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
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| 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/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi).not_ ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi.not_ | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ¬Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ¬Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
¬Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at psi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (phi.iff_ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) ((sub τ phi).iff_ (sub τ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) ((sub τ phi).iff_ (sub τ psi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (phi.iff_ psi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ (Holds D I V (head✝ :: tail✝) (sub τ phi) ↔ Holds D I V (head✝ :: tail✝) (sub τ psi)) ↔
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | congr! 1 | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ (Holds D I V (head✝ :: tail✝) (sub τ phi) ↔ Holds D I V (head✝ :: tail✝) (sub τ psi)) ↔
(Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi) | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply psi_ih | case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
psi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at phi_ih | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
V : VarAssignment D
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (sub τ (exists_ x phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (exists_ x (sub τ phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ Holds D I V (head✝ :: tail✝) (exists_ x (sub τ phi)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds =>
if ds = [] then Holds D I V (head✝ :: tail✝) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | intros d | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply phi_ih | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply forall_congr' | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∀ (d : D),
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply exists_congr | D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (sub τ phi)) ↔
∃ d,
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x d) (head✝ :: tail✝) phi | case h
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tail✝ (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tail✝ F
x : VarName
phi : Formula
V : VarAssignment D
phi_ih :
∀ (V : VarAssignment D),
Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V (head✝ :: tail✝) phi
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) (sub τ phi) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateITE V x a) (head✝ :: tail✝) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V [] (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V [] (pred_var_ (τ P) []) else I.pred_var_ P ds }
V [] (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V tl (pred_var_ (τ P) []) else I.pred_var_ P ds }
V tl F
X : DefName
xs : List VarName
V : VarAssignment D
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [Holds] | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ Holds D I V (hd :: tl) (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V (hd :: tl) (pred_var_ (τ P) []) else I.pred_var_ P ds }
V (hd :: tl) (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | split_ifs | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
⊢ (if X = hd.name ∧ xs.length = hd.args.length then Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I V tl (def_ X xs)) ↔
if X = hd.name ∧ xs.length = hd.args.length then
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | case pos
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
h✝ : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q
simp only [sub_no_predVar hd.q τ hd.h2] at ih
apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | case _ c1 =>
specialize ih V (def_ X xs)
simp only [sub] at ih
exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih (Function.updateListITE V hd.args (List.map V xs)) hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub_no_predVar hd.q τ hd.h2] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl (sub τ hd.q) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | apply ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : X = hd.name ∧ xs.length = hd.args.length
ih :
Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
(Function.updateListITE V hd.args (List.map V xs)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | specialize ih V (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
ih :
∀ (V : VarAssignment D) (F : Formula),
Holds D I V tl (sub τ F) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl F
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | simp only [sub] at ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (sub τ (def_ X xs)) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_theorem | [94, 1] | [182, 15] | exact ih | D : Type
I : Interpretation D
τ : PredName → PredName
hd : Definition
tl : List Definition
X : DefName
xs : List VarName
V : VarAssignment D
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
ih :
Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) [] else I.pred_var_ P ds }
V tl (def_ X xs)
⊢ Holds D I V tl (def_ X xs) ↔
Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
V tl (def_ X xs) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] at h1 | F : Formula
τ : PredName → PredName
h1 : F.IsValid
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [IsValid] | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | intro D I V E | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F) | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | simp only [substitution_theorem D I V E τ F] | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F) | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
V E F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Sub/Prop/All/Rec/Sub.lean | FOL.NV.Sub.Prop.All.Rec.substitution_is_valid | [185, 1] | [196, 11] | apply h1 | F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
{ nonempty := ⋯, pred_const_ := I.pred_const_,
pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
V E F | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | induction E generalizing F V | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
u v : VarName
F : Formula
a : D
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) E F ↔
Holds D I (Function.updateITE V v a) E (Sub.Var.One.Rec.fastReplaceFree u v F) | case nil
D : Type
I : Interpretation D
u v : VarName
a : D
V : VarAssignment D
F : Formula
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) [] F ↔
Holds D I (Function.updateITE V v a) [] (Sub.Var.One.Rec.fastReplaceFree u v F)
case cons
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
F : Formula
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) F ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case nil.def_ X xs =>
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
X : DefName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ X xs)
h2 : ¬isBoundIn v (def_ X xs)
⊢ Holds D I (Function.updateITE V u a) [] (def_ X xs) ↔
Holds D I (Function.updateITE V v a) [] (Sub.Var.One.Rec.fastReplaceFree u v (def_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | induction F generalizing V | case cons
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
F : Formula
h1 : ¬isFreeIn v F
h2 : ¬isBoundIn v F
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) F ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v F) | case cons.pred_const_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝)
h2 : ¬isBoundIn v (pred_const_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_const_ a✝¹ a✝))
case cons.pred_var_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝)
h2 : ¬isBoundIn v (pred_var_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_var_ a✝¹ a✝))
case cons.eq_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ a✝ : VarName
V : VarAssignment D
h1 : ¬isFreeIn v (eq_ a✝¹ a✝)
h2 : ¬isBoundIn v (eq_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (eq_ a✝¹ a✝))
case cons.true_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v true_
h2 : ¬isBoundIn v true_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) true_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v true_)
case cons.false_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v false_
h2 : ¬isBoundIn v false_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) false_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v false_)
case cons.not_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v a✝.not_
h2 : ¬isBoundIn v a✝.not_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝.not_)
case cons.imp_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝¹ →
¬isBoundIn v a✝¹ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝¹ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝¹))
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (a✝¹.imp_ a✝)
h2 : ¬isBoundIn v (a✝¹.imp_ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (a✝¹.imp_ a✝))
case cons.and_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝¹ →
¬isBoundIn v a✝¹ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝¹ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝¹))
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (a✝¹.and_ a✝)
h2 : ¬isBoundIn v (a✝¹.and_ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (a✝¹.and_ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (a✝¹.and_ a✝))
case cons.or_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝¹ →
¬isBoundIn v a✝¹ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝¹ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝¹))
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (a✝¹.or_ a✝)
h2 : ¬isBoundIn v (a✝¹.or_ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (a✝¹.or_ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (a✝¹.or_ a✝))
case cons.iff_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ a✝ : Formula
a_ih✝¹ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝¹ →
¬isBoundIn v a✝¹ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝¹ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝¹))
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (a✝¹.iff_ a✝)
h2 : ¬isBoundIn v (a✝¹.iff_ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (a✝¹.iff_ a✝))
case cons.forall_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (forall_ a✝¹ a✝)
h2 : ¬isBoundIn v (forall_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (forall_ a✝¹ a✝))
case cons.exists_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
∀ (V : VarAssignment D),
¬isFreeIn v a✝ →
¬isBoundIn v a✝ →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) a✝ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v a✝))
V : VarAssignment D
h1 : ¬isFreeIn v (exists_ a✝¹ a✝)
h2 : ¬isBoundIn v (exists_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ a✝¹ a✝))
case cons.def_
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (def_ a✝¹ a✝)
h2 : ¬isBoundIn v (def_ a✝¹ a✝)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case pred_const_ X xs | pred_var_ X xs =>
simp only [isFreeIn] at h1
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds]
congr! 1
simp
simp only [List.map_eq_map_iff]
intro x a1
simp
simp only [Function.updateITE]
simp only [eq_comm]
split_ifs
case _ c1 c2 =>
rfl
case _ c1 c2 =>
contradiction
case _ c1 c2 =>
subst c2
contradiction
case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (pred_var_ X xs)
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_var_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case true_ | false_ =>
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v false_
h2 : ¬isBoundIn v false_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) false_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v false_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case not_ phi phi_ih =>
simp only [isFreeIn] at h1
simp only [isBoundIn] at h2
simp only [Sub.Var.One.Rec.fastReplaceFree]
simp only [Holds]
congr! 1
exact phi_ih V h1 h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi.not_
h2 : ¬isBoundIn v phi.not_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : ¬isFreeIn v (pred_var_ X xs)
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_var_ X xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (pred_var_ X xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (pred_var_ X (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (pred_var_ X xs) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (pred_var_ X (List.map (fun x => if u = x then v else x) xs)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ I.pred_var_ X (List.map (Function.updateITE V u a) xs) ↔
I.pred_var_ X (List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ I.pred_var_ X (List.map (Function.updateITE V u a) xs) ↔
I.pred_var_ X (List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs)) | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ List.map (Function.updateITE V u a) xs =
List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ List.map (Function.updateITE V u a) xs =
List.map (Function.updateITE V v a) (List.map (fun x => if u = x then v else x) xs) | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ List.map (Function.updateITE V u a) xs = List.map (Function.updateITE V v a ∘ fun x => if u = x then v else x) xs |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [List.map_eq_map_iff] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ List.map (Function.updateITE V u a) xs = List.map (Function.updateITE V v a ∘ fun x => if u = x then v else x) xs | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ ∀ x ∈ xs, Function.updateITE V u a x = (Function.updateITE V v a ∘ fun x => if u = x then v else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | intro x a1 | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
⊢ ∀ x ∈ xs, Function.updateITE V u a x = (Function.updateITE V v a ∘ fun x => if u = x then v else x) x | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V u a x = (Function.updateITE V v a ∘ fun x => if u = x then v else x) x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V u a x = (Function.updateITE V v a ∘ fun x => if u = x then v else x) x | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V u a x = Function.updateITE V v a (if u = x then v else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ Function.updateITE V u a x = Function.updateITE V v a (if u = x then v else x) | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ (if x = u then a else V x) = if (if u = x then v else x) = v then a else V (if u = x then v else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [eq_comm] | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ (if x = u then a else V x) = if (if u = x then v else x) = v then a else V (if u = x then v else x) | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ (if u = x then a else V x) = if v = if u = x then v else x then a else V (if u = x then v else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | case a.h.e'_4
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
⊢ (if u = x then a else V x) = if v = if u = x then v else x then a else V (if u = x then v else x) | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
h✝¹ : u = x
h✝ : v = v
⊢ a = a
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
h✝¹ : u = x
h✝ : ¬v = v
⊢ a = V v
case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
h✝¹ : ¬u = x
h✝ : v = x
⊢ V x = a
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
h✝¹ : ¬u = x
h✝ : ¬v = x
⊢ V x = V x |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : u = x
c2 : v = v
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : u = x
c2 : ¬v = v
⊢ a = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
subst c2
contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : ¬u = x
c2 : v = x
⊢ V x = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case _ c1 c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : ¬u = x
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : u = x
c2 : v = v
⊢ a = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : u = x
c2 : ¬v = v
⊢ a = V v | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : ¬u = x
c2 : v = x
⊢ V x = a | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
a1 : v ∈ xs
c1 : ¬u = v
⊢ V v = a |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | contradiction | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
a1 : v ∈ xs
c1 : ¬u = v
⊢ V v = a | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
X : PredName
xs : List VarName
V : VarAssignment D
h1 : v ∉ xs
h2 : ¬isBoundIn v (pred_var_ X xs)
x : VarName
a1 : x ∈ xs
c1 : ¬u = x
c2 : ¬v = x
⊢ V x = V x | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬isFreeIn v (eq_ x y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ x y) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (eq_ x y)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ x y) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (eq_ x y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ x y) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (eq_ x y)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ x y) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (eq_ (if u = x then v else x) (if u = y then v else y)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (eq_ x y) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (eq_ (if u = x then v else x) (if u = y then v else y)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a x = Function.updateITE V u a y ↔
Function.updateITE V v a (if u = x then v else x) = Function.updateITE V v a (if u = y then v else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a x = Function.updateITE V u a y ↔
Function.updateITE V v a (if u = x then v else x) = Function.updateITE V v a (if u = y then v else y) | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a x = Function.updateITE V v a (if u = x then v else x)
case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a y = Function.updateITE V v a (if u = y then v else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a x = Function.updateITE V v a (if u = x then v else x) | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ (if x = u then a else V x) = if (if u = x then v else x) = v then a else V (if u = x then v else x) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs <;> tauto | case a.h.e'_2
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ (if x = u then a else V x) = if (if u = x then v else x) = v then a else V (if u = x then v else x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Function.updateITE] | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ Function.updateITE V u a y = Function.updateITE V v a (if u = y then v else y) | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ (if y = u then a else V y) = if (if u = y then v else y) = v then a else V (if u = y then v else y) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs <;> tauto | case a.h.e'_3
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x y : VarName
V : VarAssignment D
h1 : ¬(v = x ∨ v = y)
h2 : ¬isBoundIn v (eq_ x y)
⊢ (if y = u then a else V y) = if (if u = y then v else y) = v then a else V (if u = y then v else y) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v false_
h2 : ¬isBoundIn v false_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) false_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v false_) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v false_
h2 : ¬isBoundIn v false_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) false_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) false_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
V : VarAssignment D
h1 : ¬isFreeIn v false_
h2 : ¬isBoundIn v false_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) false_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi.not_
h2 : ¬isBoundIn v phi.not_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi.not_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi.not_
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi.not_) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi.not_ ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi).not_ | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ ¬Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
¬Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ ¬Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
¬Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact phi_ih V h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v phi
h2 : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1 : ¬isFreeIn v (phi.iff_ psi)
h2 : ¬isBoundIn v (phi.iff_ psi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
h2 : ¬isBoundIn v (phi.iff_ psi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi)
h2 : ¬isBoundIn v (phi.iff_ psi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬isBoundIn v (phi.iff_ psi)
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬isBoundIn v (phi.iff_ psi)
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬(isBoundIn v phi ∨ isBoundIn v psi)
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬(isBoundIn v phi ∨ isBoundIn v psi)
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
h2 : ¬isBoundIn v phi ∧ ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1 : ¬isFreeIn v phi ∧ ¬isFreeIn v psi
h2 : ¬isBoundIn v phi ∧ ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬isBoundIn v phi ∧ ¬isBoundIn v psi
left✝ : ¬isFreeIn v phi
right✝ : ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h2 : ¬isBoundIn v phi ∧ ¬isBoundIn v psi
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
left✝ : ¬isBoundIn v phi
right✝ : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (phi.iff_ psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝)
((Sub.Var.One.Rec.fastReplaceFree u v phi).iff_ (Sub.Var.One.Rec.fastReplaceFree u v psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Holds] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (phi.iff_ psi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝)
((Sub.Var.One.Rec.fastReplaceFree u v phi).iff_ (Sub.Var.One.Rec.fastReplaceFree u v psi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi) ↔
(Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | congr! 1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi) ↔
(Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi)) | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi)
case a.h.e'_2.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact phi_ih V h1_left h2_left | case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | exact psi_ih V h1_right h2_right | case a.h.e'_2.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v psi →
¬isBoundIn v psi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isFreeIn] at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬isFreeIn v (exists_ x phi)
h2 : ¬isBoundIn v (exists_ x phi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬(¬v = x ∧ isFreeIn v phi)
h2 : ¬isBoundIn v (exists_ x phi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : ¬(¬v = x ∧ isFreeIn v phi)
h2 : ¬isBoundIn v (exists_ x phi)
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2 : ¬isBoundIn v (exists_ x phi)
h1 : v ≠ x → ¬isFreeIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [isBoundIn] at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2 : ¬isBoundIn v (exists_ x phi)
h1 : v ≠ x → ¬isFreeIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2 : ¬(v = x ∨ isBoundIn v phi)
h1 : v ≠ x → ¬isFreeIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | push_neg at h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2 : ¬(v = x ∨ isBoundIn v phi)
h1 : v ≠ x → ¬isFreeIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2 : v ≠ x ∧ ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | cases h2 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2 : v ≠ x ∧ ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | case intro
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
left✝ : v ≠ x
right✝ : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | simp only [Sub.Var.One.Rec.fastReplaceFree] | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v (exists_ x phi)) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝)
(if u = x then exists_ x phi else exists_ x (Sub.Var.One.Rec.fastReplaceFree u v phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | split_ifs | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝)
(if u = x then exists_ x phi else exists_ x (Sub.Var.One.Rec.fastReplaceFree u v phi)) | case pos
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
h✝ : u = x
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ x phi)
case neg
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
h✝ : ¬u = x
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ x (Sub.Var.One.Rec.fastReplaceFree u v phi)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | case pos c1 =>
subst c1
apply Holds_coincide_Var
intro x a1
simp only [isFreeIn] at a1
cases a1
case h1.intro a1_left a1_right =>
simp only [Function.updateITE]
simp only [if_neg a1_left]
split_ifs
case pos c2 =>
subst c2
tauto
case neg c2 =>
rfl | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : u = x
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ x phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.replace_empty_Holds | [181, 1] | [345, 19] | subst c1 | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h1 : v ≠ x → ¬isFreeIn v phi
h2_left : v ≠ x
h2_right : ¬isBoundIn v phi
c1 : u = x
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ x phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ x phi) | D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
¬isFreeIn v F →
¬isBoundIn v F →
(Holds D I (Function.updateITE V u a) tail✝ F ↔
Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
¬isFreeIn v phi →
¬isBoundIn v phi →
(Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
V : VarAssignment D
h2_right : ¬isBoundIn v phi
h1 : v ≠ u → ¬isFreeIn v phi
h2_left : v ≠ u
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) (exists_ u phi) ↔
Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (exists_ u phi) |
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