Search for blocks/addresses/...
Proofgold Proof
pf
Let x0 of type
ι
be given.
Assume H0:
x0
∈
omega
.
Let x1 of type
ο
be given.
Assume H1:
∀ x2 .
and
(
x2
∈
omega
)
(
∃ x3 .
and
(
x3
∈
omega
)
(
or
(
eps_
x0
=
mul_SNo
(
eps_
x2
)
x3
)
(
eps_
x0
=
minus_SNo
(
mul_SNo
(
eps_
x2
)
x3
)
)
)
)
⟶
x1
.
Apply H1 with
x0
.
Apply andI with
x0
∈
omega
,
∃ x2 .
and
(
x2
∈
omega
)
(
or
(
eps_
x0
=
mul_SNo
(
eps_
x0
)
x2
)
(
eps_
x0
=
minus_SNo
(
mul_SNo
(
eps_
x0
)
x2
)
)
)
leaving 2 subgoals.
The subproof is completed by applying H0.
Let x2 of type
ο
be given.
Assume H2:
∀ x3 .
and
(
x3
∈
omega
)
(
or
(
eps_
x0
=
mul_SNo
(
eps_
x0
)
x3
)
(
eps_
x0
=
minus_SNo
(
mul_SNo
(
eps_
x0
)
x3
)
)
)
⟶
x2
.
Apply H2 with
1
.
Apply andI with
1
∈
omega
,
or
(
eps_
x0
=
mul_SNo
(
eps_
x0
)
1
)
(
eps_
x0
=
minus_SNo
(
mul_SNo
(
eps_
x0
)
1
)
)
leaving 2 subgoals.
Apply nat_p_omega with
1
.
The subproof is completed by applying nat_1.
Apply orIL with
eps_
x0
=
mul_SNo
(
eps_
x0
)
1
,
eps_
x0
=
minus_SNo
(
mul_SNo
(
eps_
x0
)
1
)
.
Let x3 of type
ι
→
ι
→
ο
be given.
Apply mul_SNo_oneR with
eps_
x0
,
λ x4 x5 .
x3
x5
x4
.
Apply SNo_eps_ with
x0
.
The subproof is completed by applying H0.
■