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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/NV/Margaris/Prop.lean | FOL.NV.evalPrime_prime | [136, 1] | [146, 8] | induction F | F : Formula
V : VarBoolAssignment
h1 : F.IsPrime
⊢ evalPrime V F = (V F = true) | case pred_const_
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_const_ a✝¹ a✝).IsPrime
⊢ evalPrime V (pred_const_ a✝¹ a✝) = (V (pred_const_ a✝¹ a✝) = true)
case pred_var_
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : (pred_var_ a✝¹ a✝).IsPrime
⊢ evalPrime V (pred_var_ a✝¹ a✝) = (V (pred_var_ a✝¹ a✝) = true)
case eq_
V : VarBoolAssignment
a✝¹ a✝ : VarName
h1 : (eq_ a✝¹ a✝).IsPrime
⊢ evalPrime V (eq_ a✝¹ a✝) = (V (eq_ a✝¹ a✝) = true)
case true_
V : VarBoolAssignment
h1 : true_.IsPrime
⊢ evalPrime V true_ = (V true_ = true)
case false_
V : VarBoolAssignment
h1 : false_.IsPrime
⊢ evalPrime V false_ = (V false_ = true)
case not_
V : VarBoolAssignment
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : a✝.not_.IsPrime
⊢ evalPrime V a✝.not_ = (V a✝.not_ = true)
case imp_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.imp_ a✝).IsPrime
⊢ evalPrime V (a✝¹.imp_ a✝) = (V (a✝¹.imp_ a✝) = true)
case and_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.and_ a✝).IsPrime
⊢ evalPrime V (a✝¹.and_ a✝) = (V (a✝¹.and_ a✝) = true)
case or_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.or_ a✝).IsPrime
⊢ evalPrime V (a✝¹.or_ a✝) = (V (a✝¹.or_ a✝) = true)
case iff_
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true)
case forall_
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (forall_ a✝¹ a✝).IsPrime
⊢ evalPrime V (forall_ a✝¹ a✝) = (V (forall_ a✝¹ a✝) = true)
case exists_
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (exists_ a✝¹ a✝).IsPrime
⊢ evalPrime V (exists_ a✝¹ a✝) = (V (exists_ a✝¹ a✝) = true)
case def_
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_prime | [136, 1] | [146, 8] | case true_ | false_ | not_ | imp_ | and_ | or_ | iff_ =>
simp only [Formula.IsPrime] at h1 | V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_prime | [136, 1] | [146, 8] | case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
rfl | V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_prime | [136, 1] | [146, 8] | simp only [Formula.IsPrime] at h1 | V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : a✝¹.IsPrime → evalPrime V a✝¹ = (V a✝¹ = true)
a_ih✝ : a✝.IsPrime → evalPrime V a✝ = (V a✝ = true)
h1 : (a✝¹.iff_ a✝).IsPrime
⊢ evalPrime V (a✝¹.iff_ a✝) = (V (a✝¹.iff_ a✝) = true) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_prime | [136, 1] | [146, 8] | rfl | V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : (def_ a✝¹ a✝).IsPrime
⊢ evalPrime V (def_ a✝¹ a✝) = (V (def_ a✝¹ a✝) = true) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | induction F | F : Formula
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ F) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) F | case pred_const_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (pred_const_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (pred_const_ a✝¹ a✝)
case pred_var_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (pred_var_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (pred_var_ a✝¹ a✝)
case eq_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : VarName
⊢ evalPrime V (substPrime σ (eq_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (eq_ a✝¹ a✝)
case true_
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ true_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) true_
case false_
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_
case not_
σ : Formula → Formula
V : VarBoolAssignment
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ a✝.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝.not_
case imp_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.imp_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.imp_ a✝)
case and_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.and_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.and_ a✝)
case or_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.or_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.or_ a✝)
case iff_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : evalPrime V (substPrime σ a✝¹) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝¹
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (a✝¹.iff_ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (a✝¹.iff_ a✝)
case forall_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (forall_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (forall_ a✝¹ a✝)
case exists_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : evalPrime V (substPrime σ a✝) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) a✝
⊢ evalPrime V (substPrime σ (exists_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (exists_ a✝¹ a✝)
case def_
σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | case pred_const_ | pred_var_ | eq_ | forall_ | exists_ | def_ =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
simp | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | case true_ | false_ =>
rfl | σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | case not_ phi phi_ih =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
congr! 1 | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp only [Formula.substPrime]
simp only [Formula.evalPrime]
congr! 1 | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V (substPrime σ (phi.iff_ psi)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.substPrime] | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (substPrime σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝) | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.evalPrime] | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (def_ a✝¹ a✝) | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ decide (evalPrime V (σ (def_ a✝¹ a✝))) = true |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp | σ : Formula → Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
⊢ evalPrime V (σ (def_ a✝¹ a✝)) ↔ decide (evalPrime V (σ (def_ a✝¹ a✝))) = true | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | rfl | σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ false_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) false_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.substPrime] | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi.not_) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_ | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi).not_ ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.evalPrime] | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ evalPrime V (substPrime σ phi).not_ ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi.not_ | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ ¬evalPrime V (substPrime σ phi) ↔ ¬evalPrime (fun H => decide (evalPrime V (σ H))) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | congr! 1 | σ : Formula → Formula
V : VarBoolAssignment
phi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
⊢ ¬evalPrime V (substPrime σ phi) ↔ ¬evalPrime (fun H => decide (evalPrime V (σ H))) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.substPrime] | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V (substPrime σ (phi.iff_ psi)) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi) | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V ((substPrime σ phi).iff_ (substPrime σ psi)) ↔
evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | simp only [Formula.evalPrime] | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ evalPrime V ((substPrime σ phi).iff_ (substPrime σ psi)) ↔
evalPrime (fun H => decide (evalPrime V (σ H))) (phi.iff_ psi) | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ (evalPrime V (substPrime σ phi) ↔ evalPrime V (substPrime σ psi)) ↔
(evalPrime (fun H => decide (evalPrime V (σ H))) phi ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.evalPrime_substPrime_eq_evalPrime_evalPrime | [193, 1] | [218, 13] | congr! 1 | σ : Formula → Formula
V : VarBoolAssignment
phi psi : Formula
phi_ih : evalPrime V (substPrime σ phi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) phi
psi_ih : evalPrime V (substPrime σ psi) ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi
⊢ (evalPrime V (substPrime σ phi) ↔ evalPrime V (substPrime σ psi)) ↔
(evalPrime (fun H => decide (evalPrime V (σ H))) phi ↔ evalPrime (fun H => decide (evalPrime V (σ H))) psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime | [221, 1] | [232, 11] | simp only [Formula.IsTautoPrime] at h1 | P : Formula
h1 : P.IsTautoPrime
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime | [221, 1] | [232, 11] | simp only [Formula.IsTautoPrime] | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ (substPrime σ P).IsTautoPrime | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (substPrime σ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime | [221, 1] | [232, 11] | intro V | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (substPrime σ P) | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime | [221, 1] | [232, 11] | simp only [evalPrime_substPrime_eq_evalPrime_evalPrime P σ V] | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime V (substPrime σ P) | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime (fun H => decide (evalPrime V (σ H))) P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.isTautoPrime_imp_isTautoPrime_substPrime | [221, 1] | [232, 11] | apply h1 | P : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V P
σ : Formula → Formula
V : VarBoolAssignment
⊢ evalPrime (fun H => decide (evalPrime V (σ H))) P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | simp only [IsProof] | P : Formula
⊢ IsProof (P.imp_ P) | P : Formula
⊢ IsDeduct ∅ (P.imp_ P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | apply IsDeduct.mp_ (P.imp_ (P.imp_ P)) | P : Formula
⊢ IsDeduct ∅ (P.imp_ P) | case a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))
case a
P : Formula
⊢ IsDeduct ∅ (P.imp_ (P.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | apply IsDeduct.mp_ (P.imp_ ((P.imp_ P).imp_ P)) | case a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)) | case a.a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P)))
case a.a
P : Formula
⊢ IsDeduct ∅ (P.imp_ ((P.imp_ P).imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | apply IsDeduct.axiom_ | case a.a
P : Formula
⊢ IsDeduct ∅ ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))) | case a.a.a
P : Formula
⊢ IsAxiom ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | exact IsAxiom.prop_2_ P (P.imp_ P) P | case a.a.a
P : Formula
⊢ IsAxiom ((P.imp_ ((P.imp_ P).imp_ P)).imp_ ((P.imp_ (P.imp_ P)).imp_ (P.imp_ P))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | apply IsDeduct.axiom_ | case a.a
P : Formula
⊢ IsDeduct ∅ (P.imp_ ((P.imp_ P).imp_ P)) | case a.a.a
P : Formula
⊢ IsAxiom (P.imp_ ((P.imp_ P).imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | exact IsAxiom.prop_1_ P (P.imp_ P) | case a.a.a
P : Formula
⊢ IsAxiom (P.imp_ ((P.imp_ P).imp_ P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | apply IsDeduct.axiom_ | case a
P : Formula
⊢ IsDeduct ∅ (P.imp_ (P.imp_ P)) | case a.a
P : Formula
⊢ IsAxiom (P.imp_ (P.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_5 | [253, 1] | [265, 30] | exact IsAxiom.prop_1_ P P | case a.a
P : Formula
⊢ IsAxiom (P.imp_ (P.imp_ P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.mp_ (P.not_.imp_ (Q.not_.imp_ P.not_)) | P Q : Formula
⊢ IsProof (P.not_.imp_ (P.imp_ Q)) | case a
P Q : Formula
⊢ IsDeduct ∅ ((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))
case a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ (Q.not_.imp_ P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.mp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))) | case a
P Q : Formula
⊢ IsDeduct ∅ ((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))) | case a.a
P Q : Formula
⊢ IsDeduct ∅
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q))))
case a.a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.axiom_ | case a.a
P Q : Formula
⊢ IsDeduct ∅
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))) | case a.a.a
P Q : Formula
⊢ IsAxiom
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsAxiom.prop_2_ | case a.a.a
P Q : Formula
⊢ IsAxiom
((P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))).imp_
((P.not_.imp_ (Q.not_.imp_ P.not_)).imp_ (P.not_.imp_ (P.imp_ Q)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.mp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)) | case a.a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))) | case a.a.a
P Q : Formula
⊢ IsDeduct ∅ (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q))))
case a.a.a
P Q : Formula
⊢ IsDeduct ∅ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.axiom_ | case a.a.a
P Q : Formula
⊢ IsDeduct ∅ (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))) | case a.a.a.a
P Q : Formula
⊢ IsAxiom (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsAxiom.prop_1_ | case a.a.a.a
P Q : Formula
⊢ IsAxiom (((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)).imp_ (P.not_.imp_ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.axiom_ | case a.a.a
P Q : Formula
⊢ IsDeduct ∅ ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)) | case a.a.a.a
P Q : Formula
⊢ IsAxiom ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsAxiom.prop_3_ | case a.a.a.a
P Q : Formula
⊢ IsAxiom ((Q.not_.imp_ P.not_).imp_ (P.imp_ Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsDeduct.axiom_ | case a
P Q : Formula
⊢ IsDeduct ∅ (P.not_.imp_ (Q.not_.imp_ P.not_)) | case a.a
P Q : Formula
⊢ IsAxiom (P.not_.imp_ (Q.not_.imp_ P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_13_6_no_deduct | [270, 1] | [284, 26] | apply IsAxiom.prop_1_ | case a.a
P Q : Formula
⊢ IsAxiom (P.not_.imp_ (Q.not_.imp_ P.not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | intro Γ | F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) F | F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
Γ : Set Formula
⊢ IsDeduct (Δ ∪ Γ) F |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | induction h1 | F : Formula
Δ : Set Formula
h1 : IsDeduct Δ F
Γ : Set Formula
⊢ IsDeduct (Δ ∪ Γ) F | case axiom_
F : Formula
Δ Γ : Set Formula
phi✝ : Formula
a✝ : IsAxiom phi✝
⊢ IsDeduct (Δ ∪ Γ) phi✝
case assume_
F : Formula
Δ Γ : Set Formula
phi✝ : Formula
a✝ : phi✝ ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) phi✝
case mp_
F : Formula
Δ Γ : Set Formula
phi✝ psi✝ : Formula
a✝¹ : IsDeduct Δ (phi✝.imp_ psi✝)
a✝ : IsDeduct Δ phi✝
a_ih✝¹ : IsDeduct (Δ ∪ Γ) (phi✝.imp_ psi✝)
a_ih✝ : IsDeduct (Δ ∪ Γ) phi✝
⊢ IsDeduct (Δ ∪ Γ) psi✝ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | case axiom_ h1_phi h1_1 =>
apply IsDeduct.axiom_
exact h1_1 | F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | case assume_ h1_phi h1_1 =>
apply IsDeduct.assume_
simp
left
exact h1_1 | F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | apply IsDeduct.axiom_ | F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | exact h1_1 | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | apply IsDeduct.assume_ | F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ IsDeduct (Δ ∪ Γ) h1_phi | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∪ Γ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | simp | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∪ Γ | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∨ h1_phi ∈ Γ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | left | case a
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ ∨ h1_phi ∈ Γ | case a.h
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | exact h1_1 | case a.h
F : Formula
Δ Γ : Set Formula
h1_phi : Formula
h1_1 : h1_phi ∈ Δ
⊢ h1_phi ∈ Δ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | apply IsDeduct.mp_ h1_phi | F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_psi | case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | exact h1_ih_1 | case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10 | [287, 1] | [306, 20] | exact h1_ih_2 | case a
F : Formula
Δ Γ : Set Formula
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Δ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Δ h1_phi
h1_ih_1 : IsDeduct (Δ ∪ Γ) (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct (Δ ∪ Γ) h1_phi
⊢ IsDeduct (Δ ∪ Γ) h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10_comm | [309, 1] | [316, 23] | simp only [Set.union_comm] | Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Γ ∪ Δ) Q | Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_10_comm | [309, 1] | [316, 23] | exact T_14_10 Q Δ h1 | Q : Formula
Δ : Set Formula
h1 : IsDeduct Δ Q
⊢ ∀ (Γ : Set Formula), IsDeduct (Δ ∪ Γ) Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_11 | [319, 1] | [327, 11] | intro Δ | P : Formula
h1 : IsProof P
⊢ ∀ (Δ : Set Formula), IsDeduct Δ P | P : Formula
h1 : IsProof P
Δ : Set Formula
⊢ IsDeduct Δ P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_11 | [319, 1] | [327, 11] | obtain s1 := T_14_10 P ∅ h1 Δ | P : Formula
h1 : IsProof P
Δ : Set Formula
⊢ IsDeduct Δ P | P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct (∅ ∪ Δ) P
⊢ IsDeduct Δ P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_11 | [319, 1] | [327, 11] | simp at s1 | P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct (∅ ∪ Δ) P
⊢ IsDeduct Δ P | P : Formula
h1 : IsProof P
Δ : Set Formula
s1 : IsDeduct Δ P
⊢ IsDeduct Δ P |
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