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.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E (forall_ x phi) β
Holds D I V M E (forall_ (βΟ x) (sub Ο Ο phi))
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
β’ (β (d : D), Holds D I (Function.updateITE (V β βΟ) x d) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi) β
β (d : D), Holds D I (Function.updateITE V (βΟ x) d) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply forall_congr'
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
β’ (β (d : D), Holds D I (Function.updateITE (V β βΟ) x d) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi) β
β (d : D), Holds D I (Function.updateITE V (βΟ x) d) M E (sub Ο Ο phi)
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
β’ β (a : D),
Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
intro a
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
β’ β (a : D),
Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
have s1 : Function.updateITE V (Ο.val x) a β Ο.val = Function.updateITE (V β Ο.val) x a
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
case s1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply Function.updateITE_comp_right_injective
|
case s1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
case s1.h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Function.Injective βΟ
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply Instantiation.is_injective
|
case s1.h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
β’ Function.Injective βΟ
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [β s1]
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE (V β βΟ) x a) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE V (βΟ x) a β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
exact phi_ih (Function.updateITE V (Ο.val x) a) h1
|
case h
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
β (V : Valuation D),
IsMetaVarOrAllDefInEnv E phi β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β Holds D I V M E (sub Ο Ο phi))
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E phi
h2_left : βΟ β Ο' = id
h2_right : Ο' β βΟ = id
a : D
s1 : Function.updateITE V (βΟ x) a β βΟ = Function.updateITE (V β βΟ) x a
β’ Holds D I (Function.updateITE V (βΟ x) a β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E phi β
Holds D I (Function.updateITE V (βΟ x) a) M E (sub Ο Ο phi)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
induction E
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv E (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E (Ο X')) E (def_ X xs) β Holds D I V M E (sub Ο Ο (def_ X xs))
|
case nil
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M [] (Ο X')) [] (def_ X xs) β
Holds D I V M [] (sub Ο Ο (def_ X xs))
case cons
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
headβ : Definition
tailβ : List Definition
tail_ihβ :
IsMetaVarOrAllDefInEnv tailβ (def_ X xs) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M tailβ (Ο X')) tailβ (def_ X xs) β
Holds D I V M tailβ (sub Ο Ο (def_ X xs)))
h1 : IsMetaVarOrAllDefInEnv (headβ :: tailβ) (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (headβ :: tailβ) (Ο X')) (headβ :: tailβ) (def_ X xs) β
Holds D I V M (headβ :: tailβ) (sub Ο Ο (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
case nil =>
unfold IsMetaVarOrAllDefInEnv at h1
simp at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M [] (Ο X')) [] (def_ X xs) β
Holds D I V M [] (sub Ο Ο (def_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : IsMetaVarOrAllDefInEnv [] (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M [] (Ο X')) [] (def_ X xs) β
Holds D I V M [] (sub Ο Ο (def_ X xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : β d β [], X = d.name β§ xs.length = d.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M [] (Ο X')) [] (def_ X xs) β
Holds D I V M [] (sub Ο Ο (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
h1 : β d β [], X = d.name β§ xs.length = d.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M [] (Ο X')) [] (def_ X xs) β
Holds D I V M [] (sub Ο Ο (def_ X xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold IsMetaVarOrAllDefInEnv at E_ih
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
IsMetaVarOrAllDefInEnv E_tl (def_ X xs) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (sub Ο Ο (def_ X xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (sub Ο Ο (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
have s1 : E_hd.F.metaVarSet = β
:=
no_meta_var_imp_meta_var_set_is_empty E_hd.F E_hd.args E_hd.nf
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (sub Ο Ο (def_ X xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (sub Ο Ο (def_ X xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold sub
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (sub Ο Ο (def_ X xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [Holds]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) (E_hd :: E_tl) (def_ X xs) β
Holds D I V M (E_hd :: E_tl) (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F
else Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map (βΟ) xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map (βΟ) xs))) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F
else Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs)) β
if X = E_hd.name β§ (List.map (βΟ) xs).length = E_hd.args.length then
Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map (βΟ) xs))) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F
else Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs)) 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 β βΟ) xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
split_ifs
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
β’ (if X = E_hd.name β§ xs.length = E_hd.args.length then
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F
else Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs)) 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 β βΟ) xs)) M E_tl E_hd.F
else Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
case pos
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
case neg
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
hβ : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
case _ c1 =>
unfold IsMetaVarOrAllDefInEnv at h1
simp at h1
cases h1
case inl c2 =>
contradiction
case inr c2 =>
unfold sub at E_ih
simp only [β E_ih c2]
apply holds_coincide_meta_var
unfold Formula.metaVarSet
simp
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
cases c1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case intro
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
leftβ : X = E_hd.name
rightβ : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
have s2 : Holds D I (Function.updateListITE (V β Ο.val) E_hd.args (List.map (V β Ο.val) xs)) M E_tl E_hd.F β Holds D I (Function.updateListITE V E_hd.args (List.map (V β Ο.val) xs)) M E_tl E_hd.F
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply holds_coincide_var D I (Function.updateListITE (V β Ο.val) E_hd.args (List.map (V β Ο.val) xs)) (Function.updateListITE V E_hd.args (List.map (V β Ο.val) xs)) M E_tl E_hd.F E_hd.args E_hd.nf
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ β v β E_hd.args,
Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs) v =
Function.updateListITE V E_hd.args (List.map (V β βΟ) xs) v
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
intro v a1
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
β’ β v β E_hd.args,
Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs) v =
Function.updateListITE V E_hd.args (List.map (V β βΟ) xs) v
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs) v =
Function.updateListITE V E_hd.args (List.map (V β βΟ) xs) v
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply Function.updateListITE_mem_eq_len
|
case s2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs) v =
Function.updateListITE V E_hd.args (List.map (V β βΟ) xs) v
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case s2.h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ v β E_hd.args
case s2.h2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ E_hd.args.length = (List.map (V β βΟ) xs).length
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [β s2]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply holds_coincide_meta_var_no_meta_var
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs))
(fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl E_hd.F β
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ E_hd.F.metaVarSet = β
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
exact s1
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
s2 :
Holds D I (Function.updateListITE (V β βΟ) E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F β
Holds D I (Function.updateListITE V E_hd.args (List.map (V β βΟ) xs)) M E_tl E_hd.F
β’ E_hd.F.metaVarSet = β
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
exact a1
|
case s2.h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : 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.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp
|
case s2.h2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ E_hd.args.length = (List.map (V β βΟ) xs).length
|
case s2.h2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ E_hd.args.length = xs.length
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [c1_right]
|
case s2.h2
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1_left : X = E_hd.name
c1_right : xs.length = E_hd.args.length
v : VarName
a1 : v β E_hd.args
β’ E_hd.args.length = xs.length
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold IsMetaVarOrAllDefInEnv at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
h1 : IsMetaVarOrAllDefInEnv (E_hd :: E_tl) (def_ X xs)
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
h1 : β d β E_hd :: E_tl, X = d.name β§ xs.length = d.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp at h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
h1 : β d β E_hd :: E_tl, X = d.name β§ xs.length = d.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
h1 : X = E_hd.name β§ xs.length = E_hd.args.length β¨ β a β E_tl, X = a.name β§ xs.length = a.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
cases h1
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
h1 : X = E_hd.name β§ xs.length = E_hd.args.length β¨ β a β E_tl, X = a.name β§ xs.length = a.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
case inl
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
case inr
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
hβ : β a β E_tl, X = a.name β§ xs.length = a.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
case inl c2 =>
contradiction
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
case inr c2 =>
unfold sub at E_ih
simp only [β E_ih c2]
apply holds_coincide_meta_var
unfold Formula.metaVarSet
simp
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
contradiction
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : X = E_hd.name β§ xs.length = E_hd.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold sub at E_ih
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (sub Ο Ο (def_ X xs)))
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp only [β E_ih c2]
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs))
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
apply holds_coincide_meta_var
|
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X')) E_tl (def_ X xs) β
Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs)
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ β (V' : Valuation D),
β X_1 β (def_ X xs).metaVarSet, Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X_1) β Holds D I (V' β Ο') M E_tl (Ο X_1)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
unfold Formula.metaVarSet
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ β (V' : Valuation D),
β X_1 β (def_ X xs).metaVarSet, Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X_1) β Holds D I (V' β Ο') M E_tl (Ο X_1)
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ β (V' : Valuation D), β X β β
, Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X) β Holds D I (V' β Ο') M E_tl (Ο X)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.holds_sub
|
[1465, 1]
|
[1577, 15]
|
simp
|
case h1
D : Type
I : Interpretation D
M : MetaValuation D
Ο : Instantiation
Ο' : VarName β VarName
Ο : MetaInstantiation
h2 : βΟ β Ο' = id β§ Ο' β βΟ = id
X : DefName
xs : List VarName
V : Valuation D
E_hd : Definition
E_tl : List Definition
s1 : E_hd.F.metaVarSet = β
c1 : Β¬(X = E_hd.name β§ xs.length = E_hd.args.length)
c2 : β a β E_tl, X = a.name β§ xs.length = a.args.length
E_ih :
(β d β E_tl, X = d.name β§ xs.length = d.args.length) β
(Holds D I (V β βΟ) (fun X' V' => Holds D I (V' β Ο') M E_tl (Ο X')) E_tl (def_ X xs) β
Holds D I V M E_tl (def_ X (List.map (βΟ) xs)))
β’ β (V' : Valuation D), β X β β
, Holds D I (V' β Ο') M (E_hd :: E_tl) (Ο X) β Holds D I (V' β Ο') M E_tl (Ο X)
|
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 F
|
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
F : Formula
Ξ : List (VarName Γ MetaVarName)
v : VarName
h1 : NotFree Ξ v F
h2 : β (X : MetaVarName), (v, X) β Ξ β IsNotFree D I M E (meta_var_ X) v
β’ IsNotFree D I M E F v
|
case meta_var_
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
aβ : MetaVarName
h1 : NotFree Ξ v (meta_var_ aβ)
β’ IsNotFree D I M E (meta_var_ aβ) v
case pred_
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
aβΒΉ : PredName
aβ : List VarName
h1 : NotFree Ξ v (pred_ aβΒΉ aβ)
β’ IsNotFree D I M E (pred_ aβΒΉ aβ) v
case eq_
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
aβΒΉ aβ : VarName
h1 : NotFree Ξ v (eq_ aβΒΉ aβ)
β’ IsNotFree D I M E (eq_ aβΒΉ aβ) v
case true_
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
h1 : NotFree Ξ v true_
β’ IsNotFree D I M E true_ v
case not_
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
aβ : Formula
a_ihβ : NotFree Ξ v aβ β IsNotFree D I M E aβ v
h1 : NotFree Ξ v aβ.not_
β’ IsNotFree D I M E aβ.not_ v
case imp_
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
aβΒΉ aβ : Formula
a_ihβΒΉ : NotFree Ξ v aβΒΉ β IsNotFree D I M E aβΒΉ v
a_ihβ : NotFree Ξ v aβ β IsNotFree D I M E aβ v
h1 : NotFree Ξ v (aβΒΉ.imp_ aβ)
β’ IsNotFree D I M E (aβΒΉ.imp_ aβ) v
case forall_
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
aβΒΉ : VarName
aβ : Formula
a_ihβ : NotFree Ξ v aβ β IsNotFree D I M E aβ v
h1 : NotFree Ξ v (forall_ aβΒΉ aβ)
β’ IsNotFree D I M E (forall_ aβΒΉ aβ) v
case def_
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
aβΒΉ : DefName
aβ : List VarName
h1 : NotFree Ξ v (def_ aβΒΉ aβ)
β’ IsNotFree D I M E (def_ aβΒΉ aβ) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case meta_var_ X =>
unfold NotFree at h1
exact h2 X h1
|
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 : MetaVarName
h1 : NotFree Ξ v (meta_var_ X)
β’ IsNotFree D I M E (meta_var_ X) 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]
|
case pred_ X xs =>
unfold NotFree at h1
unfold IsNotFree
simp only [Holds]
intro V d
congr! 1
apply List.map_congr
intro x a1
have s1 : Β¬ x = v
intro contra
subst contra
contradiction
unfold Function.updateITE
simp only [if_neg s1]
|
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 : PredName
xs : List VarName
h1 : NotFree Ξ v (pred_ X xs)
β’ IsNotFree D I M E (pred_ 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]
|
case eq_ x y =>
unfold NotFree at h1
unfold IsNotFree
simp only [Holds]
intro V d
cases h1
case intro h1_left h1_right =>
simp only [Function.updateITE]
simp only [if_neg h1_left]
simp only [if_neg h1_right]
|
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 y : VarName
h1 : NotFree Ξ v (eq_ x y)
β’ IsNotFree D I M E (eq_ x y) 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]
|
case true_ =>
unfold IsNotFree
intro V d
simp only [Holds]
|
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
h1 : NotFree Ξ v true_
β’ IsNotFree D I M E true_ 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]
|
case not_ phi phi_ih =>
unfold NotFree at h1
unfold IsNotFree at phi_ih
unfold IsNotFree
intro V d
simp only [Holds]
congr! 1
exact phi_ih h1 V 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
phi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
h1 : NotFree Ξ v phi.not_
β’ IsNotFree D I M E phi.not_ 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 NotFree at h1
|
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 : MetaVarName
h1 : NotFree Ξ v (meta_var_ X)
β’ IsNotFree D I M E (meta_var_ X) v
|
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 : MetaVarName
h1 : (v, X) β Ξ
β’ IsNotFree D I M E (meta_var_ X) v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
exact h2 X h1
|
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 : MetaVarName
h1 : (v, X) β Ξ
β’ IsNotFree D I M E (meta_var_ X) 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 NotFree at h1
|
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 : PredName
xs : List VarName
h1 : NotFree Ξ v (pred_ X xs)
β’ IsNotFree D I M E (pred_ X xs) v
|
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 : PredName
xs : List VarName
h1 : v β xs
β’ IsNotFree D I M E (pred_ 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
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 : PredName
xs : List VarName
h1 : v β xs
β’ IsNotFree D I M E (pred_ X xs) v
|
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 : PredName
xs : List VarName
h1 : v β xs
β’ β (V : Valuation D) (d : D), Holds D I V M E (pred_ X xs) β Holds D I (Function.updateITE V v d) M E (pred_ 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
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 : PredName
xs : List VarName
h1 : v β xs
β’ β (V : Valuation D) (d : D), Holds D I V M E (pred_ X xs) β Holds D I (Function.updateITE V v d) M E (pred_ X xs)
|
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 : PredName
xs : List VarName
h1 : v β xs
β’ β (V : Valuation D) (d : D), I X (List.map V xs) β I X (List.map (Function.updateITE V v d) 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
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 : PredName
xs : List VarName
h1 : v β xs
β’ β (V : Valuation D) (d : D), I X (List.map V xs) β I X (List.map (Function.updateITE V v d) xs)
|
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ I X (List.map V xs) β I X (List.map (Function.updateITE V v d) xs)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
congr! 1
|
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ I X (List.map V xs) β I X (List.map (Function.updateITE V v d) xs)
|
case a.h.e'_2
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ List.map V xs = List.map (Function.updateITE V v d) xs
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
apply List.map_congr
|
case a.h.e'_2
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ List.map V xs = List.map (Function.updateITE V v d) xs
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ β x β xs, V x = Function.updateITE V v d x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro x a1
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
β’ β x β xs, V x = Function.updateITE V v d x
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
β’ V x = Function.updateITE V v d x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
have s1 : Β¬ x = v
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
β’ V x = Function.updateITE V v d x
|
case s1
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
β’ Β¬x = v
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
intro contra
|
case s1
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
β’ Β¬x = v
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
case s1
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
contra : x = v
β’ False
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
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 s1
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
contra : x = v
β’ False
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
case s1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
X : PredName
xs : List VarName
V : Valuation D
d : D
x : VarName
a1 : x β xs
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
h1 : x β xs
β’ False
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
contradiction
|
case s1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Ξ : List (VarName Γ MetaVarName)
X : PredName
xs : List VarName
V : Valuation D
d : D
x : VarName
a1 : x β xs
h2 : β (X : MetaVarName), (x, X) β Ξ β IsNotFree D I M E (meta_var_ X) x
h1 : x β xs
β’ False
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
unfold Function.updateITE
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = Function.updateITE V v d x
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = if x = v then d else V x
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [if_neg s1]
|
case a.h.e'_2.a
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 : PredName
xs : List VarName
h1 : v β xs
V : Valuation D
d : D
x : VarName
a1 : x β xs
s1 : Β¬x = v
β’ V x = if x = v then d else V x
|
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
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 y : VarName
h1 : NotFree Ξ v (eq_ x y)
β’ IsNotFree D I M E (eq_ x y) v
|
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ IsNotFree D I M E (eq_ x y) 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
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ IsNotFree D I M E (eq_ x y) v
|
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ β (V : Valuation D) (d : D), Holds D I V M E (eq_ x y) β Holds D I (Function.updateITE V v d) M E (eq_ x y)
|
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
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ β (V : Valuation D) (d : D), Holds D I V M E (eq_ x y) β Holds D I (Function.updateITE V v d) M E (eq_ x y)
|
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ β (V : Valuation D) (d : D), V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
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
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
β’ β (V : Valuation D) (d : D), V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
V : Valuation D
d : D
β’ V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
cases h1
|
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 y : VarName
h1 : Β¬x = v β§ Β¬y = v
V : Valuation D
d : D
β’ V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
case intro
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 y : VarName
V : Valuation D
d : D
leftβ : Β¬x = v
rightβ : Β¬y = v
β’ V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
case intro h1_left h1_right =>
simp only [Function.updateITE]
simp only [if_neg h1_left]
simp only [if_neg h1_right]
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
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 [Function.updateITE]
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β Function.updateITE V v d x = Function.updateITE V v d y
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β (if x = v then d else V x) = if y = v then d else V y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [if_neg h1_left]
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β (if x = v then d else V x) = if y = v then d else V y
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β V x = if y = v then d else V y
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
simp only [if_neg h1_right]
|
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 y : VarName
V : Valuation D
d : D
h1_left : Β¬x = v
h1_right : Β¬y = v
β’ V x = V y β V x = if y = v then d else V y
|
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
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
h1 : NotFree Ξ v true_
β’ IsNotFree D I M E true_ v
|
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
h1 : NotFree Ξ v true_
β’ β (V : Valuation D) (d : D), Holds D I V M E true_ β Holds D I (Function.updateITE V v d) M E true_
|
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
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
h1 : NotFree Ξ v true_
β’ β (V : Valuation D) (d : D), Holds D I V M E true_ β Holds D I (Function.updateITE V v d) M E true_
|
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
h1 : NotFree Ξ v true_
V : Valuation D
d : D
β’ Holds D I V M E true_ β Holds D I (Function.updateITE V v d) M E true_
|
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
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
h1 : NotFree Ξ v true_
V : Valuation D
d : D
β’ Holds D I V M E true_ β Holds D I (Function.updateITE V v d) M E true_
|
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
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
phi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
h1 : NotFree Ξ v phi.not_
β’ IsNotFree D I M E phi.not_ v
|
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
phi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
h1 : NotFree Ξ v phi
β’ IsNotFree D I M E phi.not_ 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 phi_ih
|
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
phi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
h1 : NotFree Ξ v phi
β’ IsNotFree D I M E phi.not_ v
|
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
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 : NotFree Ξ v phi
β’ IsNotFree D I M E phi.not_ 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
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
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 : NotFree Ξ v phi
β’ IsNotFree D I M E phi.not_ v
|
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
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 : NotFree Ξ v phi
β’ β (V : Valuation D) (d : D), Holds D I V M E phi.not_ β Holds D I (Function.updateITE V v d) M E phi.not_
|
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
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
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 : NotFree Ξ v phi
β’ β (V : Valuation D) (d : D), Holds D I V M E phi.not_ β Holds D I (Function.updateITE V v d) M E phi.not_
|
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Holds D I V M E phi.not_ β Holds D I (Function.updateITE V v d) M E phi.not_
|
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
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Holds D I V M E phi.not_ β Holds D I (Function.updateITE V v d) M E phi.not_
|
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Β¬Holds D I V M E phi β Β¬Holds D I (Function.updateITE V 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]
|
congr! 1
|
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Β¬Holds D I V M E phi β Β¬Holds D I (Function.updateITE V v d) M E phi
|
case a.h.e'_1.a
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Holds D I V M E phi β Holds D I (Function.updateITE V 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 h1 V d
|
case a.h.e'_1.a
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
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 : NotFree Ξ v phi
V : Valuation D
d : D
β’ Holds D I V M E phi β Holds D I (Function.updateITE V 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]
|
unfold NotFree at h1
|
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
phi psi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
psi_ih : NotFree Ξ v psi β IsNotFree D I M E psi v
h1 : NotFree Ξ v (phi.imp_ psi)
β’ IsNotFree D I M E (phi.imp_ psi) v
|
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
phi psi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
psi_ih : NotFree Ξ v psi β IsNotFree D I M E psi v
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) 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 phi_ih
|
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
phi psi : Formula
phi_ih : NotFree Ξ v phi β IsNotFree D I M E phi v
psi_ih : NotFree Ξ v psi β IsNotFree D I M E psi v
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) v
|
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
phi psi : 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
psi_ih : NotFree Ξ v psi β IsNotFree D I M E psi v
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) 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 psi_ih
|
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
phi psi : 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
psi_ih : NotFree Ξ v psi β IsNotFree D I M E psi v
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) v
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) 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
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ IsNotFree D I M E (phi.imp_ psi) v
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ β (V : Valuation D) (d : D), Holds D I V M E (phi.imp_ psi) β Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
|
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
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
β’ β (V : Valuation D) (d : D), Holds D I V M E (phi.imp_ psi) β Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
V : Valuation D
d : D
β’ Holds D I V M E (phi.imp_ psi) β Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
|
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
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
V : Valuation D
d : D
β’ Holds D I V M E (phi.imp_ psi) β Holds D I (Function.updateITE V v d) M E (phi.imp_ psi)
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
V : Valuation D
d : D
β’ Holds D I V M E phi β Holds D I V M E psi β
Holds D I (Function.updateITE V v d) M E phi β Holds D I (Function.updateITE V v d) M E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
cases h1
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
h1 : NotFree Ξ v phi β§ NotFree Ξ v psi
V : Valuation D
d : D
β’ Holds D I V M E phi β Holds D I V M E psi β
Holds D I (Function.updateITE V v d) M E phi β Holds D I (Function.updateITE V v d) M E psi
|
case intro
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
leftβ : NotFree Ξ v phi
rightβ : NotFree Ξ v psi
β’ Holds D I V M E phi β Holds D I V M E psi β
Holds D I (Function.updateITE V v d) M E phi β Holds D I (Function.updateITE V v d) M E psi
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/MM0/MM0.lean
|
MM0.not_free_imp_is_not_free
|
[1604, 1]
|
[1734, 26]
|
congr! 1
|
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
h1_left : NotFree Ξ v phi
h1_right : NotFree Ξ v psi
β’ Holds D I V M E phi β Holds D I V M E psi β
Holds D I (Function.updateITE V v d) M E phi β Holds D I (Function.updateITE V v d) M E psi
|
case a.h.a
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
h1_left : NotFree Ξ v phi
h1_right : NotFree Ξ v psi
β’ Holds D I V M E phi β Holds D I (Function.updateITE V v d) M E phi
case a.h'.a
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
h1_left : NotFree Ξ v phi
h1_right : NotFree Ξ v psi
aβ : Holds D I (Function.updateITE V v d) M E phi
β’ Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
|
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 h1_left V d
|
case a.h.a
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
h1_left : NotFree Ξ v phi
h1_right : NotFree Ξ v psi
β’ Holds D I V M E phi β Holds D I (Function.updateITE V 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]
|
exact psi_ih h1_right V d
|
case a.h'.a
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
phi psi : 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
psi_ih :
NotFree Ξ v psi β β (V : Valuation D) (d : D), Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
V : Valuation D
d : D
h1_left : NotFree Ξ v phi
h1_right : NotFree Ξ v psi
aβ : Holds D I (Function.updateITE V v d) M E phi
β’ Holds D I V M E psi β Holds D I (Function.updateITE V v d) M E psi
|
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
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 IsNotFree D I M E phi v
h1 : NotFree Ξ v (forall_ x phi)
β’ IsNotFree D I M E (forall_ x phi) v
|
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 IsNotFree D I M E phi v
h1 : x = v β¨ NotFree Ξ v phi
β’ IsNotFree D I M E (forall_ x phi) 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 phi_ih
|
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 IsNotFree D I M E phi v
h1 : x = v β¨ NotFree Ξ v phi
β’ IsNotFree D I M E (forall_ x phi) v
|
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
β’ IsNotFree D I M E (forall_ x phi) 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
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
β’ IsNotFree D I M E (forall_ x phi) v
|
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
β’ β (V : Valuation D) (d : D),
Holds D I V M E (forall_ x phi) β Holds D I (Function.updateITE V v d) M E (forall_ x phi)
|
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
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
β’ β (V : Valuation D) (d : D),
Holds D I V M E (forall_ x phi) β Holds D I (Function.updateITE V v d) M E (forall_ x 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
h1 : x = v β¨ NotFree Ξ v phi
V : Valuation D
d : D
β’ Holds D I V M E (forall_ x phi) β Holds D I (Function.updateITE V v d) M E (forall_ x 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 [Holds]
|
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
β’ Holds D I V M E (forall_ x phi) β Holds D I (Function.updateITE V v d) M E (forall_ x 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
h1 : x = v β¨ NotFree Ξ v phi
V : Valuation D
d : D
β’ (β (d : D), Holds D I (Function.updateITE V x d) M E phi) β
β (d_1 : D), Holds D I (Function.updateITE (Function.updateITE V v d) x d_1) 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]
|
apply forall_congr'
|
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
β’ (β (d : D), Holds D I (Function.updateITE V x d) M E phi) β
β (d_1 : D), Holds D I (Function.updateITE (Function.updateITE V v d) x d_1) M E phi
|
case h
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
β’ β (a : D),
Holds D I (Function.updateITE V x a) M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x a) 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]
|
intro d'
|
case h
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
β’ β (a : D),
Holds D I (Function.updateITE V x a) M E phi β Holds D I (Function.updateITE (Function.updateITE V v d) x a) M E phi
|
case h
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
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]
|
by_cases c1 : x = v
|
case h
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
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 pos
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 : 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
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
|
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 pos
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 : 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 pos.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 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 pos.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 : 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
|
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