url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ Holds D I V (head✝ :: tail✝) (eq_ x x') ↔ Holds D I V' (head✝ :: tail✝) (eq_ y y') | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x = V x' ↔ V' y = V' y' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x = V x' ↔ V' y = V' y' | case a.h.e'_2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x = V' y
case a.h.e'_3
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x' = V' y' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact aux_1 D binders x y V V' h1 h2_left | case a.h.e'_2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x = V' y | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact aux_1 D binders x' y' V V' h1 h2_right | case a.h.e'_3
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x x' : VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y y' : VarName
h2_left : isAlphaEqvVar binders x y
h2_right : isAlphaEqvVar binders x' y'
⊢ V x' = V' y' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
h2 : True
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ Holds D I V (head✝ :: tail✝) phi.not_ ↔ Holds D I V' (head✝ :: tail✝) phi'.not_ | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ ¬Holds D I V (head✝ :: tail✝) phi ↔ ¬Holds D I V' (head✝ :: tail✝) phi' | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact phi_ih V V' phi' binders h1 h2 | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' : Formula
h2 : isAlphaEqvAux binders phi phi'
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2 : isAlphaEqvAux binders phi phi' ∧ isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi') | case intro
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
left✝ : isAlphaEqvAux binders phi phi'
right✝ : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) (phi.iff_ psi) ↔ Holds D I V' (head✝ :: tail✝) (phi'.iff_ psi') | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔
(Holds D I V' (head✝ :: tail✝) phi' ↔ Holds D I V' (head✝ :: tail✝) psi') |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | congr! 1 | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V (head✝ :: tail✝) psi) ↔
(Holds D I V' (head✝ :: tail✝) phi' ↔ Holds D I V' (head✝ :: tail✝) psi') | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi'
case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact phi_ih V V' phi' binders h1 h2_left | case a.h.e'_1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact psi_ih V V' psi' binders h1 h2_right | case a.h.e'_2.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
phi psi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
psi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders psi F' → (Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
phi' psi' : Formula
h2_left : isAlphaEqvAux binders phi phi'
h2_right : isAlphaEqvAux binders psi psi'
⊢ Holds D I V (head✝ :: tail✝) psi ↔ Holds D I V' (head✝ :: tail✝) psi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ Holds D I V (head✝ :: tail✝) (exists_ x phi) ↔ Holds D I V' (head✝ :: tail✝) (exists_ y phi') | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | first | apply forall_congr' | apply exists_congr | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | intro d | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi' | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | induction h1 | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
d : D
⊢ Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' | case h.nil
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
V✝ : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ Holds D I (Function.updateITE V✝ x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V✝ y d) (head✝ :: tail✝) phi'
case h.cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
binders✝ : List (VarName × VarName)
x✝ y✝ : VarName
V✝ V'✝ : VarAssignment D
d✝ : D
a✝ : AlphaEqvVarAssignment D binders✝ V✝ V'✝
a_ih✝ :
isAlphaEqvAux ((x, y) :: binders✝) phi phi' →
(Holds D I (Function.updateITE V✝ x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V'✝ y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (x✝, y✝) :: binders✝) phi phi'
⊢ Holds D I (Function.updateITE (Function.updateITE V✝ x✝ d✝) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE V'✝ y✝ d✝) y d) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply forall_congr' | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ (∀ (d : D), Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∀ (d : D), Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply exists_congr | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ (∃ d, Holds D I (Function.updateITE V x d) (head✝ :: tail✝) phi) ↔
∃ d, Holds D I (Function.updateITE V' y d) (head✝ :: tail✝) phi' | case h
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
y : VarName
phi' : Formula
h2 : isAlphaEqvAux ((x, y) :: binders) phi phi'
⊢ ∀ (a : D),
Holds D I (Function.updateITE V x a) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE V' y a) (head✝ :: tail✝) phi' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply phi_ih | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V y d) (head✝ :: tail✝) phi' | case h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE h1_V x d) (Function.updateITE h1_V y d)
case h2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ isAlphaEqvAux ?binders phi phi'
case binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ List (VarName × VarName) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply AlphaEqvVarAssignment.cons | case h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE h1_V x d) (Function.updateITE h1_V y d) | case h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ AlphaEqvVarAssignment D ?h1.binders h1_V h1_V
case h1.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ List (VarName × VarName) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply AlphaEqvVarAssignment.nil | case h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ AlphaEqvVarAssignment D ?h1.binders h1_V h1_V
case h1.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ List (VarName × VarName) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact h2 | case h2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_V : VarAssignment D
h2 : isAlphaEqvAux [(x, y)] phi phi'
⊢ isAlphaEqvAux [(x, y)] phi phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply phi_ih | D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ Holds D I (Function.updateITE (Function.updateITE h1_V h1_x h1_d) x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE (Function.updateITE h1_V' h1_y h1_d) y d) (head✝ :: tail✝) phi' | case h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE (Function.updateITE h1_V h1_x h1_d) x d)
(Function.updateITE (Function.updateITE h1_V' h1_y h1_d) y d)
case h2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ isAlphaEqvAux ?binders phi phi'
case binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ List (VarName × VarName) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply AlphaEqvVarAssignment.cons | case h1
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?binders (Function.updateITE (Function.updateITE h1_V h1_x h1_d) x d)
(Function.updateITE (Function.updateITE h1_V' h1_y h1_d) y d) | case h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?h1.binders (Function.updateITE h1_V h1_x h1_d) (Function.updateITE h1_V' h1_y h1_d)
case h1.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ List (VarName × VarName) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply AlphaEqvVarAssignment.cons | case h1.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?h1.binders (Function.updateITE h1_V h1_x h1_d) (Function.updateITE h1_V' h1_y h1_d)
case h1.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ List (VarName × VarName) | case h1.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?h1.a.binders h1_V h1_V'
case h1.a.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ List (VarName × VarName) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact h1_1 | case h1.a.a
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ AlphaEqvVarAssignment D ?h1.a.binders h1_V h1_V'
case h1.a.binders
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ List (VarName × VarName) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact h2 | case h2
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F')
x : VarName
phi : Formula
phi_ih :
∀ (V V' : VarAssignment D) (F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' →
isAlphaEqvAux binders phi F' → (Holds D I V (head✝ :: tail✝) phi ↔ Holds D I V' (head✝ :: tail✝) F')
V V' : VarAssignment D
binders : List (VarName × VarName)
y : VarName
phi' : Formula
d : D
h1_binders : List (VarName × VarName)
h1_x h1_y : VarName
h1_V h1_V' : VarAssignment D
h1_d : D
h1_1 : AlphaEqvVarAssignment D h1_binders h1_V h1_V'
a_ih✝ :
isAlphaEqvAux ((x, y) :: h1_binders) phi phi' →
(Holds D I (Function.updateITE h1_V x d) (head✝ :: tail✝) phi ↔
Holds D I (Function.updateITE h1_V' y d) (head✝ :: tail✝) phi')
h2 : isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi'
⊢ isAlphaEqvAux ((x, y) :: (h1_x, h1_y) :: h1_binders) phi phi' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
a✝³ : DefName
a✝² : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
a✝¹ : DefName
a✝ : List VarName
h2 : a✝³ = a✝¹ ∧ isAlphaEqvVarList binders a✝² a✝
⊢ Holds D I V [] (def_ a✝³ a✝²) ↔ Holds D I V' [] (def_ a✝¹ a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [Holds] | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
⊢ Holds D I V (hd :: tl) (def_ X xs) ↔ Holds D I V' (hd :: tl) (def_ Y ys) | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
⊢ (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 Y = hd.name ∧ ys.length = hd.args.length then
Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q
else Holds D I V' tl (def_ Y ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | split_ifs | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
⊢ (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 Y = hd.name ∧ ys.length = hd.args.length then
Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q
else Holds D I V' tl (def_ Y ys) | case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
h✝¹ : X = hd.name ∧ xs.length = hd.args.length
h✝ : Y = hd.name ∧ ys.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' ys)) tl hd.q
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
h✝¹ : X = hd.name ∧ xs.length = hd.args.length
h✝ : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys)
case pos
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
h✝¹ : ¬(X = hd.name ∧ xs.length = hd.args.length)
h✝ : Y = hd.name ∧ ys.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q
case neg
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
h✝¹ : ¬(X = hd.name ∧ xs.length = hd.args.length)
h✝ : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ Y ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case _ c1 c2 =>
cases h2
case intro h2_left h2_right =>
simp only [isAlphaEqvVarList_length binders xs ys h2_right] at c1
subst h2_left
contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case _ c1 c2 =>
cases h2
case intro h2_left h2_right =>
simp only [← isAlphaEqvVarList_length binders xs ys h2_right] at c2
subst h2_left
contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : Y = hd.name ∧ ys.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case _ c1 c2 =>
exact ih V V' (def_ X xs) (def_ Y ys) binders h1 h2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.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' ys)) tl hd.q | case intro
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
left✝ : X = Y
right✝ : isAlphaEqvVarList binders xs ys
⊢ 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' ys)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply Holds_coincide_Var | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ 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' ys)) tl hd.q | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' ys) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | intro v a1 | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ ∀ (v : VarName),
isFreeIn v hd.q →
Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' ys) v | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' ys) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [aux_2 D binders xs ys V V' h1 h2_right] | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V xs) v = Function.updateListITE V' hd.args (List.map V' ys) v | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V' ys) v = Function.updateListITE V' hd.args (List.map V' ys) v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply Function.updateListITE_mem_eq_len | case h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ Function.updateListITE V hd.args (List.map V' ys) v = Function.updateListITE V' hd.args (List.map V' ys) v | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ v ∈ hd.args
case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ hd.args.length = (List.map V' ys).length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [isFreeIn_iff_mem_freeVarSet] at a1 | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ v ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [← List.mem_toFinset] | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args.toFinset |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | apply Finset.mem_of_subset hd.h1 a1 | case h1.h1
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : v ∈ hd.q.freeVarSet
⊢ v ∈ hd.args.toFinset | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp | case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ hd.args.length = (List.map V' ys).length | case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ hd.args.length = ys.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [eq_comm] | case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ hd.args.length = ys.length | case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ ys.length = hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases c2 | case h1.h2
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
⊢ ys.length = hd.args.length | case h1.h2.intro
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
left✝ : Y = hd.name
right✝ : ys.length = hd.args.length
⊢ ys.length = hd.args.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case intro c2_left c2_right =>
exact c2_right | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
c2_left : Y = hd.name
c2_right : ys.length = hd.args.length
⊢ ys.length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact c2_right | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
v : VarName
a1 : isFreeIn v hd.q
c2_left : Y = hd.name
c2_right : ys.length = hd.args.length
⊢ ys.length = hd.args.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) | case intro
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
left✝ : X = Y
right✝ : isAlphaEqvVarList binders xs ys
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case intro h2_left h2_right =>
simp only [isAlphaEqvVarList_length binders xs ys h2_right] at c1
subst h2_left
contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [isAlphaEqvVarList_length binders xs ys h2_right] at c1 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : X = hd.name ∧ xs.length = hd.args.length
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ ys.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | subst h2_left | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ ys.length = hd.args.length
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ Y ys) | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ ys.length = hd.args.length
c2 : ¬(X = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ X ys) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
h2_right : isAlphaEqvVarList binders xs ys
c1 : X = hd.name ∧ ys.length = hd.args.length
c2 : ¬(X = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I (Function.updateListITE V hd.args (List.map V xs)) tl hd.q ↔ Holds D I V' tl (def_ X ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | cases h2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : Y = hd.name ∧ ys.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | case intro
D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : Y = hd.name ∧ ys.length = hd.args.length
left✝ : X = Y
right✝ : isAlphaEqvVarList binders xs ys
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | case intro h2_left h2_right =>
simp only [← isAlphaEqvVarList_length binders xs ys h2_right] at c2
subst h2_left
contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | simp only [← isAlphaEqvVarList_length binders xs ys h2_right] at c2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : Y = hd.name ∧ ys.length = hd.args.length
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
c2 : Y = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | subst h2_left | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
h2_left : X = Y
h2_right : isAlphaEqvVarList binders xs ys
c2 : Y = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
h2_right : isAlphaEqvVarList binders xs ys
c2 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | contradiction | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
ys : List VarName
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
h2_right : isAlphaEqvVarList binders xs ys
c2 : X = hd.name ∧ xs.length = hd.args.length
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I (Function.updateListITE V' hd.args (List.map V' ys)) tl hd.q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isAlphaEqv_Holds_aux | [624, 1] | [734, 58] | exact ih V V' (def_ X xs) (def_ Y ys) binders h1 h2 | D : Type
I : Interpretation D
hd : Definition
tl : List Definition
ih :
∀ (V V' : VarAssignment D) (F F' : Formula) (binders : List (VarName × VarName)),
AlphaEqvVarAssignment D binders V V' → isAlphaEqvAux binders F F' → (Holds D I V tl F ↔ Holds D I V' tl F')
X : DefName
xs : List VarName
V V' : VarAssignment D
binders : List (VarName × VarName)
h1 : AlphaEqvVarAssignment D binders V V'
Y : DefName
ys : List VarName
h2 : X = Y ∧ isAlphaEqvVarList binders xs ys
c1 : ¬(X = hd.name ∧ xs.length = hd.args.length)
c2 : ¬(Y = hd.name ∧ ys.length = hd.args.length)
⊢ Holds D I V tl (def_ X xs) ↔ Holds D I V' tl (def_ Y ys) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isalphaEqv_Holds | [737, 1] | [748, 76] | simp only [isAlphaEqv] at h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F F' : Formula
h1 : isAlphaEqv F F'
⊢ Holds D I V E F ↔ Holds D I V E F' | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F F' : Formula
h1 : isAlphaEqvAux [] F F'
⊢ Holds D I V E F ↔ Holds D I V E F' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Alpha.lean | FOL.NV.isalphaEqv_Holds | [737, 1] | [748, 76] | exact isAlphaEqv_Holds_aux D I V V E F F' [] AlphaEqvVarAssignment.nil h1 | D : Type
I : Interpretation D
V : VarAssignment D
E : Env
F F' : Formula
h1 : isAlphaEqvAux [] F F'
⊢ Holds D I V E F ↔ Holds D I V E F' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp only [match_char_EpsilonNFA] | α : Type
inst✝ : DecidableEq α
c : α
⊢ (match_char_EpsilonNFA c).toAbstract = match_char_AbstractEpsilonNFA c | α : Type
inst✝ : DecidableEq α
c : α
⊢ { symbol_arrow_list := [{ start_state := 0, symbol := c, stop_state_list := [1] }], epsilon_arrow_list := [],
starting_state_list := [0], accepting_state_list := [1] }.toAbstract =
match_char_AbstractEpsilonNFA c |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp only [EpsilonNFA.toAbstract] | α : Type
inst✝ : DecidableEq α
c : α
⊢ { symbol_arrow_list := [{ start_state := 0, symbol := c, stop_state_list := [1] }], epsilon_arrow_list := [],
starting_state_list := [0], accepting_state_list := [1] }.toAbstract =
match_char_AbstractEpsilonNFA c | α : Type
inst✝ : DecidableEq α
c : α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
[{ start_state := 0, symbol := c, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
match_char_AbstractEpsilonNFA c |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp only [match_char_AbstractEpsilonNFA] | α : Type
inst✝ : DecidableEq α
c : α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
[{ start_state := 0, symbol := c, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
match_char_AbstractEpsilonNFA c | α : Type
inst✝ : DecidableEq α
c : α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
[{ start_state := 0, symbol := c, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
{ symbol := fun p a q => p = 0 ∧ a = c ∧ q = 1, epsilon := fun x x => False, start := fun p => p = 0,
accepting := fun p => p = 1 } |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp | α : Type
inst✝ : DecidableEq α
c : α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
[{ start_state := 0, symbol := c, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
{ symbol := fun p a q => p = 0 ∧ a = c ∧ q = 1, epsilon := fun x x => False, start := fun p => p = 0,
accepting := fun p => p = 1 } | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, (start_state = 0 ∧ symbol = c ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p a q => p = 0 ∧ a = c ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp only [← and_assoc] | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, (start_state = 0 ∧ symbol = c ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p a q => p = 0 ∧ a = c ∧ q = 1 | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, ((start_state = 0 ∧ symbol = c) ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p a q => (p = 0 ∧ a = c) ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp only [and_right_comm] | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, ((start_state = 0 ∧ symbol = c) ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p a q => (p = 0 ∧ a = c) ∧ q = 1 | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, ((start_state = 0 ∧ symbol = c) ∧ stop_state ∈ stop_state_list) ∧ stop_state_list = [1]) =
fun p a q => (p = 0 ∧ a = c) ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_char_EpsilonNFA_toAbstract | [50, 1] | [63, 9] | simp | α : Type
inst✝ : DecidableEq α
c : α
⊢ (fun start_state symbol stop_state =>
∃ stop_state_list, ((start_state = 0 ∧ symbol = c) ∧ stop_state ∈ stop_state_list) ∧ stop_state_list = [1]) =
fun p a q => (p = 0 ∧ a = c) ∧ q = 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp only [match_epsilon_EpsilonNFA] | α : Type
inst✝ : DecidableEq α
⊢ (match_epsilon_EpsilonNFA α).toAbstract = match_epsilon_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ { symbol_arrow_list := [], epsilon_arrow_list := [{ start_state := 0, stop_state_list := [1] }],
starting_state_list := [0], accepting_state_list := [1] }.toAbstract =
match_epsilon_AbstractEpsilonNFA α |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp only [EpsilonNFA.toAbstract] | α : Type
inst✝ : DecidableEq α
⊢ { symbol_arrow_list := [], epsilon_arrow_list := [{ start_state := 0, stop_state_list := [1] }],
starting_state_list := [0], accepting_state_list := [1] }.toAbstract =
match_epsilon_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
[{ start_state := 0, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
match_epsilon_AbstractEpsilonNFA α |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp only [match_epsilon_AbstractEpsilonNFA] | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
[{ start_state := 0, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
match_epsilon_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
[{ start_state := 0, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
{ symbol := fun x x x => False, epsilon := fun p q => p = 0 ∧ q = 1, start := fun p => p = 0,
accepting := fun p => p = 1 } |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
[{ start_state := 0, stop_state_list := [1] }] ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [1] } =
{ symbol := fun x x x => False, epsilon := fun p q => p = 0 ∧ q = 1, start := fun p => p = 0,
accepting := fun p => p = 1 } | α : Type
inst✝ : DecidableEq α
⊢ (fun start_state stop_state =>
∃ stop_state_list, (start_state = 0 ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p q => p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | funext p q | α : Type
inst✝ : DecidableEq α
⊢ (fun start_state stop_state =>
∃ stop_state_list, (start_state = 0 ∧ stop_state_list = [1]) ∧ stop_state ∈ stop_state_list) =
fun p q => p = 0 ∧ q = 1 | case h.h
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) = (p = 0 ∧ q = 1) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp | case h.h
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) = (p = 0 ∧ q = 1) | case h.h
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) ↔ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | constructor | case h.h
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) ↔ p = 0 ∧ q = 1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) → p = 0 ∧ q = 1
case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ p = 0 ∧ q = 1 → ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | intro a1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ (∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list) → p = 0 ∧ q = 1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | apply Exists.elim a1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ ∀ (a : List ℕ), (p = 0 ∧ a = [1]) ∧ q ∈ a → p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | intro stop_state_list a2 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ ∀ (a : List ℕ), (p = 0 ∧ a = [1]) ∧ q ∈ a → p = 0 ∧ q = 1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
stop_state_list : List ℕ
a2 : (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | clear a1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
stop_state_list : List ℕ
a2 : (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2 : (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | cases a2 | case h.h.mp
α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2 : (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 | case h.h.mp.intro
α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
left✝ : p = 0 ∧ stop_state_list = [1]
right✝ : q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | case _ a2_left a2_right =>
cases a2_left
case _ a2_left_left a2_left_right =>
simp only [a2_left_right] at a2_right
simp at a2_right
tauto | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left : p = 0 ∧ stop_state_list = [1]
a2_right : q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | cases a2_left | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left : p = 0 ∧ stop_state_list = [1]
a2_right : q ∈ stop_state_list
⊢ p = 0 ∧ q = 1 | case intro
α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_right : q ∈ stop_state_list
left✝ : p = 0
right✝ : stop_state_list = [1]
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | case _ a2_left_left a2_left_right =>
simp only [a2_left_right] at a2_right
simp at a2_right
tauto | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_right : q ∈ stop_state_list
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
⊢ p = 0 ∧ q = 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp only [a2_left_right] at a2_right | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_right : q ∈ stop_state_list
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
⊢ p = 0 ∧ q = 1 | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
a2_right : q ∈ [1]
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp at a2_right | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
a2_right : q ∈ [1]
⊢ p = 0 ∧ q = 1 | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
a2_right : q = 1
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | tauto | α : Type
inst✝ : DecidableEq α
p q : ℕ
stop_state_list : List ℕ
a2_left_left : p = 0
a2_left_right : stop_state_list = [1]
a2_right : q = 1
⊢ p = 0 ∧ q = 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | intro a1 | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
⊢ p = 0 ∧ q = 1 → ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | apply Exists.intro [1] | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ ∃ stop_state_list, (p = 0 ∧ stop_state_list = [1]) ∧ q ∈ stop_state_list | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ (p = 0 ∧ [1] = [1]) ∧ q ∈ [1] |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | simp | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ (p = 0 ∧ [1] = [1]) ∧ q ∈ [1] | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ p = 0 ∧ q = 1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_epsilon_EpsilonNFA_toAbstract | [176, 1] | [204, 15] | exact a1 | case h.h.mpr
α : Type
inst✝ : DecidableEq α
p q : ℕ
a1 : p = 0 ∧ q = 1
⊢ p = 0 ∧ q = 1 | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_zero_EpsilonNFA_toAbstract | [237, 1] | [245, 9] | simp only [match_zero_EpsilonNFA] | α : Type
inst✝ : DecidableEq α
⊢ (match_zero_EpsilonNFA α).toAbstract = match_zero_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ { symbol_arrow_list := [], epsilon_arrow_list := [], starting_state_list := [0],
accepting_state_list := [] }.toAbstract =
match_zero_AbstractEpsilonNFA α |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_zero_EpsilonNFA_toAbstract | [237, 1] | [245, 9] | simp only [EpsilonNFA.toAbstract] | α : Type
inst✝ : DecidableEq α
⊢ { symbol_arrow_list := [], epsilon_arrow_list := [], starting_state_list := [0],
accepting_state_list := [] }.toAbstract =
match_zero_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [] } =
match_zero_AbstractEpsilonNFA α |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_zero_EpsilonNFA_toAbstract | [237, 1] | [245, 9] | simp only [match_zero_AbstractEpsilonNFA] | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [] } =
match_zero_AbstractEpsilonNFA α | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [] } =
{ symbol := fun x x x => False, epsilon := fun x x => False, start := fun p => p = 0, accepting := fun x => False } |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_zero_EpsilonNFA_toAbstract | [237, 1] | [245, 9] | simp | α : Type
inst✝ : DecidableEq α
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈ [] ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ [] ∧ stop_state ∈ stop_state_list,
start := fun state => state ∈ [0], accepting := fun state => state ∈ [] } =
{ symbol := fun x x x => False, epsilon := fun x x => False, start := fun p => p = 0, accepting := fun x => False } | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [match_union_EpsilonNFA] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ (match_union_EpsilonNFA α σ_0 σ_1 M_0 M_1).toAbstract =
match_union_AbstractEpsilonNFA α σ_0 σ_1 M_0.toAbstract M_1.toAbstract | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol_arrow_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list,
epsilon_arrow_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list,
starting_state_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting_state_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list }.toAbstract =
match_union_AbstractEpsilonNFA α σ_0 σ_1 M_0.toAbstract M_1.toAbstract |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [EpsilonNFA.toAbstract] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol_arrow_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list,
epsilon_arrow_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list,
starting_state_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting_state_list :=
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list }.toAbstract =
match_union_AbstractEpsilonNFA α σ_0 σ_1 M_0.toAbstract M_1.toAbstract | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state =>
state ∈ (EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting := fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list } =
match_union_AbstractEpsilonNFA α σ_0 σ_1
{
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
M_0.symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ M_0.starting_state_list,
accepting := fun state => state ∈ M_0.accepting_state_list }
{
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
M_1.symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ M_1.starting_state_list,
accepting := fun state => state ∈ M_1.accepting_state_list } |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [match_union_AbstractEpsilonNFA] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state =>
state ∈ (EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting := fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list } =
match_union_AbstractEpsilonNFA α σ_0 σ_1
{
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
M_0.symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ M_0.starting_state_list,
accepting := fun state => state ∈ M_0.accepting_state_list }
{
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
M_1.symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state => state ∈ M_1.starting_state_list,
accepting := fun state => state ∈ M_1.accepting_state_list } | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state =>
state ∈ (EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting := fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list } =
{
symbol := fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False,
epsilon := fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False,
start := fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list,
accepting := fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list } |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [AbstractEpsilonNFA.mk.injEq] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ {
symbol := fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list,
epsilon := fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list,
start := fun state =>
state ∈ (EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list,
accepting := fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list } =
{
symbol := fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False,
epsilon := fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False,
start := fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list,
accepting := fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list } | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [EpsilonNFA.wrapLeft] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.wrapLeft σ_1 M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.wrapLeft σ_1 M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [EpsilonNFA.wrapRight] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).symbol_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).epsilon_arrow_list ++ (EpsilonNFA.wrapRight σ_0 M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).starting_state_list ++ (EpsilonNFA.wrapRight σ_0 M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).accepting_state_list ++
(EpsilonNFA.wrapRight σ_0 M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).symbol_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).epsilon_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).starting_state_list ++ (EpsilonNFA.map Sum.inr M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).accepting_state_list ++ (EpsilonNFA.map Sum.inr M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/Parsing/RegExpToEpsilonNFA.lean | match_union_EpsilonNFA_toAbstract | [312, 1] | [506, 17] | simp only [EpsilonNFA.map] | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).symbol_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
(EpsilonNFA.map Sum.inl M_0).epsilon_arrow_list ++ (EpsilonNFA.map Sum.inr M_1).epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).starting_state_list ++ (EpsilonNFA.map Sum.inr M_1).starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state =>
state ∈
(EpsilonNFA.map Sum.inl M_0).accepting_state_list ++ (EpsilonNFA.map Sum.inr M_1).accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list | α : Type
inst✝ : DecidableEq α
σ_0 σ_1 : Type
M_0 : EpsilonNFA α σ_0
M_1 : EpsilonNFA α σ_1
⊢ ((fun start_state symbol stop_state =>
∃ stop_state_list,
{ start_state := start_state, symbol := symbol, stop_state_list := stop_state_list } ∈
List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state, symbol := arrow.symbol,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.symbol_arrow_list ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state, symbol := arrow.symbol,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.symbol_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p c q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_0.symbol_arrow_list ∧
q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', symbol := c, stop_state_list := stop_state_list } ∈ M_1.symbol_arrow_list ∧
q' ∈ stop_state_list
| x => False) ∧
((fun start_state stop_state =>
∃ stop_state_list,
{ start_state := start_state, stop_state_list := stop_state_list } ∈
List.map
(fun arrow =>
{ start_state := Sum.inl arrow.start_state,
stop_state_list := List.map Sum.inl arrow.stop_state_list })
M_0.epsilon_arrow_list ++
List.map
(fun arrow =>
{ start_state := Sum.inr arrow.start_state,
stop_state_list := List.map Sum.inr arrow.stop_state_list })
M_1.epsilon_arrow_list ∧
stop_state ∈ stop_state_list) =
fun p q =>
match (p, q) with
| (Sum.inl p', Sum.inl q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_0.epsilon_arrow_list ∧ q' ∈ stop_state_list
| (Sum.inr p', Sum.inr q') =>
∃ stop_state_list,
{ start_state := p', stop_state_list := stop_state_list } ∈ M_1.epsilon_arrow_list ∧ q' ∈ stop_state_list
| x => False) ∧
((fun state => state ∈ List.map Sum.inl M_0.starting_state_list ++ List.map Sum.inr M_1.starting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.starting_state_list
| Sum.inr p' => p' ∈ M_1.starting_state_list) ∧
(fun state => state ∈ List.map Sum.inl M_0.accepting_state_list ++ List.map Sum.inr M_1.accepting_state_list) =
fun p =>
match p with
| Sum.inl p' => p' ∈ M_0.accepting_state_list
| Sum.inr p' => p' ∈ M_1.accepting_state_list |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.