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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux | [853, 1] | [905, 10] | tauto | case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_admits | [908, 1] | [916, 44] | simp only [replaceFree] | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ admits t v (replaceFree v t F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ admits t v (replaceFreeAux v t ∅ F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_admits | [908, 1] | [916, 44] | simp only [admits] | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ admits t v (replaceFreeAux v t ∅ F) | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ admitsAux t v ∅ (replaceFreeAux v t ∅ F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.replaceFree_admits | [908, 1] | [916, 44] | exact replaceFreeAux_admitsAux F v t ∅ h1 | F : Formula
v t : VarName
h1 : ¬occursIn t F
⊢ admitsAux t v ∅ (replaceFreeAux v t ∅ F) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | induction F generalizing S | F : Formula
v u : VarName
S T : Finset VarName
h1 : admitsAux v u S F
h2 : u ∉ T
⊢ admitsAux v u (S ∪ T) F | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_const_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_var_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u S (eq_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S true_
⊢ admitsAux v u (S ∪ T) true_
case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S false_
⊢ admitsAux v u (S ∪ T) false_
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝.not_
⊢ admitsAux v u (S ∪ T) a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.imp_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.and_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.or_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.iff_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (forall_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (exists_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (def_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | all_goals
simp only [admitsAux] at h1
simp only [admitsAux] | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_const_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (pred_var_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u S (eq_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S true_
⊢ admitsAux v u (S ∪ T) true_
case false_
v u : VarName
T : Finset VarName
h2 : u ∉ T
S : Finset VarName
h1 : admitsAux v u S false_
⊢ admitsAux v u (S ∪ T) false_
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝.not_
⊢ admitsAux v u (S ∪ T) a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.imp_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.and_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.or_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (a✝¹.iff_ a✝)
⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (forall_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S (exists_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (def_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝) | case pred_const_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
case pred_var_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
case eq_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ S → u ∉ S
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ S ∪ T → u ∉ S ∪ T
case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case forall_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝
case exists_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝
case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | case pred_const_ X xs | pred_var_ X xs | eq_ x y |def_ X xs =>
simp
tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Finset.union_right_comm S T {x}]
tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ T ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | all_goals
tauto | case not_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝
case imp_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case and_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case or_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | simp only [admitsAux] at h1 | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u S (def_ a✝¹ a✝)
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | simp only [admitsAux] | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S
⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | simp | v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T | v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S → u ∉ S
⊢ v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | simp only [Finset.union_right_comm S T {x}] | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ T ∪ {x}) phi | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x} ∪ T) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | tauto | v u : VarName
T : Finset VarName
h2 : u ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x} ∪ T) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders | [920, 1] | [940, 10] | tauto | case iff_
v u : VarName
T : Finset VarName
h2 : u ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝
S : Finset VarName
h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | induction F generalizing S | F : Formula
v u : VarName
S T : Finset VarName
h1 : admitsAux v u (S ∪ T) F
h2 : v ∉ T
⊢ admitsAux v u S F | case pred_const_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
⊢ admitsAux v u S (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
⊢ admitsAux v u S (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
⊢ admitsAux v u S (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : v ∉ T
S : Finset VarName
h1 : admitsAux v u (S ∪ T) true_
⊢ admitsAux v u S true_
case false_
v u : VarName
T : Finset VarName
h2 : v ∉ T
S : Finset VarName
h1 : admitsAux v u (S ∪ T) false_
⊢ admitsAux v u S false_
case not_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝.not_
⊢ admitsAux v u S a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
⊢ admitsAux v u S (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.and_ a✝)
⊢ admitsAux v u S (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.or_ a✝)
⊢ admitsAux v u S (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
⊢ admitsAux v u S (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
⊢ admitsAux v u S (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
⊢ admitsAux v u S (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
⊢ admitsAux v u S (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | all_goals
simp only [admitsAux] at h1
simp only [admitsAux] | case pred_const_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝)
⊢ admitsAux v u S (pred_const_ a✝¹ a✝)
case pred_var_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝)
⊢ admitsAux v u S (pred_var_ a✝¹ a✝)
case eq_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (eq_ a✝¹ a✝)
⊢ admitsAux v u S (eq_ a✝¹ a✝)
case true_
v u : VarName
T : Finset VarName
h2 : v ∉ T
S : Finset VarName
h1 : admitsAux v u (S ∪ T) true_
⊢ admitsAux v u S true_
case false_
v u : VarName
T : Finset VarName
h2 : v ∉ T
S : Finset VarName
h1 : admitsAux v u (S ∪ T) false_
⊢ admitsAux v u S false_
case not_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝.not_
⊢ admitsAux v u S a✝.not_
case imp_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.imp_ a✝)
⊢ admitsAux v u S (a✝¹.imp_ a✝)
case and_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.and_ a✝)
⊢ admitsAux v u S (a✝¹.and_ a✝)
case or_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.or_ a✝)
⊢ admitsAux v u S (a✝¹.or_ a✝)
case iff_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (a✝¹.iff_ a✝)
⊢ admitsAux v u S (a✝¹.iff_ a✝)
case forall_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (forall_ a✝¹ a✝)
⊢ admitsAux v u S (forall_ a✝¹ a✝)
case exists_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (exists_ a✝¹ a✝)
⊢ admitsAux v u S (exists_ a✝¹ a✝)
case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
⊢ admitsAux v u S (def_ a✝¹ a✝) | case pred_const_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ a✝ ∧ v ∉ S → u ∉ S
case pred_var_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : PredName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ a✝ ∧ v ∉ S → u ∉ S
case eq_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : VarName
S : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ S → u ∉ S
case not_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝
⊢ admitsAux v u S a✝
case imp_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
⊢ admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
case and_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
⊢ admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
case or_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
⊢ admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
case iff_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝¹ → admitsAux v u S a✝¹
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝
⊢ admitsAux v u S a✝¹ ∧ admitsAux v u S a✝
case forall_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ {a✝¹}) a✝
case exists_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) a✝ → admitsAux v u S a✝
S : Finset VarName
h1 : admitsAux v u (S ∪ T ∪ {a✝¹}) a✝
⊢ admitsAux v u (S ∪ {a✝¹}) a✝
case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ a✝ ∧ v ∉ S → u ∉ S |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | case pred_const_ X xs | pred_var_ X xs | eq_ x y | def_ X xs =>
simp at h1
tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ xs ∧ v ∉ S → u ∉ S | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | case not_ phi phi_ih =>
exact phi_ih S h1 | v u : VarName
T : Finset VarName
h2 : v ∉ T
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ T) phi
⊢ admitsAux v u S phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
phi psi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
psi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) psi → admitsAux v u S psi
S : Finset VarName
h1 : admitsAux v u (S ∪ T) phi ∧ admitsAux v u (S ∪ T) psi
⊢ admitsAux v u S phi ∧ admitsAux v u S psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
simp only [Finset.union_right_comm S T {x}] at h1
tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ T ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | simp only [admitsAux] at h1 | case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
⊢ admitsAux v u S (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ admitsAux v u S (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | simp only [admitsAux] | case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ admitsAux v u S (def_ a✝¹ a✝) | case def_
v u : VarName
T : Finset VarName
h2 : v ∉ T
a✝¹ : DefName
a✝ : List VarName
S : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ a✝ ∧ v ∉ S → u ∉ S |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | simp at h1 | v u : VarName
T : Finset VarName
h2 : v ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T
⊢ v ∈ xs ∧ v ∉ S → u ∉ S | v u : VarName
T : Finset VarName
h2 : v ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T
⊢ v ∈ xs ∧ v ∉ S → u ∉ S |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
X : DefName
xs : List VarName
S : Finset VarName
h1 : v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T
⊢ v ∈ xs ∧ v ∉ S → u ∉ S | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | exact phi_ih S h1 | v u : VarName
T : Finset VarName
h2 : v ∉ T
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ T) phi
⊢ admitsAux v u S phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
phi psi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
psi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) psi → admitsAux v u S psi
S : Finset VarName
h1 : admitsAux v u (S ∪ T) phi ∧ admitsAux v u (S ∪ T) psi
⊢ admitsAux v u S phi ∧ admitsAux v u S psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | simp only [Finset.union_right_comm S T {x}] at h1 | v u : VarName
T : Finset VarName
h2 : v ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ T ∪ {x}) phi
⊢ admitsAux v u (S ∪ {x}) phi | v u : VarName
T : Finset VarName
h2 : v ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x} ∪ T) phi
⊢ admitsAux v u (S ∪ {x}) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_del_binders | [943, 1] | [971, 10] | tauto | v u : VarName
T : Finset VarName
h2 : v ∉ T
x : VarName
phi : Formula
phi_ih : ∀ (S : Finset VarName), admitsAux v u (S ∪ T) phi → admitsAux v u S phi
S : Finset VarName
h1 : admitsAux v u (S ∪ {x} ∪ T) phi
⊢ admitsAux v u (S ∪ {x}) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | induction F generalizing binders | F : Formula
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders F
h2 : isFreeIn v F
h3 : v ∉ binders
⊢ u ∉ binders | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : admitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case true_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders true_
h2 : isFreeIn v true_
h3 : v ∉ binders
⊢ u ∉ binders
case false_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders false_
h2 : isFreeIn v false_
h3 : v ∉ binders
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
h3 : v ∉ binders
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | all_goals
simp only [admitsAux] at h1
simp only [isFreeIn] at h2 | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_const_ a✝¹ a✝)
h2 : isFreeIn v (pred_const_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (pred_var_ a✝¹ a✝)
h2 : isFreeIn v (pred_var_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : admitsAux v u binders (eq_ a✝¹ a✝)
h2 : isFreeIn v (eq_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case true_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders true_
h2 : isFreeIn v true_
h3 : v ∉ binders
⊢ u ∉ binders
case false_
v u : VarName
binders : Finset VarName
h1 : admitsAux v u binders false_
h2 : isFreeIn v false_
h3 : v ∉ binders
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝.not_
h2 : isFreeIn v a✝.not_
h3 : v ∉ binders
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.imp_ a✝)
h2 : isFreeIn v (a✝¹.imp_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.and_ a✝)
h2 : isFreeIn v (a✝¹.and_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.or_ a✝)
h2 : isFreeIn v (a✝¹.or_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (a✝¹.iff_ a✝)
h2 : isFreeIn v (a✝¹.iff_ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (forall_ a✝¹ a✝)
h2 : isFreeIn v (forall_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders (exists_ a✝¹ a✝)
h2 : isFreeIn v (exists_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
h3 : v ∉ binders
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝
h2 : isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {a✝¹}) a✝
h2 : ¬v = a✝¹ ∧ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | all_goals
tauto | case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders
case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders
case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders
h2 : v = a✝¹ ∨ v = a✝
h3 : v ∉ binders
⊢ u ∉ binders
case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝
h2 : isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v u binders a✝¹ → isFreeIn v a✝¹ → v ∉ binders → u ∉ binders
a_ih✝ : ∀ (binders : Finset VarName), admitsAux v u binders a✝ → isFreeIn v a✝ → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝
h2 : isFreeIn v a✝¹ ∨ isFreeIn v a✝
h3 : v ∉ binders
⊢ u ∉ binders
case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | simp only [admitsAux] at h1 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : admitsAux v u binders (def_ a✝¹ a✝)
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | simp only [isFreeIn] at h2 | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : isFreeIn v (def_ a✝¹ a✝)
h3 : v ∉ binders
⊢ u ∉ binders | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | cases h2 | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h2 : ¬v = x ∧ isFreeIn v phi
h3 : v ∉ binders
⊢ u ∉ binders | case intro
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
left✝ : ¬v = x
right✝ : isFreeIn v phi
⊢ u ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | apply phi_ih binders | v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ u ∉ binders | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ admitsAux v u binders phi
case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ isFreeIn v phi
case h3
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ v ∉ binders |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | apply admitsAux_del_binders phi v u binders {x} h1 | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ admitsAux v u binders phi | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ v ∉ {x} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | simp | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ v ∉ {x} | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ ¬v = x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | exact h2_left | case h1
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ ¬v = x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | exact h2_right | case h2
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ isFreeIn v phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | exact h3 | case h3
v u x : VarName
phi : Formula
phi_ih : ∀ (binders : Finset VarName), admitsAux v u binders phi → isFreeIn v phi → v ∉ binders → u ∉ binders
binders : Finset VarName
h1 : admitsAux v u (binders ∪ {x}) phi
h3 : v ∉ binders
h2_left : ¬v = x
h2_right : isFreeIn v phi
⊢ v ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.admitsAux_isFreeIn | [974, 1] | [998, 10] | tauto | case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders
h2 : v ∈ a✝
h3 : v ∉ binders
⊢ u ∉ binders | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | induction E generalizing F binders V | D : Type
I : Interpretation D
V V' : VarAssignment D
E : Env
v t : VarName
binders : Finset VarName
F : Formula
h1 : fastAdmitsAux v t binders F
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) E F ↔ Holds D I V E (fastReplaceFree v t F) | case nil
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
V : VarAssignment D
binders : Finset VarName
F : Formula
h1 : fastAdmitsAux v t binders F
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) [] F ↔ Holds D I V [] (fastReplaceFree v t F)
case cons
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
F : Formula
h1 : fastAdmitsAux v t binders F
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) F ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | induction F generalizing binders V | case cons
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
F : Formula
h1 : fastAdmitsAux v t binders F
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) F ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t F) | case cons.pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_const_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_var_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (eq_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders true_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) true_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t true_)
case cons.false_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders false_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) false_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t false_)
case cons.not_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝.not_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝.not_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.imp_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.and_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.and_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.or_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.or_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.iff_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (forall_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (exists_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (def_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | all_goals
simp only [fastAdmitsAux] at h1
simp only [fastReplaceFree]
simp only [Holds] | case cons.pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_const_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_var_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (eq_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders true_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) true_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t true_)
case cons.false_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders false_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) false_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t false_)
case cons.not_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝.not_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝.not_ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.imp_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.and_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.and_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.or_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.or_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.iff_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (forall_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (exists_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (def_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝)) | case cons.pred_const_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_const_ a✝¹ (List.map (Function.updateITE V v (V' t)) a✝) ↔
I.pred_const_ a✝¹ (List.map V (List.map (fun x => if v = x then t else x) a✝))
case cons.pred_var_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_var_ a✝¹ (List.map (Function.updateITE V v (V' t)) a✝) ↔
I.pred_var_ a✝¹ (List.map V (List.map (fun x => if v = x then t else x) a✝))
case cons.eq_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = a✝¹ ∨ v = a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Function.updateITE V v (V' t) a✝¹ = Function.updateITE V v (V' t) a✝ ↔
V (if v = a✝¹ then t else a✝¹) = V (if v = a✝ then t else a✝)
case cons.not_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ ¬Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
¬Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝)
case cons.imp_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝¹ ∧ fastAdmitsAux v t binders a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ →
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹) → Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝)
case cons.and_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝¹ ∧ fastAdmitsAux v t binders a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ∧
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹) ∧ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝)
case cons.or_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝¹ ∧ fastAdmitsAux v t binders a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ∨
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹) ∨ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝)
case cons.iff_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝¹ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝¹ ∧ fastAdmitsAux v t binders a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝) ↔
(Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹) ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
case cons.forall_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v t (binders ∪ {a✝¹}) a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V v (V' t)) a✝¹ d) (head✝ :: tail✝) a✝) ↔
Holds D I V (head✝ :: tail✝) (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝))
case cons.exists_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders a✝ →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : v = a✝¹ ∨ fastAdmitsAux v t (binders ∪ {a✝¹}) a✝
h2 : ∀ v ∉ binders, V' v = V v
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) a✝¹ d) (head✝ :: tail✝) a✝) ↔
Holds D I V (head✝ :: tail✝) (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝))
case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V v (V' t)) head✝.args
(List.map (Function.updateITE V v (V' t)) a✝))
tail✝ head✝.q
else Holds D I (Function.updateITE V v (V' t)) tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ (List.map (fun x => if v = x then t else x) a✝).length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V (List.map (fun x => if v = x then t else x) a✝))) tail✝
head✝.q
else Holds D I V tail✝ (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case pred_const_ X xs | pred_var_ X xs =>
simp
congr! 1
simp only [List.map_eq_map_iff]
intro x a1
simp
simp only [Function.updateITE]
split_ifs
case _ c1 c2 =>
subst c1
tauto
case _ c1 c2 =>
subst c1
contradiction
case _ c1 c2 =>
subst c2
contradiction
case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_var_ X (List.map (Function.updateITE V v (V' t)) xs) ↔
I.pred_var_ X (List.map V (List.map (fun x => if v = x then t else x) xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case eq_ x y =>
simp only [Function.updateITE]
simp only [eq_comm]
congr! 1
all_goals
split_ifs
case _ c1 =>
subst c1
tauto
case _ c1 =>
rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Function.updateITE V v (V' t) x = Function.updateITE V v (V' t) y ↔
V (if v = x then t else x) = V (if v = y then t else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case not_ phi phi_ih =>
congr! 1
exact phi_ih V binders h1 h2 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ ¬Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
¬Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case forall_ x phi phi_ih | exists_ x phi phi_ih =>
split_ifs
case _ c1 =>
subst c1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
congr! 1
funext x
simp only [Function.updateITE]
split_ifs <;> rfl
case _ c1 =>
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
cases h1
case inl h1 =>
contradiction
case inr h1 =>
simp only [Function.updateITE_comm V v x d (V' t) c1]
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
simp only [Function.updateITE]
simp
push_neg
intros v' a1 a2
simp only [if_neg a2]
exact h2 v' a1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [fastAdmitsAux] at h1 | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (def_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝)) | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [fastReplaceFree] | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝)) | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Holds] | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I V (head✝ :: tail✝) (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) | case cons.def_
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ a✝ → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V v (V' t)) head✝.args
(List.map (Function.updateITE V v (V' t)) a✝))
tail✝ head✝.q
else Holds D I (Function.updateITE V v (V' t)) tail✝ (def_ a✝¹ a✝)) ↔
if a✝¹ = head✝.name ∧ (List.map (fun x => if v = x then t else x) a✝).length = head✝.args.length then
Holds D I (Function.updateListITE V head✝.args (List.map V (List.map (fun x => if v = x then t else x) a✝))) tail✝
head✝.q
else Holds D I V tail✝ (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_var_ X (List.map (Function.updateITE V v (V' t)) xs) ↔
I.pred_var_ X (List.map V (List.map (fun x => if v = x then t else x) xs)) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_var_ X (List.map (Function.updateITE V v (V' t)) xs) ↔
I.pred_var_ X (List.map (V ∘ fun x => if v = x then t else x) xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ I.pred_var_ X (List.map (Function.updateITE V v (V' t)) xs) ↔
I.pred_var_ X (List.map (V ∘ fun x => if v = x then t else x) xs) | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ List.map (Function.updateITE V v (V' t)) xs = List.map (V ∘ fun x => if v = x then t else x) xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ List.map (Function.updateITE V v (V' t)) xs = List.map (V ∘ fun x => if v = x then t else x) xs | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ∀ x ∈ xs, Function.updateITE V v (V' t) x = (V ∘ fun x => if v = x then t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ∀ x ∈ xs, Function.updateITE V v (V' t) x = (V ∘ fun x => if v = x then t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V v (V' t) x = (V ∘ fun x => if v = x then t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V v (V' t) x = (V ∘ fun x => if v = x then t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V v (V' t) x = V (if v = x then t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Function.updateITE] | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V v (V' t) x = V (if v = x then t else x) | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ (if x = v then V' t else V x) = V (if v = x then t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | split_ifs | case a.h.e'_4
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
⊢ (if x = v then V' t else V x) = V (if v = x then t else x) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
h✝¹ : x = v
h✝ : v = x
⊢ V' t = V t
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
h✝¹ : x = v
h✝ : ¬v = x
⊢ V' t = V x
case pos
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
h✝¹ : ¬x = v
h✝ : v = x
⊢ V x = V t
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
h✝¹ : ¬x = v
h✝ : ¬v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 c2 =>
subst c1
tauto | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : x = v
c2 : v = x
⊢ V' t = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 c2 =>
subst c1
contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : x = v
c2 : ¬v = x
⊢ V' t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 c2 =>
subst c2
contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : ¬x = v
c2 : v = x
⊢ V x = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : ¬x = v
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : x = v
c2 : v = x
⊢ V' t = V t | D : Type
I : Interpretation D
V' : VarAssignment D
t : VarName
head✝ : Definition
tail✝ : List Definition
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux x t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V x (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree x t F))
h1 : x ∈ xs → t ∉ binders
c2 : x = x
⊢ V' t = V t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | tauto | D : Type
I : Interpretation D
V' : VarAssignment D
t : VarName
head✝ : Definition
tail✝ : List Definition
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux x t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V x (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree x t F))
h1 : x ∈ xs → t ∉ binders
c2 : x = x
⊢ V' t = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : x = v
c2 : ¬v = x
⊢ V' t = V x | D : Type
I : Interpretation D
V' : VarAssignment D
t : VarName
head✝ : Definition
tail✝ : List Definition
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux x t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V x (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree x t F))
h1 : x ∈ xs → t ∉ binders
c2 : ¬x = x
⊢ V' t = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
t : VarName
head✝ : Definition
tail✝ : List Definition
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux x t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V x (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree x t F))
h1 : x ∈ xs → t ∉ binders
c2 : ¬x = x
⊢ V' t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c2 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : ¬x = v
c2 : v = x
⊢ V x = V t | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a1 : v ∈ xs
c1 : ¬v = v
⊢ V v = V t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | contradiction | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
a1 : v ∈ xs
c1 : ¬v = v
⊢ V v = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
X : PredName
xs : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : v ∈ xs → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
x : VarName
a1 : x ∈ xs
c1 : ¬x = v
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Function.updateITE] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ Function.updateITE V v (V' t) x = Function.updateITE V v (V' t) y ↔
V (if v = x then t else x) = V (if v = y then t else y) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ((if x = v then V' t else V x) = if y = v then V' t else V y) ↔
V (if v = x then t else x) = V (if v = y then t else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [eq_comm] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ((if x = v then V' t else V x) = if y = v then V' t else V y) ↔
V (if v = x then t else x) = V (if v = y then t else y) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ((if v = x then V' t else V x) = if v = y then V' t else V y) ↔
V (if v = x then t else x) = V (if v = y then t else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ ((if v = x then V' t else V x) = if v = y then V' t else V y) ↔
V (if v = x then t else x) = V (if v = y then t else y) | case a.h.e'_2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if v = x then V' t else V x) = V (if v = x then t else x)
case a.h.e'_3
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if v = y then V' t else V y) = V (if v = y then t else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | all_goals
split_ifs
case _ c1 =>
subst c1
tauto
case _ c1 =>
rfl | case a.h.e'_2
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if v = x then V' t else V x) = V (if v = x then t else x)
case a.h.e'_3
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if v = y then V' t else V y) = V (if v = y then t else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | split_ifs | case a.h.e'_3
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
⊢ (if v = y then V' t else V y) = V (if v = y then t else y) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
h✝ : v = y
⊢ V' t = V t
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
h✝ : ¬v = y
⊢ V y = V y |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 =>
subst c1
tauto | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : v = y
⊢ V' t = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 =>
rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = y
⊢ V y = V y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : v = y
⊢ V' t = V t | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = x ∨ v = v → t ∉ binders
⊢ V' t = V t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | tauto | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = x ∨ v = v → t ∉ binders
⊢ V' t = V t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x y : VarName
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ v = y → t ∉ binders
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = y
⊢ V y = V y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ ¬Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
¬Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact phi_ih V binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | cases h1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders phi ∧ fastAdmitsAux v t binders psi
h2 : ∀ v ∉ binders, V' v = V v
⊢ (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi) ↔
(Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi)) | case intro
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
left✝ : fastAdmitsAux v t binders phi
right✝ : fastAdmitsAux v t binders psi
⊢ (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi) ↔
(Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1_left : fastAdmitsAux v t binders phi
h1_right : fastAdmitsAux v t binders psi
⊢ (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi) ↔
(Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi)) | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1_left : fastAdmitsAux v t binders phi
h1_right : fastAdmitsAux v t binders psi
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1_left : fastAdmitsAux v t binders phi
h1_right : fastAdmitsAux v t binders psi
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact phi_ih V binders h1_left h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1_left : fastAdmitsAux v t binders phi
h1_right : fastAdmitsAux v t binders psi
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | exact psi_ih V binders h1_right h2 | case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
psi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders psi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1_left : fastAdmitsAux v t binders phi
h1_right : fastAdmitsAux v t binders psi
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) psi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | split_ifs | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) | case pos
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
h✝ : v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x phi)
case neg
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
h✝ : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x (fastReplaceFree v t phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 =>
subst c1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
congr! 1
funext x
simp only [Function.updateITE]
split_ifs <;> rfl | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | case _ c1 =>
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
cases h1
case inl h1 =>
contradiction
case inr h1 =>
simp only [Function.updateITE_comm V v x d (V' t) c1]
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
simp only [Function.updateITE]
simp
push_neg
intros v' a1 a2
simp only [if_neg a2]
exact h2 v' a1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x (fastReplaceFree v t phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | subst c1 | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h1 : v = x ∨ fastAdmitsAux v t (binders ∪ {x}) phi
h2 : ∀ v ∉ binders, V' v = V v
c1 : v = x
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Holds] | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
Holds D I V (head✝ :: tail✝) (exists_ v phi) | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | intro d | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | congr! 1 | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi | case h.a.h.e'_3
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
⊢ Function.updateITE (Function.updateITE V v (V' t)) v d = Function.updateITE V v d |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | funext x | case h.a.h.e'_3
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
⊢ Function.updateITE (Function.updateITE V v (V' t)) v d = Function.updateITE V v d | case h.a.h.e'_3.h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
x : VarName
⊢ Function.updateITE (Function.updateITE V v (V' t)) v d x = Function.updateITE V v d x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | simp only [Function.updateITE] | case h.a.h.e'_3.h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
x : VarName
⊢ Function.updateITE (Function.updateITE V v (V' t)) v d x = Function.updateITE V v d x | case h.a.h.e'_3.h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
x : VarName
⊢ (if x = v then d else if x = v then V' t else V x) = if x = v then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | split_ifs <;> rfl | case h.a.h.e'_3.h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
d : D
x : VarName
⊢ (if x = v then d else if x = v then V' t else V x) = if x = v then d else V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply forall_congr' | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Rec/Admits.lean | FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux | [1001, 1] | [1136, 17] | apply exists_congr | D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V v d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
fastAdmitsAux v t binders F →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D) (binders : Finset VarName),
fastAdmitsAux v t binders phi →
(∀ v ∉ binders, V' v = V v) →
(Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
h1 : v = v ∨ fastAdmitsAux v t (binders ∪ {v}) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V v (V' t)) v a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) phi |
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