url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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} |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
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