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/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case _ c2 =>
subst c1
simp only [Function.updateITE_idem]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 c2 : x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case _ c2 =>
subst c1
simp only [Function.updateITE_idem]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
subst c1
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 c2 : x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
x : VarName
phi : Formula
V : Valuation D
d d' : D
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
phi_ih :
NotFree Ξ x phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V x d) M E phi
c2 : x = x
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Function.updateITE_idem]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
x : VarName
phi : Formula
V : Valuation D
d d' : D
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
phi_ih :
NotFree Ξ x phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V x d) M E phi
c2 : x = x
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
subst c1
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
x : VarName
phi : Formula
V : Valuation D
d d' : D
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
phi_ih :
NotFree Ξ x phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V x d) M E phi
c2 : NotFree Ξ x phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Function.updateITE_idem]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
x : VarName
phi : Formula
V : Valuation D
d d' : D
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
phi_ih :
NotFree Ξ x phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V x d) M E phi
c2 : NotFree Ξ x phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
cases h1
|
case neg
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
h1 : x = v β¨ NotFree Ξ v phi
V : Valuation D
d d' : D
c1 : Β¬x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
case neg.inl
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
hβ : x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
case neg.inr
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
hβ : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case _ c2 =>
contradiction
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case _ c2 =>
simp only [β Function.updateITE_comm V x v d d' c1]
exact phi_ih c2 (Function.updateITE V x d') d
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
contradiction
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : x = v
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [β Function.updateITE_comm V x v d d' c1]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x d') M E phi
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d') v d) M E phi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
exact phi_ih c2 (Function.updateITE V x d') d
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
x : VarName
phi : Formula
phi_ih :
NotFree Ξ v phi β β (V : Valuation D) (d : D), Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
V : Valuation D
d d' : D
c1 : Β¬x = v
c2 : NotFree Ξ v phi
β’ Holds D I (Function.updateITE V x d') M E phi β Holds D I (Function.updateITE (Function.updateITE V x d') v d) M E phi
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
induction E
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
v : VarName
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
β’ IsNotFree D I M E (def_ X xs) v
|
case nil
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
β’ IsNotFree D I M [] (def_ X xs) v
case cons
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
headβ : Definition
tailβ : List Definition
tail_ihβ :
(β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M tailβ (meta_var_ X) v) β IsNotFree D I M tailβ (def_ X xs) v
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M (headβ :: tailβ) (meta_var_ X) v
β’ IsNotFree D I M (headβ :: tailβ) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case nil =>
unfold IsNotFree
intro V d
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
β’ IsNotFree D I M [] (def_ X xs) v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold IsNotFree
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
β’ IsNotFree D I M [] (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
β’ β (V : Valuation D) (d : D), Holds D I V M [] (def_ X xs) β Holds D I (Function.updateITE V v d) M [] (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro V d
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
β’ β (V : Valuation D) (d : D), Holds D I V M [] (def_ X xs) β Holds D I (Function.updateITE V v d) M [] (def_ X xs)
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
V : Valuation D
d : D
β’ Holds D I V M [] (def_ X xs) β Holds D I (Function.updateITE V v d) M [] (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M [] (meta_var_ X) v
V : Valuation D
d : D
β’ Holds D I V M [] (def_ X xs) β Holds D I (Function.updateITE V v d) M [] (def_ X xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold NotFree at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
h1 : NotFree Ξ v (def_ X xs)
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v
h1 : v β xs
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold IsNotFree at h2
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M (E_hd :: E_tl) (meta_var_ X) v
h1 : v β xs
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h2 :
β (X : MetaVarName),
(v, X) β Ξ β
β (V : Valuation D) (d : D),
Holds D I V M (E_hd :: E_tl) (meta_var_ X) β Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (meta_var_ X)
h1 : v β xs
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Holds] at h2
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h2 :
β (X : MetaVarName),
(v, X) β Ξ β
β (V : Valuation D) (d : D),
Holds D I V M (E_hd :: E_tl) (meta_var_ X) β Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (meta_var_ X)
h1 : v β xs
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold IsNotFree at E_ih
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih : (β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E_tl (meta_var_ X) v) β IsNotFree D I M E_tl (def_ X xs) v
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
(β (X : MetaVarName),
(v, X) β Ξ β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (meta_var_ X) β Holds D I (Function.updateITE V v d) M E_tl (meta_var_ X)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Holds] at E_ih
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
(β (X : MetaVarName),
(v, X) β Ξ β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (meta_var_ X) β Holds D I (Function.updateITE V v d) M E_tl (meta_var_ X)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold IsNotFree
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ IsNotFree D I M (E_hd :: E_tl) (def_ X xs) v
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ β (V : Valuation D) (d : D),
Holds D I V M (E_hd :: E_tl) (def_ X xs) β Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ β (V : Valuation D) (d : D),
Holds D I V M (E_hd :: E_tl) (def_ X xs) β Holds D I (Function.updateITE V v d) M (E_hd :: E_tl) (def_ X xs)
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ β (V : Valuation D) (d : D),
(if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V v d) E_hd.args (List.map (Function.updateITE V v d) xs))
M E_tl E_hd.F
else Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro V a
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
β’ β (V : Valuation D) (d : D),
(if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V v d) E_hd.args (List.map (Function.updateITE V v d) xs))
M E_tl E_hd.F
else Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M
E_tl E_hd.F
else Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
split_ifs
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X xs)) β
if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M
E_tl E_hd.F
else Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
|
case pos
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M
E_tl E_hd.F
case neg
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
hβ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case _ c1 =>
exact E_ih h2 V a
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
apply holds_coincide_var D I (Function.updateListITE V E_hd.args (List.map V xs)) (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M E_tl E_hd.F E_hd.args E_hd.nf
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs)) M
E_tl E_hd.F
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ β v_1 β E_hd.args,
Function.updateListITE V E_hd.args (List.map V xs) v_1 =
Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v_1
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro v' a1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ β v_1 β E_hd.args,
Function.updateListITE V E_hd.args (List.map V xs) v_1 =
Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v_1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ Function.updateListITE V E_hd.args (List.map V xs) v' =
Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v'
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
apply Function.updateListITE_map_updateIte V (Function.updateITE V v a)
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ Function.updateListITE V E_hd.args (List.map V xs) v' =
Function.updateListITE (Function.updateITE V v a) E_hd.args (List.map (Function.updateITE V v a) xs) v'
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ β y β xs, Β¬y = v
case h2
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ E_hd.args.length = xs.length
case h3
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ v' β E_hd.args
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro y a2 contra
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ β y β xs, Β¬y = v
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
y : VarName
a2 : y β xs
contra : y = v
β’ False
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
subst contra
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
y : VarName
a2 : y β xs
contra : y = v
β’ False
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
y : VarName
a2 : y β xs
h1 : y β xs
h2 : β (X : MetaVarName), (y, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V y d)
E_ih :
(β (X : MetaVarName), (y, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V y d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V y d) M E_tl (def_ X xs)
β’ False
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
contradiction
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
y : VarName
a2 : y β xs
h1 : y β xs
h2 : β (X : MetaVarName), (y, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V y d)
E_ih :
(β (X : MetaVarName), (y, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V y d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V y d) M E_tl (def_ X xs)
β’ False
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
cases c1
|
case h2
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ E_hd.args.length = xs.length
|
case h2.intro
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
v' : VarName
a1 : v' β E_hd.args
leftβ : X = E_hd.name
rightβ : xs.length = E_hd.args.length
β’ E_hd.args.length = xs.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case intro c1_left c1_right =>
simp only [c1_right]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
v' : VarName
a1 : v' β E_hd.args
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ E_hd.args.length = xs.length
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [c1_right]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
v' : VarName
a1 : v' β E_hd.args
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ E_hd.args.length = xs.length
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
exact a1
|
case h3
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
v' : VarName
a1 : v' β E_hd.args
β’ v' β E_hd.args
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
exact E_ih h2 V a
|
D : Type
I : Interpretation D
M : MetaValuation D
Ξ : List (VarName Γ MetaVarName)
v : VarName
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
h1 : v β xs
h2 : β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)
E_ih :
(β (X : MetaVarName), (v, X) β Ξ β β (V : Valuation D) (d : D), M X V β M X (Function.updateITE V v d)) β
β (V : Valuation D) (d : D),
Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v d) M E_tl (def_ X xs)
V : Valuation D
a : D
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I V M E_tl (def_ X xs) β Holds D I (Function.updateITE V v a) M E_tl (def_ X xs)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
intro v X a1
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
β’ β (v : VarName) (X : MetaVarName),
(v, X) β Ξ β IsNotFree D I (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X) v
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ IsNotFree D I (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
unfold IsNotFree
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ IsNotFree D I (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X) v
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ β (V : Valuation D) (d : D),
Holds D I V (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X) β
Holds D I (Function.updateITE V v d) (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ β (V : Valuation D) (d : D),
Holds D I V (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X) β
Holds D I (Function.updateITE V v d) (fun X V' => Holds D I (V' β Ο') M E (Ο X)) E (meta_var_ X)
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ β (V : Valuation D) (d : D), Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
intro V d
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
β’ β (V : Valuation D) (d : D), Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
cases h1
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h1 : βΟ β Ο' = id β§ Ο' β βΟ = id
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
case intro
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
leftβ : βΟ β Ο' = id
rightβ : Ο' β βΟ = id
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
simp only [Function.updateITE_comp_right Ο' Ο.1 V v d h1_left h1_right]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE V v d β Ο') M E (Ο X)
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE (V β Ο') (βΟ v) d) M E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
apply not_free_imp_is_not_free D I M E (Ο X) Ξ' (Ο.1 v)
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ Holds D I (V β Ο') M E (Ο X) β Holds D I (Function.updateITE (V β Ο') (βΟ v) d) M E (Ο X)
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ NotFree Ξ' (βΟ v) (Ο X)
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ β (X : MetaVarName), (βΟ v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) (βΟ v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
exact h3 v X a1
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ NotFree Ξ' (βΟ v) (Ο X)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
intro X' a2
|
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
β’ β (X : MetaVarName), (βΟ v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) (βΟ v)
|
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
X' : MetaVarName
a2 : (βΟ v, X') β Ξ'
β’ IsNotFree D I M E (meta_var_ X') (βΟ v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_1
|
[1737, 1]
|
[1763, 29]
|
exact h2 (Ο.1 v) X' a2
|
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ Ξ' : List (VarName Γ MetaVarName)
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ' β IsNotFree D I M E (meta_var_ X) v
h3 : β (v : VarName) (X : MetaVarName), (v, X) β Ξ β NotFree Ξ' (βΟ v) (Ο X)
v : VarName
X : MetaVarName
a1 : (v, X) β Ξ
V : Valuation D
d : D
h1_left : βΟ β Ο' = id
h1_right : Ο' β βΟ = id
X' : MetaVarName
a2 : (βΟ v, X') β Ξ'
β’ IsNotFree D I M E (meta_var_ X') (βΟ v)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
induction F
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
F : Formula
h1 : IsMetaVarOrAllDefInEnv E F
h2 : β X β F.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο F)
|
case meta_var_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβ : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ aβ)
h2 : β X β (meta_var_ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ aβ))
case pred_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : PredName
aβ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ aβΒΉ aβ)
h2 : β X β (pred_ aβΒΉ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ aβΒΉ aβ))
case eq_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ aβ : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ aβΒΉ aβ)
h2 : β X β (eq_ aβΒΉ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ aβΒΉ aβ))
case true_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
case not_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβ : Formula
a_ihβ :
IsMetaVarOrAllDefInEnv E aβ β
(β X β aβ.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ)
h1 : IsMetaVarOrAllDefInEnv E aβ.not_
h2 : β X β aβ.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ.not_)
case imp_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ aβ : Formula
a_ihβΒΉ :
IsMetaVarOrAllDefInEnv E aβΒΉ β
(β X β aβΒΉ.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο aβΒΉ)
a_ihβ :
IsMetaVarOrAllDefInEnv E aβ β
(β X β aβ.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ)
h1 : IsMetaVarOrAllDefInEnv E (aβΒΉ.imp_ aβ)
h2 : β X β (aβΒΉ.imp_ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (aβΒΉ.imp_ aβ))
case forall_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : VarName
aβ : Formula
a_ihβ :
IsMetaVarOrAllDefInEnv E aβ β
(β X β aβ.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο aβ)
h1 : IsMetaVarOrAllDefInEnv E (forall_ aβΒΉ aβ)
h2 : β X β (forall_ aβΒΉ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ aβΒΉ aβ))
case def_
E : Env
Ο : Instantiation
Ο : MetaInstantiation
aβΒΉ : DefName
aβ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ aβΒΉ aβ)
h2 : β X β (def_ aβΒΉ aβ).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ aβΒΉ aβ))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case meta_var_ X =>
unfold Formula.metaVarSet at h2
simp at h2
unfold sub
exact h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : β X_1 β (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case pred_ X xs =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case eq_ x y =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case true_ =>
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case not_ phi phi_ih =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold Formula.metaVarSet at h2
unfold sub
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
h2 : β X β phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case forall_ x phi phi_ih =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold Formula.metaVarSet at h2
unfold sub
unfold IsMetaVarOrAllDefInEnv
exact phi_ih h1 h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi)
h2 : β X β (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
case def_ X xs =>
unfold IsMetaVarOrAllDefInEnv at h1
unfold sub
unfold IsMetaVarOrAllDefInEnv
simp
exact h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X xs)
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold Formula.metaVarSet at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : β X_1 β (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : β X_1 β {X}, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : β X_1 β {X}, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (meta_var_ X))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
exact h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (pred_ X xs))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (βΟ) xs))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ True
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : β X_1 β (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ True
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (eq_ x y))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (eq_ (βΟ x) (βΟ y))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (eq_ (βΟ x) (βΟ y))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ True
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : β X β (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ True
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο true_)
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E true_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E true_
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ True
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp only
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : β X β true_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ True
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
h2 : β X β phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h2 : β X β phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold Formula.metaVarSet at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h2 : β X β phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi.not_)
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi).not_
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
exact phi_ih h1 h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi)
h2 : β X β (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β X β (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold Formula.metaVarSet at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β X β (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β X β phi.metaVarSet βͺ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β X β phi.metaVarSet βͺ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (phi.imp_ psi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E ((sub Ο Ο phi).imp_ (sub Ο Ο psi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E ((sub Ο Ο phi).imp_ (sub Ο Ο psi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
cases h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h1 : IsMetaVarOrAllDefInEnv E phi β§ IsMetaVarOrAllDefInEnv E psi
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
case intro
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
leftβ : IsMetaVarOrAllDefInEnv E phi
rightβ : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
constructor
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi) β§ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
apply phi_ih h1_left
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
intro X a1
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
apply h2
|
case left
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
case left.a
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ X β phi.metaVarSet β¨ X β psi.metaVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
left
|
case left.a
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ X β phi.metaVarSet β¨ X β psi.metaVarSet
|
case left.a.h
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ X β phi.metaVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
exact a1
|
case left.a.h
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β phi.metaVarSet
β’ X β phi.metaVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
apply psi_ih h1_right
|
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
|
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
intro X a1
|
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
β’ β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
|
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
apply h2
|
case right
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ IsMetaVarOrAllDefInEnv E (Ο X)
|
case right.a
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ X β phi.metaVarSet β¨ X β psi.metaVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
right
|
case right.a
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ X β phi.metaVarSet β¨ X β psi.metaVarSet
|
case right.a.h
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ X β psi.metaVarSet
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
exact a1
|
case right.a.h
E : Env
Ο : Instantiation
Ο : MetaInstantiation
phi psi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
psi_ih :
IsMetaVarOrAllDefInEnv E psi β
(β X β psi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο psi)
h2 : β (X : MetaVarName), X β phi.metaVarSet β¨ X β psi.metaVarSet β IsMetaVarOrAllDefInEnv E (Ο X)
h1_left : IsMetaVarOrAllDefInEnv E phi
h1_right : IsMetaVarOrAllDefInEnv E psi
X : MetaVarName
a1 : X β psi.metaVarSet
β’ X β psi.metaVarSet
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi)
h2 : β X β (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h2 : β X β (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold Formula.metaVarSet at h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h2 : β X β (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
h1 : IsMetaVarOrAllDefInEnv E phi
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (forall_ x phi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (forall_ (βΟ x) (sub Ο Ο phi))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (forall_ (βΟ x) (sub Ο Ο phi))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
exact phi_ih h1 h2
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
IsMetaVarOrAllDefInEnv E phi β
(β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)) β IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : β X β phi.metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X xs)
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold sub
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (sub Ο Ο (def_ X xs))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
unfold IsMetaVarOrAllDefInEnv
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ IsMetaVarOrAllDefInEnv E (def_ X (List.map (βΟ) xs))
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ (List.map (βΟ) xs).length = d.args.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.lem_2_a
|
[1766, 1]
|
[1837, 13]
|
simp
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ (List.map (βΟ) xs).length = d.args.length
|
E : Env
Ο : Instantiation
Ο : MetaInstantiation
X : DefName
xs : List VarName
h2 : β X_1 β (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (Ο X_1)
h1 : β d β E, X = d.name β§ xs.length = d.args.length
β’ β d β E, X = d.name β§ xs.length = d.args.length
|
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