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/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c1 =>
split at a1
case _ c2 =>
simp at c2
exact lt_add_of_tsub_lt_left c2
case _ c2 =>
simp only [Var.lc_at] at a1
simp at a1 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
a1 :
Var.lc_at j
(if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j))
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | linarith | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : i < j
a1 : Var.lc_at j (bound_ i)
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | split at a1 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
a1 :
Var.lc_at j
(if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j]
else bound_ (i - j - (List.map free_ zs).length + j))
β’ i < j + zs.length | case inl
X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
hβ : i - j < (List.map free_ zs).length
a1 : Var.lc_at j (List.map free_ zs)[i - j]
β’ i < j + zs.length
case inr
X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
hβ : Β¬i - j < (List.map free_ zs).length
a1 : Var.lc_at j (bound_ (i - j - (List.map free_ zs).length + j))
β’ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
simp at c2
exact lt_add_of_tsub_lt_left c2 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : i - j < (List.map free_ zs).length
a1 : Var.lc_at j (List.map free_ zs)[i - j]
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
simp only [Var.lc_at] at a1
simp at a1 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : Β¬i - j < (List.map free_ zs).length
a1 : Var.lc_at j (bound_ (i - j - (List.map free_ zs).length + j))
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at c2 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : i - j < (List.map free_ zs).length
a1 : Var.lc_at j (List.map free_ zs)[i - j]
β’ i < j + zs.length | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2β : i - j < (List.map free_ zs).length
a1 : Var.lc_at j (List.map free_ zs)[i - j]
c2 : i - j < zs.length
β’ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exact lt_add_of_tsub_lt_left c2 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2β : i - j < (List.map free_ zs).length
a1 : Var.lc_at j (List.map free_ zs)[i - j]
c2 : i - j < zs.length
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] at a1 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : Β¬i - j < (List.map free_ zs).length
a1 : Var.lc_at j (bound_ (i - j - (List.map free_ zs).length + j))
β’ i < j + zs.length | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : Β¬i - j < (List.map free_ zs).length
a1 : i - j - (List.map free_ zs).length + j < j
β’ i < j + zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a1 | X : String
vs : List Var
j : β
zs : List String
i : β
a2 : bound_ i β vs
c1 : Β¬i < j
c2 : Β¬i - j < (List.map free_ zs).length
a1 : i - j - (List.map free_ zs).length + j < j
β’ i < j + zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | intro a1 v a2 | case mpr
X : String
vs : List Var
j : β
zs : List String
β’ (β v β vs, Var.lc_at (j + zs.length) v) β β v β List.map (Var.openList j (List.map free_ zs)) vs, Var.lc_at j v | case mpr
X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
v : Var
a2 : v β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j v |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | cases v | case mpr
X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
v : Var
a2 : v β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j v | case mpr.free_
X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
aβ : String
a2 : free_ aβ β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (free_ aβ)
case mpr.bound_
X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
aβ : β
a2 : bound_ aβ β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (bound_ aβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ x =>
simp only [Var.lc_at] | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
x : String
a2 : free_ x β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ i =>
simp only [Var.lc_at]
simp at a2
apply Exists.elim a2
intro z a3
clear a2
cases a3
case _ a3_left a3_right =>
specialize a1 z a3_left
cases z
case _ x =>
simp only [Var.openList] at a3_right
case _ i' =>
simp only [Var.lc_at] at a1
simp only [Var.openList] at a3_right
split at a3_right
case _ c1 =>
simp at a3_right
subst a3_right
exact c1
case _ c1 =>
simp at c1
simp at a3_right
split at a3_right
case _ c2 =>
contradiction
case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : bound_ i β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (bound_ i) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
x : String
a2 : free_ x β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (free_ x) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : bound_ i β List.map (Var.openList j (List.map free_ zs)) vs
β’ Var.lc_at j (bound_ i) | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : bound_ i β List.map (Var.openList j (List.map free_ zs)) vs
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a2 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : bound_ i β List.map (Var.openList j (List.map free_ zs)) vs
β’ i < j | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply Exists.elim a2 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
β’ β (a : Var), a β vs β§ Var.openList j (List.map free_ zs) a = bound_ i β i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | intro z a3 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
β’ β (a : Var), a β vs β§ Var.openList j (List.map free_ zs) a = bound_ i β i < j | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
z : Var
a3 : z β vs β§ Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | clear a2 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
a2 : β a β vs, Var.openList j (List.map free_ zs) a = bound_ i
z : Var
a3 : z β vs β§ Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
z : Var
a3 : z β vs β§ Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | cases a3 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
z : Var
a3 : z β vs β§ Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j | case intro
X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
z : Var
leftβ : z β vs
rightβ : Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ a3_left a3_right =>
specialize a1 z a3_left
cases z
case _ x =>
simp only [Var.openList] at a3_right
case _ i' =>
simp only [Var.lc_at] at a1
simp only [Var.openList] at a3_right
split at a3_right
case _ c1 =>
simp at a3_right
subst a3_right
exact c1
case _ c1 =>
simp at c1
simp at a3_right
split at a3_right
case _ c2 =>
contradiction
case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
z : Var
a3_left : z β vs
a3_right : Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | specialize a1 z a3_left | X : String
vs : List Var
j : β
zs : List String
a1 : β v β vs, Var.lc_at (j + zs.length) v
i : β
z : Var
a3_left : z β vs
a3_right : Var.openList j (List.map free_ zs) z = bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i : β
z : Var
a3_left : z β vs
a3_right : Var.openList j (List.map free_ zs) z = bound_ i
a1 : Var.lc_at (j + zs.length) z
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | cases z | X : String
vs : List Var
j : β
zs : List String
i : β
z : Var
a3_left : z β vs
a3_right : Var.openList j (List.map free_ zs) z = bound_ i
a1 : Var.lc_at (j + zs.length) z
β’ i < j | case free_
X : String
vs : List Var
j : β
zs : List String
i : β
aβ : String
a3_left : free_ aβ β vs
a3_right : Var.openList j (List.map free_ zs) (free_ aβ) = bound_ i
a1 : Var.lc_at (j + zs.length) (free_ aβ)
β’ i < j
case bound_
X : String
vs : List Var
j : β
zs : List String
i aβ : β
a3_left : bound_ aβ β vs
a3_right : Var.openList j (List.map free_ zs) (bound_ aβ) = bound_ i
a1 : Var.lc_at (j + zs.length) (bound_ aβ)
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ x =>
simp only [Var.openList] at a3_right | X : String
vs : List Var
j : β
zs : List String
i : β
x : String
a3_left : free_ x β vs
a3_right : Var.openList j (List.map free_ zs) (free_ x) = bound_ i
a1 : Var.lc_at (j + zs.length) (free_ x)
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ i' =>
simp only [Var.lc_at] at a1
simp only [Var.openList] at a3_right
split at a3_right
case _ c1 =>
simp at a3_right
subst a3_right
exact c1
case _ c1 =>
simp at c1
simp at a3_right
split at a3_right
case _ c2 =>
contradiction
case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a3_right : Var.openList j (List.map free_ zs) (bound_ i') = bound_ i
a1 : Var.lc_at (j + zs.length) (bound_ i')
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.openList] at a3_right | X : String
vs : List Var
j : β
zs : List String
i : β
x : String
a3_left : free_ x β vs
a3_right : Var.openList j (List.map free_ zs) (free_ x) = bound_ i
a1 : Var.lc_at (j + zs.length) (free_ x)
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.lc_at] at a1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a3_right : Var.openList j (List.map free_ zs) (bound_ i') = bound_ i
a1 : Var.lc_at (j + zs.length) (bound_ i')
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a3_right : Var.openList j (List.map free_ zs) (bound_ i') = bound_ i
a1 : i' < j + zs.length
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Var.openList] at a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a3_right : Var.openList j (List.map free_ zs) (bound_ i') = bound_ i
a1 : i' < j + zs.length
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
a3_right :
(if i' < j then bound_ i'
else
if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | split at a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
a3_right :
(if i' < j then bound_ i'
else
if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
β’ i < j | case inl
X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
hβ : i' < j
a3_right : bound_ i' = bound_ i
β’ i < j
case inr
X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
hβ : Β¬i' < j
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c1 =>
simp at a3_right
subst a3_right
exact c1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
a3_right : bound_ i' = bound_ i
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c1 =>
simp at c1
simp at a3_right
split at a3_right
case _ c2 =>
contradiction
case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : Β¬i' < j
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
a3_right : bound_ i' = bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
a3_right : i' = i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | subst a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
a3_right : i' = i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
β’ i' < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exact c1 | X : String
vs : List Var
j : β
zs : List String
i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : i' < j
β’ i' < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at c1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : Β¬i' < j
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
c1 : j β€ i'
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp at a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
a3_right :
(if x : i' - j < (List.map free_ zs).length then (List.map free_ zs)[i' - j]
else bound_ (i' - j - (List.map free_ zs).length + j)) =
bound_ i
c1 : j β€ i'
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
a3_right : (if h : i' - j < zs.length then free_ (zs.get β¨i' - j, β―β©) else bound_ (i' - j - zs.length + j)) = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | split at a3_right | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
a3_right : (if h : i' - j < zs.length then free_ (zs.get β¨i' - j, β―β©) else bound_ (i' - j - zs.length + j)) = bound_ i
β’ i < j | case inl
X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
hβ : i' - j < zs.length
a3_right : False
β’ i < j
case inr
X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
hβ : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ i < j |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
contradiction | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : i' - j < zs.length
a3_right : False
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | case _ c2 =>
exfalso
apply c2
exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | contradiction | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : i' - j < zs.length
a3_right : False
β’ i < j | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exfalso | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ i < j | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ False |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply c2 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ False | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ i' - j < zs.length |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | exact Nat.sub_lt_left_of_lt_add c1 a1 | X : String
vs : List Var
j : β
zs : List String
i i' : β
a3_left : bound_ i' β vs
a1 : i' < j + zs.length
c1 : j β€ i'
c2 : Β¬i' - j < zs.length
a3_right : bound_ (i' - j - zs.length + j) = bound_ i
β’ i' - j < zs.length | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi.not_) β Formula.lc_at (j + zs.length) phi.not_ | phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi).not_ β Formula.lc_at (j + zs.length) phi.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi).not_ β Formula.lc_at (j + zs.length) phi.not_ | phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply phi_ih | phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) (phi.imp_ psi)) β Formula.lc_at (j + zs.length) (phi.imp_ psi) | phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)) β
Formula.lc_at (j + zs.length) (phi.imp_ psi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)) β
Formula.lc_at (j + zs.length) (phi.imp_ psi) | phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β§
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β
Formula.lc_at (j + zs.length) phi β§ Formula.lc_at (j + zs.length) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | congr! 1 | phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β§
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β
Formula.lc_at (j + zs.length) phi β§ Formula.lc_at (j + zs.length) psi | case a.h.e'_1.a
phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
case a.h.e'_2.a
phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply phi_ih | case a.h.e'_1.a
phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | apply psi_ih | case a.h.e'_2.a
phi psi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
psi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) psi) β Formula.lc_at (j + zs.length) psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.openList] | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (Formula.openList j (List.map free_ zs) (forall_ aβ phi)) β
Formula.lc_at (j + zs.length) (forall_ aβ phi) | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (forall_ aβ (Formula.openList (j + 1) (List.map free_ zs) phi)) β
Formula.lc_at (j + zs.length) (forall_ aβ phi) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [Formula.lc_at] | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at j (forall_ aβ (Formula.openList (j + 1) (List.map free_ zs) phi)) β
Formula.lc_at (j + zs.length) (forall_ aβ phi) | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at (j + 1) (Formula.openList (j + 1) (List.map free_ zs) phi) β Formula.lc_at (j + zs.length + 1) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [phi_ih] | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at (j + 1) (Formula.openList (j + 1) (List.map free_ zs) phi) β Formula.lc_at (j + zs.length + 1) phi | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | have s1 : j + 1 + List.length zs = j + List.length zs + 1 | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi | case s1
aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ j + 1 + zs.length = j + zs.length + 1
aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | linarith | case s1
aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
β’ j + 1 + zs.length = j + zs.length + 1
aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/LN/Paper.lean | LN.lc_at_instantiate | [1259, 1] | [1342, 19] | simp only [s1] | aβ : String
phi : Formula
phi_ih :
β (j : β) (zs : List String),
Formula.lc_at j (Formula.openList j (List.map free_ zs) phi) β Formula.lc_at (j + zs.length) phi
j : β
zs : List String
s1 : j + 1 + zs.length = j + zs.length + 1
β’ Formula.lc_at (j + 1 + zs.length) phi β Formula.lc_at (j + zs.length + 1) phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | induction h1 | Ξ : List Formula
F : Formula
h1 : IsDeduct Ξ F
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ, Holds D I V E H) β Holds D I V E F | case struct_1_
Ξ : List Formula
F : Formula
Ξβ : List Formula
Hβ phiβ : Formula
aβ : IsDeduct Ξβ phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Hβ :: Ξβ, Holds D I V E H) β Holds D I V E phiβ
case struct_2_
Ξ : List Formula
F : Formula
Ξβ : List Formula
Hβ phiβ : Formula
aβ : IsDeduct (Hβ :: Hβ :: Ξβ) phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Hβ :: Hβ :: Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Hβ :: Ξβ, Holds D I V E H) β Holds D I V E phiβ
case struct_3_
Ξ : List Formula
F : Formula
Ξ_1β Ξ_2β : List Formula
H_1β H_2β phiβ : Formula
aβ : IsDeduct (Ξ_1β ++ [H_1β] ++ [H_2β] ++ Ξ_2β) phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1β ++ [H_1β] ++ [H_2β] ++ Ξ_2β, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1β ++ [H_2β] ++ [H_1β] ++ Ξ_2β, Holds D I V E H) β Holds D I V E phiβ
case assume_
Ξ : List Formula
F phiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [phiβ], Holds D I V E H) β Holds D I V E phiβ
case prop_0_
Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E true_
case prop_1_
Ξ : List Formula
F phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phiβ.imp_ (psiβ.imp_ phiβ))
case prop_2_
Ξ : List Formula
F phiβ psiβ chiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((phiβ.imp_ (psiβ.imp_ chiβ)).imp_ ((phiβ.imp_ psiβ).imp_ (phiβ.imp_ chiβ)))
case prop_3_
Ξ : List Formula
F phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phiβ.not_.imp_ psiβ.not_).imp_ (psiβ.imp_ phiβ))
case mp_
Ξ : List Formula
F : Formula
Ξβ : List Formula
phiβ psiβ : Formula
aβΒΉ : IsDeduct Ξβ (phiβ.imp_ psiβ)
aβ : IsDeduct Ξβ phiβ
a_ihβΒΉ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξβ, Holds D I V E H) β Holds D I V E (phiβ.imp_ psiβ)
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξβ, Holds D I V E H) β Holds D I V E psiβ
case dt_
Ξ : List Formula
F : Formula
Ξβ : List Formula
Hβ phiβ : Formula
aβ : IsDeduct (Hβ :: Ξβ) phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Hβ :: Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξβ, Holds D I V E H) β Holds D I V E (Hβ.imp_ phiβ)
case pred_1_
Ξ : List Formula
F : Formula
vβ : VarName
phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((forall_ vβ (phiβ.imp_ psiβ)).imp_ ((forall_ vβ phiβ).imp_ (forall_ vβ psiβ)))
case pred_2_
Ξ : List Formula
F : Formula
vβ tβ : VarName
phiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((forall_ vβ phiβ).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id vβ tβ) freshChar phiβ))
case pred_3_
Ξ : List Formula
F : Formula
vβ : VarName
phiβ : Formula
aβ : Β¬isFreeIn vβ phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phiβ.imp_ (forall_ vβ phiβ))
case gen_
Ξ : List Formula
F : Formula
vβ : VarName
phiβ : Formula
aβ : IsDeduct [] phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (forall_ vβ phiβ)
case eq_1_
Ξ : List Formula
F : Formula
vβ : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (eq_ vβ vβ)
case eq_2_pred_var_
Ξ : List Formula
F : Formula
nameβ : PredName
xsβ ysβ : List VarName
aβ : xsβ.length = ysβ.length
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
((List.foldr and_ true_ (List.zipWith eq_ xsβ ysβ)).imp_ ((pred_var_ nameβ xsβ).iff_ (pred_var_ nameβ ysβ)))
case eq_2_eq_
Ξ : List Formula
F : Formula
x_0β x_1β y_0β y_1β : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E (((eq_ x_0β y_0β).and_ (eq_ x_1β y_1β)).imp_ ((eq_ x_0β x_1β).iff_ (eq_ y_0β y_1β)))
case def_false_
Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (false_.iff_ true_.not_)
case def_and_
Ξ : List Formula
F phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phiβ.and_ psiβ).iff_ (phiβ.imp_ psiβ.not_).not_)
case def_or_
Ξ : List Formula
F phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phiβ.or_ psiβ).iff_ (phiβ.not_.imp_ psiβ))
case def_iff_
Ξ : List Formula
F phiβ psiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phiβ.iff_ psiβ).imp_ ((phiβ.imp_ psiβ).imp_ (psiβ.imp_ phiβ).not_).not_).imp_
(((phiβ.imp_ psiβ).imp_ (psiβ.imp_ phiβ).not_).not_.imp_ (phiβ.iff_ psiβ)).not_).not_
case def_exists_
Ξ : List Formula
F : Formula
vβ : VarName
phiβ : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ vβ phiβ).iff_ (forall_ vβ phiβ.not_).not_)
case sub_
Ξ : List Formula
F : Formula
Ξβ : List Formula
phiβ : Formula
Οβ : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξβ phiβ
a_ihβ :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξβ, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Οβ) Ξβ, Holds D I V E H) β
Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Οβ phiβ) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_1_ Ξ' H _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
cases a1
case _ a1_left a1_right =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_2_ Ξ' H _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
cases a1
case _ a1_left a1_right =>
simp at a2
cases a2
case _ a2 =>
simp only [a2]
exact a1_left
case _ a2 =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case struct_3_ Ξ_1 Ξ_2 H_1 H_2 _ _ ih_2 =>
intro D I V E a1
apply ih_2
intro H' a2
simp at a1
apply a1
simp at a2
tauto | Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case assume_ phi =>
intro D I V E a1
simp at a1
exact a1 | Ξ : List Formula
F phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [phi], Holds D I V E H) β Holds D I V E phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_0_ =>
intro D I V E _
simp only [Holds] | Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_1_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (psi.imp_ phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_2_ phi psi chi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi chi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.imp_ (psi.imp_ chi)).imp_ ((phi.imp_ psi).imp_ (phi.imp_ chi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case prop_3_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.not_.imp_ psi.not_).imp_ (psi.imp_ phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_1_ v phi psi =>
intro D I V E _
simp only [Holds]
intro a2 a3 d
apply a2 d
exact a3 d | Ξ : List Formula
F : Formula
v : VarName
phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((forall_ v (phi.imp_ psi)).imp_ ((forall_ v phi).imp_ (forall_ v psi))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_2_ v t phi =>
intro D I V E _
obtain s1 := FOL.NV.Sub.Var.All.Rec.Fresh.substitution_theorem D I V E (Function.updateITE id v t) freshChar phi
simp only [Holds]
intro a2
simp only [s1]
simp only [Function.updateITE_comp_left]
simp
exact a2 (V t) | Ξ : List Formula
F : Formula
v t : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E ((forall_ v phi).imp_ (Sub.Var.All.Rec.Fresh.sub (Function.updateITE id v t) freshChar phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case pred_3_ v phi ih =>
intro D I V E _
simp only [Holds]
intro a2 d
have s1 : Holds D I (Function.updateITE V v d) E phi β Holds D I V E phi
{
apply Holds_coincide_Var
intro v' a1
simp only [Function.updateITE]
split_ifs
case _ c1 =>
subst c1
contradiction
case _ c1 =>
rfl
}
simp only [s1]
exact a2 | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
ih : Β¬isFreeIn v phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (phi.imp_ (forall_ v phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case gen_ v phi _ ih_2 =>
intro D I V E _
simp only [Holds]
intro d
apply ih_2
simp | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
aβ : IsDeduct [] phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β [], Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (forall_ v phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case eq_1_ v =>
intro D I V E _
simp only [Holds] | Ξ : List Formula
F : Formula
v : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (eq_ v v) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case eq_2_eq_ x_0 x_1 y_0 y_1 =>
intro D I V E _
simp only [Holds]
intro a2
cases a2
case _ a2_left a2_right =>
simp only [a2_left]
simp only [a2_right] | Ξ : List Formula
F : Formula
x_0 x_1 y_0 y_1 : VarName
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E (((eq_ x_0 y_0).and_ (eq_ x_1 y_1)).imp_ ((eq_ x_0 x_1).iff_ (eq_ y_0 y_1))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_false_ =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E (false_.iff_ true_.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_and_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.and_ psi).iff_ (phi.imp_ psi.not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_or_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((phi.or_ psi).iff_ (phi.not_.imp_ psi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_iff_ phi psi =>
intro D I V E _
simp only [Holds]
tauto | Ξ : List Formula
F phi psi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β
Holds D I V E
(((phi.iff_ psi).imp_ ((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_).imp_
(((phi.imp_ psi).imp_ (psi.imp_ phi).not_).not_.imp_ (phi.iff_ psi)).not_).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case def_exists_ v phi =>
intro D I V E _
simp only [Holds]
simp | Ξ : List Formula
F : Formula
v : VarName
phi : Formula
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β [], Holds D I V E H) β Holds D I V E ((exists_ v phi).iff_ (forall_ v phi.not_).not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case sub_ Ξ' phi Ο _ ih_2 =>
intro D I V E a1
simp at a1
obtain s1 := Sub.Pred.All.Rec.Option.Fresh.substitution_theorem D I V E freshChar Ο
simp only [β s1] at a1
simp only [β s1]
apply ih_2
exact a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
phi : Formula
Ο : PredName β β β Option (List VarName Γ Formula)
aβ : IsDeduct Ξ' phi
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phi
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β List.map (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο) Ξ', Holds D I V E H) β
Holds D I V E (Sub.Pred.All.Rec.Option.Fresh.sub freshChar Ο phi) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D I V E phiβ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D I V E phiβ | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ β H β Ξ', Holds D I V E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ β H β Ξ', Holds D I V E H | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
H' : Formula
a2 : H' β Ξ'
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
H' : Formula
a2 : H' β Ξ'
β’ Holds D I V E H' | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ'
a1 : Holds D I V E H β§ β a β Ξ', Holds D I V E a
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ'
a1 : Holds D I V E H β§ β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | case a.intro
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ'
leftβ : Holds D I V E H
rightβ : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a1_left a1_right =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ'
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct Ξ' phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), (β H β Ξ', Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β Ξ'
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D I V E phiβ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ Holds D I V E phiβ | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ β H_1 β H :: H :: Ξ', Holds D I V E H_1 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro H' a2 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
β’ β H_1 β H :: H :: Ξ', Holds D I V E H_1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
H' : Formula
a2 : H' β H :: H :: Ξ'
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H_1 β H :: Ξ', Holds D I V E H_1
H' : Formula
a2 : H' β H :: H :: Ξ'
β’ Holds D I V E H' | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β H :: H :: Ξ'
a1 : Holds D I V E H β§ β a β Ξ', Holds D I V E a
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a1 | case a
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β H :: H :: Ξ'
a1 : Holds D I V E H β§ β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | case a.intro
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β H :: H :: Ξ'
leftβ : Holds D I V E H
rightβ : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a1_left a1_right =>
simp at a2
cases a2
case _ a2 =>
simp only [a2]
exact a1_left
case _ a2 =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β H :: H :: Ξ'
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp at a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a2 : H' β H :: H :: Ξ'
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
β’ Holds D I V E H' | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H β¨ H' β Ξ'
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | cases a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H β¨ H' β Ξ'
β’ Holds D I V E H' | case inl
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
hβ : H' = H
β’ Holds D I V E H'
case inr
Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
hβ : H' β Ξ'
β’ Holds D I V E H' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2 =>
simp only [a2]
exact a1_left | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | case _ a2 =>
exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' β Ξ'
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | simp only [a2] | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H
β’ Holds D I V E H' | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H
β’ Holds D I V E H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_left | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' = H
β’ Holds D I V E H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | exact a1_right H' a2 | Ξ : List Formula
F : Formula
Ξ' : List Formula
H phiβ : Formula
aβ : IsDeduct (H :: H :: Ξ') phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H_1 β H :: H :: Ξ', Holds D I V E H_1) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
H' : Formula
a1_left : Holds D I V E H
a1_right : β a β Ξ', Holds D I V E a
a2 : H' β Ξ'
β’ Holds D I V E H' | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | intro D I V E a1 | Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
β’ β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ | Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
β’ Holds D I V E phiβ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Program/Backend.lean | FOL.NV.Program.Backend.soundness | [793, 1] | [979, 13] | apply ih_2 | Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
β’ Holds D I V E phiβ | case a
Ξ : List Formula
F : Formula
Ξ_1 Ξ_2 : List Formula
H_1 H_2 phiβ : Formula
aβ : IsDeduct (Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2) phiβ
ih_2 :
β (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env),
(β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H) β Holds D I V E phiβ
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
a1 : β H β Ξ_1 ++ [H_2] ++ [H_1] ++ Ξ_2, Holds D I V E H
β’ β H β Ξ_1 ++ [H_1] ++ [H_2] ++ Ξ_2, Holds D I V E H |
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