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Param
SNo
SNo
:
ι
→
ο
Param
SNoLe
SNoLe
:
ι
→
ι
→
ο
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Param
minus_SNo
minus_SNo
:
ι
→
ι
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Param
SNoLt
SNoLt
:
ι
→
ι
→
ο
Known
SNoLeE
SNoLeE
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
x0
x1
⟶
or
(
SNoLt
x0
x1
)
(
x0
=
x1
)
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Known
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
x0
)
Known
SNoLtLe
SNoLtLe
:
∀ x0 x1 .
SNoLt
x0
x1
⟶
SNoLe
x0
x1
Known
add_SNo_minus_Lt1
add_SNo_minus_Lt1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
(
add_SNo
x0
(
minus_SNo
x1
)
)
x2
⟶
SNoLt
x0
(
add_SNo
x2
x1
)
Known
add_SNo_minus_R2'
add_SNo_minus_R2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
(
add_SNo
x0
(
minus_SNo
x1
)
)
x1
=
x0
Known
SNoLe_ref
SNoLe_ref
:
∀ x0 .
SNoLe
x0
x0
Theorem
05aff..
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
(
add_SNo
x0
(
minus_SNo
x1
)
)
x2
⟶
SNoLe
x0
(
add_SNo
x2
x1
)
(proof)
Known
add_SNo_minus_Lt1b
add_SNo_minus_Lt1b
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
(
add_SNo
x2
x1
)
⟶
SNoLt
(
add_SNo
x0
(
minus_SNo
x1
)
)
x2
Known
add_SNo_minus_R2
add_SNo_minus_R2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
(
add_SNo
x0
x1
)
(
minus_SNo
x1
)
=
x0
Theorem
add_SNo_minus_Le1b
add_SNo_minus_Le1b
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
(
add_SNo
x2
x1
)
⟶
SNoLe
(
add_SNo
x0
(
minus_SNo
x1
)
)
x2
(proof)
Known
add_SNo_minus_Lt2
add_SNo_minus_Lt2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x2
(
add_SNo
x0
(
minus_SNo
x1
)
)
⟶
SNoLt
(
add_SNo
x2
x1
)
x0
Theorem
add_SNo_minus_Le2
add_SNo_minus_Le2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x2
(
add_SNo
x0
(
minus_SNo
x1
)
)
⟶
SNoLe
(
add_SNo
x2
x1
)
x0
(proof)
Known
add_SNo_minus_Lt2b
add_SNo_minus_Lt2b
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
(
add_SNo
x2
x1
)
x0
⟶
SNoLt
x2
(
add_SNo
x0
(
minus_SNo
x1
)
)
Theorem
add_SNo_minus_Le2b
add_SNo_minus_Le2b
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
(
add_SNo
x2
x1
)
x0
⟶
SNoLe
x2
(
add_SNo
x0
(
minus_SNo
x1
)
)
(proof)
Param
nat_primrec
nat_primrec
:
ι
→
(
ι
→
ι
→
ι
) →
ι
→
ι
Param
ordsucc
ordsucc
:
ι
→
ι
Param
mul_SNo
mul_SNo
:
ι
→
ι
→
ι
Definition
exp_SNo_nat
exp_SNo_nat
:=
λ x0 .
nat_primrec
1
(
λ x1 .
mul_SNo
x0
)
Known
nat_primrec_0
nat_primrec_0
:
∀ x0 .
∀ x1 :
ι →
ι → ι
.
nat_primrec
x0
x1
0
=
x0
Theorem
exp_SNo_nat_0
exp_SNo_nat_0
:
∀ x0 .
SNo
x0
⟶
exp_SNo_nat
x0
0
=
1
(proof)
Param
nat_p
nat_p
:
ι
→
ο
Known
nat_primrec_S
nat_primrec_S
:
∀ x0 .
∀ x1 :
ι →
ι → ι
.
∀ x2 .
nat_p
x2
⟶
nat_primrec
x0
x1
(
ordsucc
x2
)
=
x1
x2
(
nat_primrec
x0
x1
x2
)
Theorem
exp_SNo_nat_S
exp_SNo_nat_S
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
nat_p
x1
⟶
exp_SNo_nat
x0
(
ordsucc
x1
)
=
mul_SNo
x0
(
exp_SNo_nat
x0
x1
)
(proof)
Known
nat_ind
nat_ind
:
∀ x0 :
ι → ο
.
x0
0
⟶
(
∀ x1 .
nat_p
x1
⟶
x0
x1
⟶
x0
(
ordsucc
x1
)
)
⟶
∀ x1 .
nat_p
x1
⟶
x0
x1
Known
SNo_1
SNo_1
:
SNo
1
Known
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
Theorem
SNo_exp_SNo_nat
SNo_exp_SNo_nat
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
nat_p
x1
⟶
SNo
(
exp_SNo_nat
x0
x1
)
(proof)
Known
nat_p_SNo
nat_p_SNo
:
∀ x0 .
nat_p
x0
⟶
SNo
x0
Known
nat_1
nat_1
:
nat_p
1
Param
omega
omega
:
ι
Param
mul_nat
mul_nat
:
ι
→
ι
→
ι
Known
mul_nat_mul_SNo
mul_nat_mul_SNo
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
omega
⟶
mul_nat
x0
x1
=
mul_SNo
x0
x1
Known
nat_p_omega
nat_p_omega
:
∀ x0 .
nat_p
x0
⟶
x0
∈
omega
Known
mul_nat_p
mul_nat_p
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
nat_p
(
mul_nat
x0
x1
)
Theorem
nat_exp_SNo_nat
nat_exp_SNo_nat
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
nat_p
(
exp_SNo_nat
x0
x1
)
(proof)
Param
eps_
eps_
:
ι
→
ι
Known
eps_0_1
eps_0_1
:
eps_
0
=
1
Known
SNo_2
SNo_2
:
SNo
2
Known
mul_SNo_oneR
mul_SNo_oneR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
1
=
x0
Known
mul_SNo_assoc
mul_SNo_assoc
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
x0
(
mul_SNo
x1
x2
)
=
mul_SNo
(
mul_SNo
x0
x1
)
x2
Known
add_SNo_1_1_2
add_SNo_1_1_2
:
add_SNo
1
1
=
2
Known
mul_SNo_distrL
mul_SNo_distrL
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
x0
(
add_SNo
x1
x2
)
=
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
Known
eps_ordsucc_half_add
eps_ordsucc_half_add
:
∀ x0 .
nat_p
x0
⟶
add_SNo
(
eps_
(
ordsucc
x0
)
)
(
eps_
(
ordsucc
x0
)
)
=
eps_
x0
Known
SNo_eps_
SNo_eps_
:
∀ x0 .
x0
∈
omega
⟶
SNo
(
eps_
x0
)
Known
nat_ordsucc
nat_ordsucc
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
x0
)
Theorem
mul_SNo_eps_power_2
mul_SNo_eps_power_2
:
∀ x0 .
nat_p
x0
⟶
mul_SNo
(
eps_
x0
)
(
exp_SNo_nat
2
x0
)
=
1
(proof)
Known
mul_SNo_com
mul_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
x0
x1
=
mul_SNo
x1
x0
Theorem
mul_SNo_eps_power_2'
mul_SNo_eps_power_2
:
∀ x0 .
nat_p
x0
⟶
mul_SNo
(
exp_SNo_nat
2
x0
)
(
eps_
x0
)
=
1
(proof)
Known
add_SNo_0R
add_SNo_0R
:
∀ x0 .
SNo
x0
⟶
add_SNo
x0
0
=
x0
Param
add_nat
add_nat
:
ι
→
ι
→
ι
Known
add_nat_add_SNo
add_nat_add_SNo
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
omega
⟶
add_nat
x0
x1
=
add_SNo
x0
x1
Known
omega_ordsucc
omega_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
ordsucc
x0
∈
omega
Known
add_nat_SR
add_nat_SR
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
x1
)
=
ordsucc
(
add_nat
x0
x1
)
Known
add_nat_p
add_nat_p
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
nat_p
(
add_nat
x0
x1
)
Theorem
exp_SNo_nat_mul_add
exp_SNo_nat_mul_add
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
nat_p
x1
⟶
∀ x2 .
nat_p
x2
⟶
mul_SNo
(
exp_SNo_nat
x0
x1
)
(
exp_SNo_nat
x0
x2
)
=
exp_SNo_nat
x0
(
add_SNo
x1
x2
)
(proof)
Known
omega_nat_p
omega_nat_p
:
∀ x0 .
x0
∈
omega
⟶
nat_p
x0
Theorem
exp_SNo_nat_mul_add'
exp_SNo_nat_mul_add
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
omega
⟶
∀ x2 .
x2
∈
omega
⟶
mul_SNo
(
exp_SNo_nat
x0
x1
)
(
exp_SNo_nat
x0
x2
)
=
exp_SNo_nat
x0
(
add_SNo
x1
x2
)
(proof)
Known
SNoLt_0_1
SNoLt_0_1
:
SNoLt
0
1
Known
mul_SNo_zeroR
mul_SNo_zeroR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
0
=
0
Known
pos_mul_SNo_Lt
pos_mul_SNo_Lt
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNoLt
0
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x1
x2
⟶
SNoLt
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
Known
SNo_0
SNo_0
:
SNo
0
Theorem
exp_SNo_nat_pos
exp_SNo_nat_pos
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
∀ x1 .
nat_p
x1
⟶
SNoLt
0
(
exp_SNo_nat
x0
x1
)
(proof)
Known
SNoLt_trichotomy_or_impred
SNoLt_trichotomy_or_impred
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
∀ x2 : ο .
(
SNoLt
x0
x1
⟶
x2
)
⟶
(
x0
=
x1
⟶
x2
)
⟶
(
SNoLt
x1
x0
⟶
x2
)
⟶
x2
Definition
False
False
:=
∀ x0 : ο .
x0
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Known
SNoLt_irref
SNoLt_irref
:
∀ x0 .
not
(
SNoLt
x0
x0
)
Known
neg_mul_SNo_Lt
neg_mul_SNo_Lt
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNoLt
x0
0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x2
x1
⟶
SNoLt
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
Theorem
mul_SNo_nonzero_cancel
mul_SNo_nonzero_cancel_L
:
∀ x0 x1 x2 .
SNo
x0
⟶
(
x0
=
0
⟶
∀ x3 : ο .
x3
)
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
x0
x1
=
mul_SNo
x0
x2
⟶
x1
=
x2
(proof)
Known
SNoLt_0_2
SNoLt_0_2
:
SNoLt
0
2
Known
add_SNo_In_omega
add_SNo_In_omega
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
omega
⟶
add_SNo
x0
x1
∈
omega
Known
mul_SNo_oneL
mul_SNo_oneL
:
∀ x0 .
SNo
x0
⟶
mul_SNo
1
x0
=
x0
Theorem
mul_SNo_eps_eps_add_SNo
mul_SNo_eps_eps_add_SNo
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
omega
⟶
mul_SNo
(
eps_
x0
)
(
eps_
x1
)
=
eps_
(
add_SNo
x0
x1
)
(proof)