url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [isFreeIn] at h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | cases h1_1 | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | case intro
F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
left✝ : ¬h1_v = h1_x
right✝ : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | case intro h1_1_left h1_1_right =>
simp only [if_neg h1_1_left]
subst h1_ih
rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | simp only [if_neg h1_1_left] | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ (if h1_v = h1_x then exists_ h1_x h1_phi else exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi)) =
exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | subst h1_ih | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝¹ : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : Rec.fastReplaceFree h1_v h1_t h1_phi = h1_phi'
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x h1_phi' | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
a✝¹ : ¬h1_x = h1_t
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
a✝ : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastReplaceFree | [234, 1] | [275, 10] | rfl | F F' : Formula
v u h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
a✝¹ : ¬h1_x = h1_t
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
a✝ : IsSub h1_phi h1_v h1_t (Rec.fastReplaceFree h1_v h1_t h1_phi)
⊢ exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) = exists_ h1_x (Rec.fastReplaceFree h1_v h1_t h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | induction h1 generalizing V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
v t : VarName
F F' : Formula
h1 : IsSub F v t F'
⊢ Holds D I (Function.updateITE V v (V t)) E F ↔ Holds D I V E F' | case pred_const_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (pred_const_ X✝ xs✝) ↔
Holds D I V E (pred_const_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝))
case pred_var_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : PredName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (pred_var_ X✝ xs✝) ↔
Holds D I V E (pred_var_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝))
case eq_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ y✝ v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (eq_ x✝ y✝) ↔
Holds D I V E (eq_ (if v✝ = x✝ then t✝ else x✝) (if v✝ = y✝ then t✝ else y✝))
case true_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝.not_ ↔ Holds D I V E phi'✝.not_
case imp_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.imp_ psi✝) ↔ Holds D I V E (phi'✝.imp_ psi'✝)
case and_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.and_ psi✝) ↔ Holds D I V E (phi'✝.and_ psi'✝)
case or_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.or_ psi✝) ↔ Holds D I V E (phi'✝.or_ psi'✝)
case iff_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' phi✝ psi✝ : Formula
v✝ t✝ : VarName
phi'✝ psi'✝ : Formula
a✝¹ : IsSub phi✝ v✝ t✝ phi'✝
a✝ : IsSub psi✝ v✝ t✝ psi'✝
a_ih✝¹ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E psi✝ ↔ Holds D I V E psi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (phi✝.iff_ psi✝) ↔ Holds D I V E (phi'✝.iff_ psi'✝)
case forall_not_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ x✝ phi✝)
case forall_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (forall_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (forall_ x✝ phi✝) ↔ Holds D I V E (forall_ x✝ phi'✝)
case exists_not_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ x✝ phi✝)
case exists_free_in
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
x✝ : VarName
phi✝ : Formula
v✝ t✝ : VarName
phi'✝ : Formula
a✝² : isFreeIn v✝ (exists_ x✝ phi✝)
a✝¹ : ¬x✝ = t✝
a✝ : IsSub phi✝ v✝ t✝ phi'✝
a_ih✝ : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V v✝ (V t✝)) E phi✝ ↔ Holds D I V E phi'✝
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (exists_ x✝ phi✝) ↔ Holds D I V E (exists_ x✝ phi'✝)
case def_
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
X✝ : DefName
xs✝ : List VarName
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E (def_ X✝ xs✝) ↔
Holds D I V E (def_ X✝ (List.map (fun x => if v✝ = x then t✝ else x) xs✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case pred_const_ h1_X h1_xs h1_v h1_t | pred_var_ h1_X h1_xs h1_v h1_t =>
simp only [Holds]
congr! 1
simp
simp only [List.map_eq_map_iff]
intro x _
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 =>
simp
simp only [if_pos c1]
case _ c1 =>
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (pred_var_ h1_X h1_xs) ↔
Holds D I V E (pred_var_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case eq_ h1_x h1_y h1_v h1_t =>
simp only [Holds]
simp only [Function.updateITE]
simp only [eq_comm]
congr! 1 <;> { split_ifs <;> rfl } | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (eq_ h1_x h1_y) ↔
Holds D I V E (eq_ (if h1_v = h1_x then h1_t else h1_x) (if h1_v = h1_y then h1_t else h1_y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case true_ _ _ | false_ _ _ =>
simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case not_ h1_phi h1_v h1_t h1_phi' _ h1_ih =>
simp only [Holds]
congr! 1
apply h1_ih | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case
forall_not_free_in h1_x h1_phi h1_v h1_t h1_1
| exists_not_free_in h1_x h1_phi h1_v h1_t h1_1 =>
simp only [isFreeIn] at h1_1
simp at h1_1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
apply Holds_coincide_Var
intro x a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
rfl
case _ c1 c2 =>
subst c2
tauto
case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case
forall_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 h1_2 _ h1_ih
| exists_free_in h1_x h1_phi h1_v h1_t h1_phi' h1_1 h1_2 _ h1_ih =>
simp only [isFreeIn] at h1_1
simp only [Holds]
first | apply forall_congr' | apply exists_congr
intro d
specialize h1_ih (Function.updateITE V h1_x d)
simp only [← h1_ih]
apply Holds_coincide_Var
intro x _
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 c2 c3 =>
subst c2
cases h1_1
case intro h1_1_left h1_1_right =>
contradiction
case _ | _ | _ =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (pred_var_ h1_X h1_xs) ↔
Holds D I V E (pred_var_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ I.pred_var_ h1_X (List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) ↔
I.pred_var_ h1_X (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ I.pred_var_ h1_X (List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) ↔
I.pred_var_ h1_X (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs =
List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs =
List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs) | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x _ | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case a.h.e'_4
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
h✝ : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
h✝ : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
simp
simp only [if_pos c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
simp
simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = (V ∘ fun x => if h1_v = x then h1_t else x) x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = V (if h1_v = x then h1_t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [if_pos c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : h1_v = x
⊢ V h1_t = V (if h1_v = x then h1_t else x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = (V ∘ fun x => if h1_v = x then h1_t else x) x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = V (if h1_v = x then h1_t else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [if_neg c1] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : PredName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
x : VarName
a✝ : x ∈ h1_xs
c1 : ¬h1_v = x
⊢ V x = V (if h1_v = x then h1_t else x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (eq_ h1_x h1_y) ↔
Holds D I V E (eq_ (if h1_v = h1_x then h1_t else h1_x) (if h1_v = h1_y then h1_t else h1_y)) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Function.updateITE V h1_v (V h1_t) h1_x = Function.updateITE V h1_v (V h1_t) h1_y ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ Function.updateITE V h1_v (V h1_t) h1_x = Function.updateITE V h1_v (V h1_t) h1_y ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_v = h1_x then h1_t else h1_x) = V (if h1_v = h1_y then h1_t else h1_y) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_x = h1_v then h1_t else h1_x) = V (if h1_y = h1_v then h1_t else h1_y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 <;> { split_ifs <;> rfl } | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ ((if h1_x = h1_v then V h1_t else V h1_x) = if h1_y = h1_v then V h1_t else V h1_y) ↔
V (if h1_x = h1_v then h1_t else h1_x) = V (if h1_y = h1_v then h1_t else h1_y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs <;> rfl | case a.h.e'_3
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x h1_y h1_v h1_t : VarName
V : VarAssignment D
⊢ (if h1_y = h1_v then V h1_t else V h1_y) = V (if h1_y = h1_v then h1_t else h1_y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
v✝ t✝ : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V v✝ (V t✝)) E false_ ↔ Holds D I V E false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi.not_ ↔ Holds D I V E h1_phi'.not_ | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ ¬Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ¬Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ ¬Holds D I V E h1_phi' | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (h1_phi.iff_ h1_psi) ↔ Holds D I V E (h1_phi'.iff_ h1_psi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi) ↔
(Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | congr! 1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ (Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi) ↔
(Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih_1 | case a.h.e'_1.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply h1_ih_2 | case a.h.e'_2.a
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' h1_phi h1_psi : Formula
h1_v h1_t : VarName
h1_phi' h1_psi' : Formula
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
a✝ : IsSub h1_psi h1_v h1_t h1_psi'
h1_ih_1 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
h1_ih_2 : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_psi ↔ Holds D I V E h1_psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [isFreeIn] at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬isFreeIn h1_v (exists_ h1_x h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_1 : ¬(¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi)
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi) | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro d | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Holds_coincide_Var | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ ∀ (v : VarName),
isFreeIn v h1_phi → Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v = Function.updateITE V h1_x d v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x a1 | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
⊢ ∀ (v : VarName),
isFreeIn v h1_phi → Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v = Function.updateITE V h1_x d v | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x = Function.updateITE V h1_x d x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x = Function.updateITE V h1_x d x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) = if x = h1_x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) = if x = h1_x then d else V x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝ : x = h1_x
⊢ d = d
case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝¹ : ¬x = h1_x
h✝ : x = h1_v
⊢ V h1_t = V x
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
h✝¹ : ¬x = h1_x
h✝ : ¬x = h1_v
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : x = h1_x
⊢ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
subst c2
tauto | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : x = h1_v
⊢ V h1_t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : ¬x = h1_v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : x = h1_x
⊢ d = d | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | subst c2 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : x = h1_v
⊢ V h1_t = V x | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_t : VarName
V : VarAssignment D
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
h1_1 : ¬x = h1_x → ¬isFreeIn x h1_phi
⊢ V h1_t = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | tauto | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_t : VarName
V : VarAssignment D
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
h1_1 : ¬x = h1_x → ¬isFreeIn x h1_phi
⊢ V h1_t = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
V : VarAssignment D
h1_1 : ¬h1_v = h1_x → ¬isFreeIn h1_v h1_phi
d : D
x : VarName
a1 : isFreeIn x h1_phi
c1 : ¬x = h1_x
c2 : ¬x = h1_v
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [isFreeIn] at h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : isFreeIn h1_v (exists_ h1_x h1_phi)
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (exists_ h1_x h1_phi) ↔ Holds D I V E (exists_ h1_x h1_phi') | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro d | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | specialize h1_ih (Function.updateITE V h1_x d) | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
d : D
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [← h1_ih] | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Holds_coincide_Var | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi ↔
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ ∀ (v : VarName),
isFreeIn v h1_phi →
Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x _ | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
⊢ ∀ (v : VarName),
isFreeIn v h1_phi →
Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d v =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) v | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d x =
Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t) x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) =
if x = h1_v then if h1_t = h1_x then d else V h1_t else if x = h1_x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if x = h1_x then d else if x = h1_v then V h1_t else V x) =
if x = h1_v then if h1_t = h1_x then d else V h1_t else if x = h1_x then d else V x | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if h1_x = x then d else if h1_v = x then V h1_t else V x) =
if h1_v = x then if h1_x = h1_t then d else V h1_t else if h1_x = x then d else V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case h.h1
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
⊢ (if h1_x = x then d else if h1_v = x then V h1_t else V x) =
if h1_v = x then if h1_x = h1_t then d else V h1_t else if h1_x = x then d else V x | case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : h1_x = x
h✝ : h1_v = x
⊢ d = V h1_t
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : h1_x = x
h✝ : ¬h1_v = x
⊢ d = d
case pos
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : h1_v = x
⊢ V h1_t = V h1_t
case neg
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 c3 =>
subst c2
cases h1_1
case intro h1_1_left h1_1_right =>
contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
c1 : isFreeIn x h1_phi
c2 : h1_x = x
c3 : h1_v = x
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ | _ | _ =>
rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply forall_congr' | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∀ (d : D), Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∀ (d : D), Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply exists_congr | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ (∃ d, Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x d) E h1_phi) ↔
∃ d, Holds D I (Function.updateITE V h1_x d) E h1_phi' | case h
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
h1_ih : ∀ (V : VarAssignment D), Holds D I (Function.updateITE V h1_v (V h1_t)) E h1_phi ↔ Holds D I V E h1_phi'
V : VarAssignment D
⊢ ∀ (a : D),
Holds D I (Function.updateITE (Function.updateITE V h1_v (V h1_t)) h1_x a) E h1_phi ↔
Holds D I (Function.updateITE V h1_x a) E h1_phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | subst c2 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
c1 : isFreeIn x h1_phi
c2 : h1_x = x
c3 : h1_v = x
⊢ d = V h1_t | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
⊢ d = V h1_t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | cases h1_1 | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
⊢ d = V h1_t | case intro
D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
left✝ : ¬h1_v = h1_x
right✝ : isFreeIn h1_v h1_phi
⊢ d = V h1_t |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case intro h1_1_left h1_1_right =>
contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | contradiction | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_2 : ¬h1_x = h1_t
a✝ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
c1 : isFreeIn h1_x h1_phi
c3 : h1_v = h1_x
h1_1_left : ¬h1_v = h1_x
h1_1_right : isFreeIn h1_v h1_phi
⊢ d = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | rfl | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_x : VarName
h1_phi : Formula
h1_v h1_t : VarName
h1_phi' : Formula
h1_1 : ¬h1_v = h1_x ∧ isFreeIn h1_v h1_phi
h1_2 : ¬h1_x = h1_t
a✝¹ : IsSub h1_phi h1_v h1_t h1_phi'
V : VarAssignment D
d : D
h1_ih :
Holds D I (Function.updateITE (Function.updateITE V h1_x d) h1_v (Function.updateITE V h1_x d h1_t)) E h1_phi ↔
Holds D I (Function.updateITE V h1_x d) E h1_phi'
x : VarName
a✝ : isFreeIn x h1_phi
h✝¹ : ¬h1_x = x
h✝ : ¬h1_v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | induction E | D : Type
I : Interpretation D
E : Env
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) E (def_ h1_X h1_xs) ↔
Holds D I V E (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | case nil
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
case cons
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
Holds D I (Function.updateITE V h1_v (V h1_t)) tail✝ (def_ h1_X h1_xs) ↔
Holds D I V tail✝ (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) (head✝ :: tail✝) (def_ h1_X h1_xs) ↔
Holds D I V (head✝ :: tail✝) (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case nil =>
simp only [Holds] | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) [] (def_ h1_X h1_xs) ↔
Holds D I V [] (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Holds] | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) (hd :: tl) (def_ h1_X h1_xs) ↔
Holds D I V (hd :: tl) (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ (if h1_X = hd.name ∧ h1_xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q
else Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs)) ↔
if h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
else Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
⊢ (if h1_X = hd.name ∧ h1_xs.length = hd.args.length then
Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q
else Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs)) ↔
if h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length then
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
else Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : h1_X = hd.name ∧ h1_xs.length = hd.args.length
h✝ : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : h1_X = hd.name ∧ h1_xs.length = hd.args.length
h✝ : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
h✝ : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
h✝¹ : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
h✝ : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
simp only [List.length_map] at c2
contradiction | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
simp at c2
contradiction | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c1 c2 =>
exact ih | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : ¬(h1_X = hd.name ∧ h1_xs.length = hd.args.length)
c2 : ¬(h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length)
⊢ Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map V (List.map (fun x => if h1_v = x then h1_t else x) h1_xs)))
tl hd.q | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Holds_coincide_Var | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ Holds D I
(Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs))
tl hd.q ↔
Holds D I (Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs)) tl hd.q | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro v' a1 | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | have s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun (x : VarName) => if h1_v = x then h1_t else x) h1_xs | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [List.map_eq_map_iff] | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | intro x _ | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
⊢ ∀ x ∈ h1_xs, Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [Function.updateITE] | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ Function.updateITE V h1_v (V h1_t) x = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [eq_comm] | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if x = h1_v then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = (V ∘ fun x => if h1_v = x then h1_t else x) x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = V (if h1_v = x then h1_t else x)
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | split_ifs | case s1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
⊢ (if h1_v = x then V h1_t else V x) = V (if h1_v = x then h1_t else x)
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case pos
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
h✝ : h1_v = x
⊢ V h1_t = V h1_t
case neg
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
h✝ : ¬h1_v = x
⊢ V x = V x
case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c3 =>
simp only [if_pos c3] | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : h1_v = x
⊢ V h1_t = V h1_t | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | case _ c3 =>
simp only [if_neg c3] | D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
x : VarName
a✝ : x ∈ h1_xs
c3 : ¬h1_v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | simp only [s1] | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (Function.updateITE V h1_v (V h1_t)) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Var/One/Ind/Sub.lean | FOL.NV.Sub.Var.One.Ind.substitution_theorem | [298, 1] | [424, 17] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ Function.updateListITE (Function.updateITE V h1_v (V h1_t)) hd.args
(List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' =
Function.updateListITE V hd.args (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs) v' | case h1.h1
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ v' ∈ hd.args
case h1.h2
D : Type
I : Interpretation D
v t : VarName
F F' : Formula
h1_X : DefName
h1_xs : List VarName
h1_v h1_t : VarName
V : VarAssignment D
hd : Definition
tl : List Definition
ih :
Holds D I (Function.updateITE V h1_v (V h1_t)) tl (def_ h1_X h1_xs) ↔
Holds D I V tl (def_ h1_X (List.map (fun x => if h1_v = x then h1_t else x) h1_xs))
c1 : h1_X = hd.name ∧ h1_xs.length = hd.args.length
c2 : h1_X = hd.name ∧ (List.map (fun x => if h1_v = x then h1_t else x) h1_xs).length = hd.args.length
v' : VarName
a1 : isFreeIn v' hd.q
s1 : List.map (Function.updateITE V h1_v (V h1_t)) h1_xs = List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs
⊢ hd.args.length = (List.map (V ∘ fun x => if h1_v = x then h1_t else x) h1_xs).length |
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