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Pierre Letouzey
natded
Commits
3573759d
Commit
3573759d
authored
Jun 11, 2020
by
Samuel Ben Hamou
Browse files
Suite de la preuve de SuccRight.
parent
9ebd3ff8
Changes
1
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Side-by-side
Peano.v
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3573759d
...
...
@@ -236,9 +236,26 @@ Proof.
bsubst
0
(
Succ
(#
0
))
(
∀
Succ
(#
1
+
#
0
)
=
#
1
+
Succ
(#
0
))))).
*
apply
R_Ax
.
unfold
induction_schema
.
apply
in_eq
.
*
simpl
.
apply
R_And_i
.
++
assert
((
∀
bsubst
0
Zero
(
∀
Succ
(#
1
+
#
0
)
=
#
1
+
Succ
(#
0
)))
=
(
∀
Succ
(#
0
+
Zero
)
=
#
0
+
Succ
(
Zero
))).
{
auto
.
(
*
EST
-
CE
QUE
C
'
EST
AU
MOINS
VRAI
???
*
)
++
compute
.
change
(
Fun
"O"
[])
with
Zero
.
apply
R_All_i
with
(
x
:=
"x"
);
[
|
compute
;
change
(
Fun
"O"
[])
with
Zero
].
--
unfold
Names
.
In
.
compute
.
inversion
1.
--
apply
R_All_i
with
(
x
:=
"y"
);
[
compute
;
inversion
1
|
].
compute
.
change
(
Fun
"O"
[])
with
Zero
.
set
(
Γ
:=
[(
∀
∀
Fun
"S"
[
Zero
+
#
0
]
=
Zero
+
Fun
"S"
[#
0
])
/
\
(
∀
(
∀
Fun
"S"
[#
1
+
#
0
]
=
#
1
+
Fun
"S"
[#
0
])
->
∀
Fun
"S"
[
Fun
"S"
[#
1
]
+
#
0
]
=
Fun
"S"
[#
1
]
+
Fun
"S"
[#
0
])
->
∀
∀
Fun
"S"
[#
1
+
#
0
]
=
#
1
+
Fun
"S"
[#
0
];
∀
#
0
=
#
0
;
∀
∀
#
1
=
#
0
->
#
0
=
#
1
;
∀
∀
∀
#
2
=
#
1
/
\
#
1
=
#
0
->
#
2
=
#
0
;
∀
∀
#
1
=
#
0
->
Fun
"S"
[#
1
]
=
Fun
"S"
[#
0
];
∀
∀
∀
#
2
=
#
1
->
#
2
+
#
0
=
#
1
+
#
0
;
∀
∀
∀
#
1
=
#
0
->
#
2
+
#
1
=
#
2
+
#
0
;
∀
∀
∀
#
2
=
#
1
->
#
2
*
#
0
=
#
1
*
#
0
;
∀
∀
∀
#
1
=
#
0
->
#
2
*
#
1
=
#
2
*
#
0
;
∀
Zero
+
#
0
=
#
0
;
∀
∀
Fun
"S"
[#
1
]
+
#
0
=
Fun
"S"
[#
1
+
#
0
];
∀
Zero
*
#
0
=
Zero
;
∀
∀
Fun
"S"
[#
1
]
*
#
0
=
#
1
*
#
0
+
#
0
;
∀
∀
Fun
"S"
[#
1
]
=
Fun
"S"
[#
0
]
->
#
1
=
#
0
;
∀
~
Fun
"S"
[#
0
]
=
Zero
]).
assert
(
Pr
Intuiti
(
Γ
⊢
Zero
+
FVar
"y"
=
FVar
"y"
)).
{
assert
(
AX9
:
Pr
Intuiti
(
Γ
⊢
ax9
)).
{
apply
R_Ax
.
compute
;
intuition
.
}
unfold
ax9
in
AX9
.
apply
R_All_e
with
(
t
:=
FVar
"y"
)
in
AX9
;
[
|
auto
].
compute
in
AX9
.
assumption
.
}
Print
bsubst
.
apply
Lemma
Comm
:
IsTheorem
Intuiti
PeanoTheory
(
∀∀
(#
0
+
#
1
=
#
1
+
#
0
)).
Admitted
.
...
...
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