vs ∧ False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ a1_left a1_right => contradiction

X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

contradiction

X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var x : String a1_left : free_ x ∈ vs a1_right : False ⊢ free_ x ∈ vs ∧ free_ x ∈ (free_ x).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Var.boundVarSet]

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i}

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ (bound_ i).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i}

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs ∧ bound_ i ∈ {bound_ i} TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

cases a1

X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs

case intro X : String vs : List Var i : ℕ left✝ : bound_ i ∈ vs right✝ : (bound_ i).isBound ⊢ bound_ i ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1 : bound_ i ∈ vs ∧ (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ a1_left a1_right => exact a1_left

X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

exact a1_left

X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a1_left : bound_ i ∈ vs a1_right : (bound_ i).isBound ⊢ bound_ i ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

intro a1

case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var ⊢ (∃ a ∈ vs, v ∈ a.boundVarSet) → v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

apply Exists.elim a1

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

intro u a2

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet ⊢ ∀ (a : Var), a ∈ vs ∧ v ∈ a.boundVarSet → v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

cases u

case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound

case mpr.free_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet a✝ : String a2 : free_ a✝ ∈ vs ∧ v ∈ (free_ a✝).boundVarSet ⊢ v ∈ vs ∧ v.isBound case mpr.bound_ v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet a✝ : ℕ a2 : bound_ a✝ ∈ vs ∧ v ∈ (bound_ a✝).boundVarSet ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: case mpr v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet u : Var a2 : u ∈ vs ∧ v ∈ u.boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ x => simp only [Var.boundVarSet] at a2 simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ i => simp only [Var.boundVarSet] at a2 simp at a2 cases a2 case _ a2_left a2_right => subst a2_right simp only [isBound] simp exact a2_left

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Var.boundVarSet] at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ (free_ x).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet x : String a2 : free_ x ∈ vs ∧ v ∈ ∅ ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Var.boundVarSet] at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ (bound_ i).boundVarSet ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp at a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v ∈ {bound_ i} ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

cases a2

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound

case intro v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ left✝ : bound_ i ∈ vs right✝ : v = bound_ i ⊢ v ∈ vs ∧ v.isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2 : bound_ i ∈ vs ∧ v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

case _ a2_left a2_right => subst a2_right simp only [isBound] simp exact a2_left

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

subst a2_right

v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound

Please generate a tactic in lean4 to solve the state. STATE: v : Var X : String vs : List Var a1 : ∃ a ∈ vs, v ∈ a.boundVarSet i : ℕ a2_left : bound_ i ∈ vs a2_right : v = bound_ i ⊢ v ∈ vs ∧ v.isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [isBound]

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ (bound_ i).isBound TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs ∧ True TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

exact a2_left

X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs

no goals

Please generate a tactic in lean4 to solve the state. STATE: X : String vs : List Var i : ℕ a2_left : bound_ i ∈ vs a1 : ∃ a ∈ vs, bound_ i ∈ a.boundVarSet ⊢ bound_ i ∈ vs TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Formula.boundVarSet]

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.not_.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [occursIn]

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi.not_ ∧ v.isBound ↔ v ∈ phi.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

exact phi_ih

v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet

no goals

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [Formula.boundVarSet]

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ (phi.imp_ psi).boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp only [occursIn]

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ occursIn v (phi.imp_ psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet TACTIC:

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

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Binders.lean

LN.isBoundIn_iff_mem_boundVarSet

[267, 1]

[320, 17]

simp

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet

v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet

Please generate a tactic in lean4 to solve the state. STATE: v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∪ psi.boundVarSet TACTIC: