Search for blocks/addresses/...
Proofgold Proof
pf
Let x0 of type
ι
be given.
Let x1 of type
ι
be given.
Let x2 of type
ι
be given.
Let x3 of type
ι
be given.
Assume H0:
nat_p
x2
.
Assume H1:
nat_p
x3
.
Claim L2:
...
...
Claim L3:
...
...
Assume H4:
TwoRamseyProp_atleastp
(
ordsucc
x0
)
x1
(
ordsucc
x2
)
.
Assume H5:
TwoRamseyProp_atleastp
x0
(
ordsucc
x1
)
(
ordsucc
x3
)
.
Let x4 of type
ι
→
ι
→
ο
be given.
Assume H6:
∀ x5 x6 .
x4
x5
x6
⟶
x4
x6
x5
.
Claim L7:
...
...
Claim L8:
...
...
Apply unknownprop_e15eccd5f8a3702936e1124d59891ca5d635ffe75b915541be475fc6fad587bd with
x2
,
x3
,
{x5 ∈
ordsucc
(
add_nat
x2
x3
)
|
not
(
x4
x5
(
ordsucc
(
add_nat
x2
x3
)
)
)
}
,
{x5 ∈
ordsucc
(
add_nat
x2
x3
)
|
x4
x5
(
ordsucc
(
add_nat
x2
x3
)
)
}
,
or
(
∃ x5 .
and
(
x5
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x0
)
x5
)
(
∀ x6 .
x6
∈
x5
⟶
∀ x7 .
x7
∈
x5
⟶
(
x6
=
x7
⟶
∀ x8 : ο .
x8
)
⟶
x4
x6
x7
)
)
)
(
∃ x5 .
and
(
x5
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x1
)
x5
)
(
∀ x6 .
x6
∈
x5
⟶
∀ x7 .
x7
∈
x5
⟶
(
x6
=
x7
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x4
x6
x7
)
)
)
)
leaving 5 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying L8.
Assume H9:
atleastp
(
ordsucc
x2
)
{x5 ∈
ordsucc
(
add_nat
x2
x3
)
|
not
(
x4
x5
(
ordsucc
(
add_nat
x2
x3
)
)
)
}
.
Apply H9 with
or
(
∃ x5 .
and
(
x5
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x0
)
x5
)
(
∀ x6 .
x6
∈
x5
⟶
∀ x7 .
x7
∈
x5
⟶
(
x6
=
x7
⟶
∀ x8 : ο .
x8
)
⟶
x4
x6
x7
)
)
)
(
∃ x5 .
and
(
x5
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x1
)
x5
)
(
∀ x6 .
x6
∈
x5
⟶
∀ x7 .
x7
∈
x5
⟶
(
x6
=
x7
⟶
∀ x8 : ο .
x8
)
⟶
not
(
x4
x6
x7
)
)
)
)
.
Let x5 of type
ι
→
ι
be given.
Assume H10:
inj
(
ordsucc
x2
)
{x6 ∈
ordsucc
(
add_nat
x2
x3
)
|
not
(
x4
x6
(
ordsucc
(
add_nat
x2
x3
)
)
)
}
x5
.
Apply H10 with
or
(
∃ x6 .
and
(
x6
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x0
)
x6
)
(
∀ x7 .
x7
∈
x6
⟶
∀ x8 .
x8
∈
x6
⟶
(
x7
=
x8
⟶
∀ x9 : ο .
x9
)
⟶
x4
x7
x8
)
)
)
(
∃ x6 .
and
(
x6
⊆
ordsucc
(
ordsucc
(
add_nat
x2
x3
)
)
)
(
and
(
atleastp
(
ordsucc
x1
)
x6
)
(
∀ x7 .
...
⟶
∀ x8 .
...
⟶
(
x7
=
...
⟶
∀ x9 : ο .
x9
)
⟶
not
(
x4
x7
x8
)
)
)
)
.
...
...
■