url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
apply h2
|
case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
case a.h.e'_2.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
right
|
case a.h.e'_2.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi
|
case a.h.e'_2.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
exact a1
|
case a.h.e'_2.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a.h2.a.h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
phi psi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
psi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) psi ↔ Holds D I' V (head✝ :: tail✝) psi)
V : VarAssignment D
h2 :
∀ (P : PredName) (ds : List D),
predVarOccursIn P ds.length phi ∨ predVarOccursIn P ds.length psi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length psi
⊢ predVarOccursIn P ds.length psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
first | apply forall_congr' | apply exists_congr
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
intro d
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
exact phi_ih (Function.updateITE V x d) h2
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔ Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
apply forall_congr'
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
apply exists_congr
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
|
case h
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I' (Function.updateITE V x a) (head✝ :: tail✝) phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tail✝ F ↔ Holds D I' V tail✝ F)
x : VarName
phi : Formula
phi_ih :
∀ (V : VarAssignment D),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V (head✝ :: tail✝) phi ↔ Holds D I' V (head✝ :: tail✝) phi)
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length phi → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I' (Function.updateITE V x d) (head✝ :: tail✝) phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
split_ifs
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I' V tl (def_ X xs)
|
case pos
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
case neg
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
h✝ : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
⊢ (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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
else Holds D I' V tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
case pos c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn_iff_mem_predVarSet P ds.length] at a1
simp only [hd.h2] at a1
simp at a1
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
case neg c1 =>
apply ih
intro P ds a1
simp only [predVarOccursIn] at a1
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
apply ih
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
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 I' (Function.updateListITE V hd.args (List.map V xs)) tl hd.q
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
intro P ds a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length hd.q
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length hd.q → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
simp only [predVarOccursIn_iff_mem_predVarSet P ds.length] at a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length hd.q
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ hd.q.predVarSet
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length hd.q
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
simp only [hd.h2] at a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ hd.q.predVarSet
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ ∅
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ hd.q.predVarSet
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
simp at a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ ∅
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : X = hd.name ∧ xs.length = hd.args.length
P : PredName
ds : List D
a1 : (P, ds.length) ∈ ∅
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
apply ih
|
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs)
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I' V tl (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
intro P ds a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
⊢ ∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length (def_ X xs) → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_PredVar
|
[175, 1]
|
[236, 40]
|
simp only [predVarOccursIn] at a1
|
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
D : Type
I I' : Interpretation D
h1 : I.pred_const_ = I'.pred_const_
hd : Definition
tl : List Definition
ih :
∀ (V : VarAssignment D) (F : Formula),
(∀ (P : PredName) (ds : List D), predVarOccursIn P ds.length F → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)) →
(Holds D I V tl F ↔ Holds D I' V tl F)
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∀ (P : PredName) (ds : List D), False → (I.pred_var_ P ds ↔ I'.pred_var_ P ds)
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
P : PredName
ds : List D
a1 : predVarOccursIn P ds.length (def_ X xs)
⊢ I.pred_var_ P ds ↔ I'.pred_var_ P ds
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
induction F generalizing V
|
D : Type
I : Interpretation D
V : VarAssignment D
E E' : Env
F : Formula
h1 : ∃ E1, E' = E1 ++ E
h2 : all_def_in_env E F
h3 : E'.nodup_
⊢ Holds D I V E' F ↔ Holds D I V E F
|
case pred_const_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_const_ a✝¹ a✝)
⊢ Holds D I V E' (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_var_ a✝¹ a✝)
⊢ Holds D I V E' (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : all_def_in_env E (eq_ a✝¹ a✝)
⊢ Holds D I V E' (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E true_
⊢ Holds D I V E' true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E false_
⊢ Holds D I V E' false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝.not_
⊢ Holds D I V E' a✝.not_ ↔ Holds D I V E a✝.not_
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.imp_ a✝)
⊢ Holds D I V E' (a✝¹.imp_ a✝) ↔ Holds D I V E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.and_ a✝)
⊢ Holds D I V E' (a✝¹.and_ a✝) ↔ Holds D I V E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.or_ a✝)
⊢ Holds D I V E' (a✝¹.or_ a✝) ↔ Holds D I V E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.iff_ a✝)
⊢ Holds D I V E' (a✝¹.iff_ a✝) ↔ Holds D I V E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (forall_ a✝¹ a✝)
⊢ Holds D I V E' (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (exists_ a✝¹ a✝)
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
V : VarAssignment D
E E' : Env
F : Formula
h1 : ∃ E1, E' = E1 ++ E
h2 : all_def_in_env E F
h3 : E'.nodup_
⊢ Holds D I V E' F ↔ Holds D I V E F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
any_goals
simp only [all_def_in_env] at h2
simp only [Holds]
|
case pred_const_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_const_ a✝¹ a✝)
⊢ Holds D I V E' (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_var_ a✝¹ a✝)
⊢ Holds D I V E' (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : all_def_in_env E (eq_ a✝¹ a✝)
⊢ Holds D I V E' (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E true_
⊢ Holds D I V E' true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E false_
⊢ Holds D I V E' false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝.not_
⊢ Holds D I V E' a✝.not_ ↔ Holds D I V E a✝.not_
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.imp_ a✝)
⊢ Holds D I V E' (a✝¹.imp_ a✝) ↔ Holds D I V E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.and_ a✝)
⊢ Holds D I V E' (a✝¹.and_ a✝) ↔ Holds D I V E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.or_ a✝)
⊢ Holds D I V E' (a✝¹.or_ a✝) ↔ Holds D I V E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.iff_ a✝)
⊢ Holds D I V E' (a✝¹.iff_ a✝) ↔ Holds D I V E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (forall_ a✝¹ a✝)
⊢ Holds D I V E' (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (exists_ a✝¹ a✝)
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
|
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ ¬Holds D I V E' a✝ ↔ ¬Holds D I V E a✝
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ → Holds D I V E' a✝ ↔ Holds D I V E a✝¹ → Holds D I V E a✝
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ ∧ Holds D I V E' a✝ ↔ Holds D I V E a✝¹ ∧ Holds D I V E a✝
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ Holds D I V E' a✝¹ ∨ Holds D I V E' a✝ ↔ Holds D I V E a✝¹ ∨ Holds D I V E a✝
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝¹ ∧ all_def_in_env E a✝
⊢ (Holds D I V E' a✝¹ ↔ Holds D I V E' a✝) ↔ (Holds D I V E a✝¹ ↔ Holds D I V E a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) E a✝
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∃ d, Holds D I (Function.updateITE V a✝¹ d) E a✝
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pred_const_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_const_ a✝¹ a✝)
⊢ Holds D I V E' (pred_const_ a✝¹ a✝) ↔ Holds D I V E (pred_const_ a✝¹ a✝)
case pred_var_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (pred_var_ a✝¹ a✝)
⊢ Holds D I V E' (pred_var_ a✝¹ a✝) ↔ Holds D I V E (pred_var_ a✝¹ a✝)
case eq_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : VarName
V : VarAssignment D
h2 : all_def_in_env E (eq_ a✝¹ a✝)
⊢ Holds D I V E' (eq_ a✝¹ a✝) ↔ Holds D I V E (eq_ a✝¹ a✝)
case true_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E true_
⊢ Holds D I V E' true_ ↔ Holds D I V E true_
case false_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
V : VarAssignment D
h2 : all_def_in_env E false_
⊢ Holds D I V E' false_ ↔ Holds D I V E false_
case not_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝.not_
⊢ Holds D I V E' a✝.not_ ↔ Holds D I V E a✝.not_
case imp_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.imp_ a✝)
⊢ Holds D I V E' (a✝¹.imp_ a✝) ↔ Holds D I V E (a✝¹.imp_ a✝)
case and_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.and_ a✝)
⊢ Holds D I V E' (a✝¹.and_ a✝) ↔ Holds D I V E (a✝¹.and_ a✝)
case or_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.or_ a✝)
⊢ Holds D I V E' (a✝¹.or_ a✝) ↔ Holds D I V E (a✝¹.or_ a✝)
case iff_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (V : VarAssignment D), all_def_in_env E a✝¹ → (Holds D I V E' a✝¹ ↔ Holds D I V E a✝¹)
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (a✝¹.iff_ a✝)
⊢ Holds D I V E' (a✝¹.iff_ a✝) ↔ Holds D I V E (a✝¹.iff_ a✝)
case forall_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (forall_ a✝¹ a✝)
⊢ Holds D I V E' (forall_ a✝¹ a✝) ↔ Holds D I V E (forall_ a✝¹ a✝)
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E (exists_ a✝¹ a✝)
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
case not_ phi phi_ih =>
congr! 1
exact phi_ih V h2
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
case forall_ x phi phi_ih | exists_ x phi phi_ih =>
first | apply forall_congr' | apply exists_congr
intro d
apply phi_ih
exact h2
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [all_def_in_env] at h2
|
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
|
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, a✝¹ = d.name ∧ a✝.length = d.args.length
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case def_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ a✝¹ a✝)
⊢ Holds D I V E' (def_ a✝¹ a✝) ↔ Holds D I V E (def_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [Holds]
|
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
|
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) E' a✝) ↔ ∃ d, Holds D I (Function.updateITE V a✝¹ d) E a✝
|
Please generate a tactic in lean4 to solve the state.
STATE:
case exists_
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (V : VarAssignment D), all_def_in_env E a✝ → (Holds D I V E' a✝ ↔ Holds D I V E a✝)
V : VarAssignment D
h2 : all_def_in_env E a✝
⊢ Holds D I V E' (exists_ a✝¹ a✝) ↔ Holds D I V E (exists_ a✝¹ a✝)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
congr! 1
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ¬Holds D I V E' phi ↔ ¬Holds D I V E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact phi_ih V h2
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
cases h2
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2 : all_def_in_env E phi ∧ all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi)
|
case intro
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
left✝ : all_def_in_env E phi
right✝ : all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2 : all_def_in_env E phi ∧ all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
congr! 1
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi)
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
case a.h.e'_2.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' psi ↔ Holds D I V E psi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ (Holds D I V E' phi ↔ Holds D I V E' psi) ↔ (Holds D I V E phi ↔ Holds D I V E psi)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact phi_ih V h2_left
|
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_1.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' phi ↔ Holds D I V E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact psi_ih V h2_right
|
case a.h.e'_2.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' psi ↔ Holds D I V E psi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.e'_2.a
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
phi psi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
psi_ih : ∀ (V : VarAssignment D), all_def_in_env E psi → (Holds D I V E' psi ↔ Holds D I V E psi)
V : VarAssignment D
h2_left : all_def_in_env E phi
h2_right : all_def_in_env E psi
⊢ Holds D I V E' psi ↔ Holds D I V E psi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
first | apply forall_congr' | apply exists_congr
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
intro d
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ Holds D I (Function.updateITE V x d) E' phi ↔ Holds D I (Function.updateITE V x d) E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply phi_ih
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ Holds D I (Function.updateITE V x d) E' phi ↔ Holds D I (Function.updateITE V x d) E phi
|
case h.h2
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ all_def_in_env E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ Holds D I (Function.updateITE V x d) E' phi ↔ Holds D I (Function.updateITE V x d) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact h2
|
case h.h2
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ all_def_in_env E phi
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.h2
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
d : D
⊢ all_def_in_env E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply forall_congr'
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) E' phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V x d) E phi
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) E' phi) ↔ ∀ (d : D), Holds D I (Function.updateITE V x d) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply exists_congr
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
|
case h
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ ∀ (a : D), Holds D I (Function.updateITE V x a) E' phi ↔ Holds D I (Function.updateITE V x a) E phi
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
x : VarName
phi : Formula
phi_ih : ∀ (V : VarAssignment D), all_def_in_env E phi → (Holds D I V E' phi ↔ Holds D I V E phi)
V : VarAssignment D
h2 : all_def_in_env E phi
⊢ (∃ d, Holds D I (Function.updateITE V x d) E' phi) ↔ ∃ d, Holds D I (Function.updateITE V x d) E phi
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply Exists.elim h1
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ ∀ (a : Env), E' = a ++ E → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
intro E1 h1_1
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ ∀ (a : Env), E' = a ++ E → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
⊢ ∀ (a : Env), E' = a ++ E → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
clear h1
|
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h1 : ∃ E1, E' = E1 ++ E
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [all_def_in_env] at h2
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : all_def_in_env E (def_ X xs)
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply Exists.elim h2
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ ∀ (a : Definition),
a ∈ E ∧ X = a.name ∧ xs.length = a.args.length → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
intro a h2_1
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ ∀ (a : Definition),
a ∈ E ∧ X = a.name ∧ xs.length = a.args.length → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
⊢ ∀ (a : Definition),
a ∈ E ∧ X = a.name ∧ xs.length = a.args.length → (Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs))
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
clear h2
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
h2 : ∃ d ∈ E, X = d.name ∧ xs.length = d.args.length
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [Env.nodup_] at h3
|
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E E' : Env
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) E'
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : E'.nodup_
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
subst h1_1
|
D : Type
I : Interpretation D
E E' : Env
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) E'
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1 ++ E)
⊢ Holds D I V (E1 ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E E' : Env
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) E'
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
h1_1 : E' = E1 ++ E
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V E' (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
induction E1
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1 ++ E)
⊢ Holds D I V (E1 ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
case nil
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
case cons
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (tail✝ ++ E) →
(Holds D I V (tail✝ ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (head✝ :: tail✝ ++ E)
⊢ Holds D I V (head✝ :: tail✝ ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
E1 : Env
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1 ++ E)
⊢ Holds D I V (E1 ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
case nil =>
simp
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) ([] ++ E)
⊢ Holds D I V ([] ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp at h3
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_hd :: E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 : List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_hd :: E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
cases h2_1
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
left✝ : a ∈ E
right✝ : X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a : Definition
h2_1 : a ∈ E ∧ X = a.name ∧ xs.length = a.args.length
E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
cases h2_1_right
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right : X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
left✝ : X = a.name
right✝ : xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right : X = a.name ∧ xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
cases h3
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
left✝ : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
right✝ : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h3 :
(∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length) ∧
List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [Holds]
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then
Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q
else Holds D I V (E1_tl ++ E) (def_ X xs)) ↔
Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ Holds D I V (E1_hd :: (E1_tl ++ E)) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
split_ifs
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then
Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q
else Holds D I V (E1_tl ++ E) (def_ X xs)) ↔
Holds D I V E (def_ X xs)
|
case pos
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h✝ : X = E1_hd.name ∧ xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
case neg
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
h✝ : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
⊢ (if X = E1_hd.name ∧ xs.length = E1_hd.args.length then
Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q
else Holds D I V (E1_tl ++ E) (def_ X xs)) ↔
Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
case _ c1 =>
exact E1_ih h3_right
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
cases c1
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : X = E1_hd.name ∧ xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
|
case intro
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
left✝ : X = E1_hd.name
right✝ : xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : X = E1_hd.name ∧ xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exfalso
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ False
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ Holds D I (Function.updateListITE V E1_hd.args (List.map V xs)) (E1_tl ++ E) E1_hd.q ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
apply h3_left a
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ False
|
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E1_tl ∨ a ∈ E
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.name = a.name
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = a.args.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ False
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
right
|
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E1_tl ∨ a ∈ E
|
case a.h
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E1_tl ∨ a ∈ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact h2_1_left
|
case a.h
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ a ∈ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
subst c1_left
|
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.name = a.name
|
case a
D : Type
I : Interpretation D
E : Env
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
h2_1_left : a ∈ E
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_right : xs.length = E1_hd.args.length
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ E1_hd.name xs) ↔ Holds D I V E (def_ E1_hd.name xs))
h2_1_right_left : E1_hd.name = a.name
⊢ E1_hd.name = a.name
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.name = a.name
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact h2_1_right_left
|
case a
D : Type
I : Interpretation D
E : Env
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
h2_1_left : a ∈ E
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_right : xs.length = E1_hd.args.length
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ E1_hd.name xs) ↔ Holds D I V E (def_ E1_hd.name xs))
h2_1_right_left : E1_hd.name = a.name
⊢ E1_hd.name = a.name
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
E : Env
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
h2_1_left : a ∈ E
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_right : xs.length = E1_hd.args.length
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ E1_hd.name xs) ↔ Holds D I V E (def_ E1_hd.name xs))
h2_1_right_left : E1_hd.name = a.name
⊢ E1_hd.name = a.name
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
trans List.length xs
|
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = a.args.length
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = xs.length
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = a.args.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = a.args.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
simp only [eq_comm]
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = xs.length
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = E1_hd.args.length
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ E1_hd.args.length = xs.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact c1_right
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = E1_hd.args.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = E1_hd.args.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact h2_1_right_right
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = a.args.length
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1_left : X = E1_hd.name
c1_right : xs.length = E1_hd.args.length
⊢ xs.length = a.args.length
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Semantics.lean
|
FOL.NV.Holds_coincide_Env
|
[239, 1]
|
[316, 35]
|
exact E1_ih h3_right
|
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
D : Type
I : Interpretation D
E : Env
X : DefName
xs : List VarName
V : VarAssignment D
a E1_hd : Definition
E1_tl : List Definition
E1_ih :
List.Pairwise (fun d1 d2 => d1.name = d2.name → d1.args.length = d2.args.length → False) (E1_tl ++ E) →
(Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs))
h2_1_left : a ∈ E
h2_1_right_left : X = a.name
h2_1_right_right : xs.length = a.args.length
h3_left : ∀ (a' : Definition), a' ∈ E1_tl ∨ a' ∈ E → E1_hd.name = a'.name → ¬E1_hd.args.length = a'.args.length
h3_right : List.Pairwise (fun d1 d2 => d1.name = d2.name → ¬d1.args.length = d2.args.length) (E1_tl ++ E)
c1 : ¬(X = E1_hd.name ∧ xs.length = E1_hd.args.length)
⊢ Holds D I V (E1_tl ++ E) (def_ X xs) ↔ Holds D I V E (def_ X xs)
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
apply Finset.union_subset_iff.mpr
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ∪ B ⊆ C ∪ D
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ∪ B ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
constructor
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D
|
case left
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D
case right
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D ∧ B ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
trans C
|
case left
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ C ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
exact h1
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ A ⊆ C
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
exact Finset.subset_union_left C D
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ C ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ C ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
trans D
|
case right
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ C ∪ D
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ D
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ D ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
exact h2
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ B ⊆ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right
|
[8, 1]
|
[23, 44]
|
exact Finset.subset_union_right C D
|
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ D ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D : Finset α
h1 : A ⊆ C
h2 : B ⊆ D
⊢ D ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
apply Finset.union_subset_iff.mpr
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ∪ B ⊆ C ∪ D ∪ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ∪ B ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
constructor
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ C ∪ D ∪ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E ∧ B ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
trans C ∪ E
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ∪ E ⊆ C ∪ D ∪ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
exact h1
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ A ⊆ C ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
apply Finset.union_subset_union_left
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ∪ E ⊆ C ∪ D ∪ E
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ∪ E ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
exact Finset.subset_union_left C D
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ C ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
trans D ∪ E
|
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ C ∪ D ∪ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ D ∪ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ∪ E ⊆ C ∪ D ∪ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
exact h2
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ D ∪ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ B ⊆ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
apply Finset.union_subset_union_left
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ∪ E ⊆ C ∪ D ∪ E
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ⊆ C ∪ D
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ∪ E ⊆ C ∪ D ∪ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_union_left_right
|
[26, 1]
|
[43, 42]
|
exact Finset.subset_union_right C D
|
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C ∪ E
h2 : B ⊆ D ∪ E
⊢ D ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
apply Finset.union_subset_iff.mpr
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ∪ B ⊆ (C ∪ D) \ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ∪ B ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
constructor
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ (C ∪ D) \ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E ∧ B ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
trans C \ E
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ C \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C \ E ⊆ (C ∪ D) \ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
exact h1
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ C \ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ A ⊆ C \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
apply Finset.sdiff_subset_sdiff
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C \ E ⊆ (C ∪ D) \ E
|
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C ⊆ C ∪ D
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C \ E ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
exact Finset.subset_union_left C D
|
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ C ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
rfl
|
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
trans D \ E
|
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ (C ∪ D) \ E
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ D \ E
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D \ E ⊆ (C ∪ D) \ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
exact h2
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ D \ E
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ B ⊆ D \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
apply Finset.sdiff_subset_sdiff
|
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D \ E ⊆ (C ∪ D) \ E
|
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D ⊆ C ∪ D
case hvu
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ E ⊆ E
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D \ E ⊆ (C ∪ D) \ E
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_diff
|
[46, 1]
|
[65, 12]
|
exact Finset.subset_union_right C D
|
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D ⊆ C ∪ D
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case hst
α : Type
inst✝ : DecidableEq α
A B C D E : Finset α
h1 : A ⊆ C \ E
h2 : B ⊆ D \ E
⊢ D ⊆ C ∪ D
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
apply Finset.union_subset_iff.mpr
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ∪ B ⊆ E ∪ (C ∪ D) \ F
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ∪ B ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/Finset.lean
|
Finset.union_subset_left_right_diff
|
[68, 1]
|
[89, 12]
|
constructor
|
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F
|
case left
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F
case right
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ B ⊆ E ∪ (C ∪ D) \ F
|
Please generate a tactic in lean4 to solve the state.
STATE:
α : Type
inst✝ : DecidableEq α
A B C D E F : Finset α
h1 : A ⊆ E ∪ C \ F
h2 : B ⊆ E ∪ D \ F
⊢ A ⊆ E ∪ (C ∪ D) \ F ∧ B ⊆ E ∪ (C ∪ D) \ F
TACTIC:
|
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