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Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
omega
omega
:
ι
Param
SNoS_
SNoS_
:
ι
→
ι
Param
ordinal
ordinal
:
ι
→
ο
Param
SNo_
SNo_
:
ι
→
ι
→
ο
Known
SNoS_I
SNoS_I
:
∀ x0 .
ordinal
x0
⟶
∀ x1 x2 .
x2
∈
x0
⟶
SNo_
x2
x1
⟶
x1
∈
SNoS_
x0
Known
omega_ordinal
omega_ordinal
:
ordinal
omega
Param
SNoLev
SNoLev
:
ι
→
ι
Known
ordinal_SNoLev
ordinal_SNoLev
:
∀ x0 .
ordinal
x0
⟶
SNoLev
x0
=
x0
Param
nat_p
nat_p
:
ι
→
ο
Known
nat_p_ordinal
nat_p_ordinal
:
∀ x0 .
nat_p
x0
⟶
ordinal
x0
Known
omega_nat_p
omega_nat_p
:
∀ x0 .
x0
∈
omega
⟶
nat_p
x0
Param
SNo
SNo
:
ι
→
ο
Known
SNoLev_
SNoLev_
:
∀ x0 .
SNo
x0
⟶
SNo_
(
SNoLev
x0
)
x0
Known
omega_SNo
omega_SNo
:
∀ x0 .
x0
∈
omega
⟶
SNo
x0
Theorem
omega_SNoS_omega
omega_SNoS_omega
:
omega
⊆
SNoS_
omega
(proof)
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Param
ordsucc
ordsucc
:
ι
→
ι
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
nat_p_omega
nat_p_omega
:
∀ x0 .
nat_p
x0
⟶
x0
∈
omega
Known
nat_1
nat_1
:
nat_p
1
Known
add_nat_SR
add_nat_SR
:
∀ x0 x1 .
nat_p
x1
⟶
add_nat
x0
(
ordsucc
x1
)
=
ordsucc
(
add_nat
x0
x1
)
Known
nat_0
nat_0
:
nat_p
0
Known
add_nat_0R
add_nat_0R
:
∀ x0 .
add_nat
x0
0
=
x0
Theorem
add_SNo_1_ordsucc
add_SNo_1_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
add_SNo
x0
1
=
ordsucc
x0
(proof)
Param
SNoLt
SNoLt
:
ι
→
ι
→
ο
Param
abs_SNo
abs_SNo
:
ι
→
ι
Param
minus_SNo
minus_SNo
:
ι
→
ι
Param
eps_
eps_
:
ι
→
ι
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
SNoS_E2
SNoS_E2
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
x1
∈
SNoS_
x0
⟶
∀ x2 : ο .
(
SNoLev
x1
∈
x0
⟶
ordinal
(
SNoLev
x1
)
⟶
SNo
x1
⟶
SNo_
(
SNoLev
x1
)
x1
⟶
x2
)
⟶
x2
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Param
SNoLe
SNoLe
:
ι
→
ι
→
ο
Known
SNoLtLe_or
SNoLtLe_or
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
or
(
SNoLt
x0
x1
)
(
SNoLe
x1
x0
)
Known
SNo_0
SNo_0
:
SNo
0
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Known
xm
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
SNoLt_irref
SNoLt_irref
:
∀ x0 .
not
(
SNoLt
x0
x0
)
Known
add_SNo_0L
add_SNo_0L
:
∀ x0 .
SNo
x0
⟶
add_SNo
0
x0
=
x0
Known
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
x0
)
Known
abs_SNo_minus
abs_SNo_minus
:
∀ x0 .
SNo
x0
⟶
abs_SNo
(
minus_SNo
x0
)
=
abs_SNo
x0
Known
pos_abs_SNo
pos_abs_SNo
:
∀ x0 .
SNoLt
0
x0
⟶
abs_SNo
x0
=
x0
Known
dneg
dneg
:
∀ x0 : ο .
not
(
not
x0
)
⟶
x0
Known
SNo_eps_
SNo_eps_
:
∀ x0 .
x0
∈
omega
⟶
SNo
(
eps_
x0
)
Known
SNo_eps_SNoS_omega
SNo_eps_SNoS_omega
:
∀ x0 .
x0
∈
omega
⟶
eps_
x0
∈
SNoS_
omega
Known
omega_ordsucc
omega_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
ordsucc
x0
∈
omega
Known
SNo_eps_pos
SNo_eps_pos
:
∀ x0 .
x0
∈
omega
⟶
SNoLt
0
(
eps_
x0
)
Known
SNoLtLe_tra
SNoLtLe_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLe
x1
x2
⟶
SNoLt
x0
x2
Known
SNo_eps_decr
SNo_eps_decr
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
x0
⟶
SNoLt
(
eps_
x0
)
(
eps_
x1
)
Known
ordsuccI2
ordsuccI2
:
∀ x0 .
x0
∈
ordsucc
x0
Known
SNoLt_tra
SNoLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
x0
x2
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Known
add_SNo_Lt1
add_SNo_Lt1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x2
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x2
x1
)
Known
SNoLeLt_tra
SNoLeLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
x0
x2
Theorem
SNo_prereal_incr_lower_pos
SNo_prereal_incr_lower_pos
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
(
∀ x1 .
x1
∈
SNoS_
omega
⟶
(
∀ x2 .
x2
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
)
(
eps_
x2
)
)
⟶
x1
=
x0
)
⟶
(
∀ x1 .
x1
∈
omega
⟶
∀ x2 : ο .
(
∀ x3 .
and
(
x3
∈
SNoS_
omega
)
(
and
(
SNoLt
x3
x0
)
(
SNoLt
x0
(
add_SNo
x3
(
eps_
x1
)
)
)
)
⟶
x2
)
⟶
x2
)
⟶
∀ x1 .
x1
∈
omega
⟶
∀ x2 : ο .
(
∀ x3 .
x3
∈
SNoS_
omega
⟶
SNoLt
0
x3
⟶
SNoLt
x3
x0
⟶
SNoLt
x0
(
add_SNo
x3
(
eps_
x1
)
)
⟶
x2
)
⟶
x2
(proof)
Param
mul_SNo
mul_SNo
:
ι
→
ι
→
ι
Known
mul_SNo_com
mul_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
x0
x1
=
mul_SNo
x1
x0
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
)
Theorem
pos_mul_SNo_Lt'
pos_mul_SNo_Lt
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
0
x2
⟶
SNoLt
x0
x1
⟶
SNoLt
(
mul_SNo
x0
x2
)
(
mul_SNo
x1
x2
)
(proof)
Known
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
Theorem
pos_mul_SNo_Lt2
pos_mul_SNo_Lt2
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
SNoLt
x0
x2
⟶
SNoLt
x1
x3
⟶
SNoLt
(
mul_SNo
x0
x1
)
(
mul_SNo
x2
x3
)
(proof)
Known
nonneg_mul_SNo_Le
nonneg_mul_SNo_Le
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNoLe
0
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x1
x2
⟶
SNoLe
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
Theorem
nonneg_mul_SNo_Le'
nonneg_mul_SNo_Le
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
0
x2
⟶
SNoLe
x0
x1
⟶
SNoLe
(
mul_SNo
x0
x2
)
(
mul_SNo
x1
x2
)
(proof)
Known
SNoLe_tra
SNoLe_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x1
⟶
SNoLe
x1
x2
⟶
SNoLe
x0
x2
Theorem
nonneg_mul_SNo_Le2
nonneg_mul_SNo_Le2
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLe
0
x0
⟶
SNoLe
0
x1
⟶
SNoLe
x0
x2
⟶
SNoLe
x1
x3
⟶
SNoLe
(
mul_SNo
x0
x1
)
(
mul_SNo
x2
x3
)
(proof)
Param
exp_SNo_nat
exp_SNo_nat
:
ι
→
ι
→
ι
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
exp_SNo_nat_0
exp_SNo_nat_0
:
∀ x0 .
SNo
x0
⟶
exp_SNo_nat
x0
0
=
1
Known
SNoLe_ref
SNoLe_ref
:
∀ x0 .
SNoLe
x0
x0
Known
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
)
Known
mul_SNo_oneL
mul_SNo_oneL
:
∀ x0 .
SNo
x0
⟶
mul_SNo
1
x0
=
x0
Known
SNo_1
SNo_1
:
SNo
1
Known
SNo_exp_SNo_nat
SNo_exp_SNo_nat
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
nat_p
x1
⟶
SNo
(
exp_SNo_nat
x0
x1
)
Known
SNoLtLe
SNoLtLe
:
∀ x0 x1 .
SNoLt
x0
x1
⟶
SNoLe
x0
x1
Known
SNoLt_0_1
SNoLt_0_1
:
SNoLt
0
1
Theorem
exp_SNo_1_bd
exp_SNo_1_bd
:
∀ x0 .
SNo
x0
⟶
SNoLe
1
x0
⟶
∀ x1 .
nat_p
x1
⟶
SNoLe
1
(
exp_SNo_nat
x0
x1
)
(proof)
Known
SNo_2
SNo_2
:
SNo
2
Known
add_SNo_1_1_2
add_SNo_1_1_2
:
add_SNo
1
1
=
2
Known
mul_SNo_distrR
mul_SNo_distrR
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
(
add_SNo
x0
x1
)
x2
=
add_SNo
(
mul_SNo
x0
x2
)
(
mul_SNo
x1
x2
)
Known
nat_p_SNo
nat_p_SNo
:
∀ x0 .
nat_p
x0
⟶
SNo
x0
Known
add_SNo_Le2
add_SNo_Le2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x1
x2
⟶
SNoLe
(
add_SNo
x0
x1
)
(
add_SNo
x0
x2
)
Known
SNoLt_1_2
SNoLt_1_2
:
SNoLt
1
2
Theorem
exp_SNo_2_bd
exp_SNo_2_bd
:
∀ x0 .
nat_p
x0
⟶
SNoLt
x0
(
exp_SNo_nat
2
x0
)
(proof)
Known
mul_SNo_oneR
mul_SNo_oneR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
1
=
x0
Known
mul_SNo_eps_power_2
mul_SNo_eps_power_2
:
∀ x0 .
nat_p
x0
⟶
mul_SNo
(
eps_
x0
)
(
exp_SNo_nat
2
x0
)
=
1
Theorem
eps_bd_1
eps_bd_1
:
∀ x0 .
x0
∈
omega
⟶
SNoLe
(
eps_
x0
)
1
(proof)
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
add_SNo_com_4_inner_mid
add_SNo_com_4_inner_mid
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
add_SNo
(
add_SNo
x0
x1
)
(
add_SNo
x2
x3
)
=
add_SNo
(
add_SNo
x0
x2
)
(
add_SNo
x1
x3
)
Known
add_SNo_assoc
add_SNo_assoc
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
add_SNo
x0
(
add_SNo
x1
x2
)
=
add_SNo
(
add_SNo
x0
x1
)
x2
Theorem
SNo_foil
SNo_foil
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
mul_SNo
(
add_SNo
x0
x1
)
(
add_SNo
x2
x3
)
=
add_SNo
(
mul_SNo
x0
x2
)
(
add_SNo
(
mul_SNo
x0
x3
)
(
add_SNo
(
mul_SNo
x1
x2
)
(
mul_SNo
x1
x3
)
)
)
(proof)
Known
mul_SNo_minus_distrL
mul_SNo_minus_distrL
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
(
minus_SNo
x0
)
x1
=
minus_SNo
(
mul_SNo
x0
x1
)
Known
mul_SNo_minus_distrR
mul_minus_SNo_distrR
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
x0
(
minus_SNo
x1
)
=
minus_SNo
(
mul_SNo
x0
x1
)
Known
minus_SNo_invol
minus_SNo_invol
:
∀ x0 .
SNo
x0
⟶
minus_SNo
(
minus_SNo
x0
)
=
x0
Theorem
mul_SNo_minus_minus
mul_SNo_minus_minus
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
(
minus_SNo
x0
)
(
minus_SNo
x1
)
=
mul_SNo
x0
x1
(proof)
Known
add_SNo_Lt3
add_SNo_Lt3
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLt
x0
x2
⟶
SNoLt
x1
x3
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x2
x3
)
Theorem
add_SNo_Lt4
add_SNo_Lt4
:
∀ x0 x1 x2 x3 x4 x5 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNoLt
x0
x3
⟶
SNoLt
x1
x4
⟶
SNoLt
x2
x5
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x1
x2
)
)
(
add_SNo
x3
(
add_SNo
x4
x5
)
)
(proof)
Theorem
mul_SNo_Lt1_pos_Lt
mul_SNo_Lt1_pos_Lt
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
x0
1
⟶
SNoLt
0
x1
⟶
SNoLt
(
mul_SNo
x0
x1
)
x1
(proof)
Theorem
mul_SNo_Le1_nonneg_Le
mul_SNo_Le1_nonneg_Le
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
x0
1
⟶
SNoLe
0
x1
⟶
SNoLe
(
mul_SNo
x0
x1
)
x1
(proof)
Theorem
SNo_mul_SNo_3
SNo_mul_SNo_3
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
(
mul_SNo
x0
(
mul_SNo
x1
x2
)
)
(proof)
Theorem
82b86..
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
(
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
x3
)
)
)
(proof)
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
Theorem
624bf..
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
x3
)
)
=
mul_SNo
(
mul_SNo
x0
(
mul_SNo
x1
x2
)
)
x3
(proof)
Theorem
mul_SNo_com_3_0_1
mul_SNo_com_3_0_1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
x0
(
mul_SNo
x1
x2
)
=
mul_SNo
x1
(
mul_SNo
x0
x2
)
(proof)
Theorem
85f24..
:
∀ x0 x1 x2 x3 .
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
x3
)
)
=
mul_SNo
x0
(
mul_SNo
x2
(
mul_SNo
x1
x3
)
)
(proof)
Theorem
mul_SNo_com_3b_1_2
mul_SNo_com_3b_1_2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
(
mul_SNo
x0
x1
)
x2
=
mul_SNo
(
mul_SNo
x0
x2
)
x1
(proof)
Theorem
mul_SNo_com_4_inner_mid
mul_SNo_com_4_inner_mid
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
mul_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x2
x3
)
=
mul_SNo
(
mul_SNo
x0
x2
)
(
mul_SNo
x1
x3
)
(proof)
Theorem
mul_SNo_rotate_3_1
mul_SNo_rotate_3_1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
mul_SNo
x0
(
mul_SNo
x1
x2
)
=
mul_SNo
x2
(
mul_SNo
x0
x1
)
(proof)
Theorem
mul_SNo_rotate_4_1
mul_SNo_rotate_4_1
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
x3
)
)
=
mul_SNo
x3
(
mul_SNo
x0
(
mul_SNo
x1
x2
)
)
(proof)
Theorem
b4aa3..
:
∀ x0 x1 x2 x3 x4 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
(
mul_SNo
x3
x4
)
)
)
=
mul_SNo
x4
(
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
x3
)
)
)
(proof)
Theorem
1b9b9..
:
∀ x0 x1 x2 x3 x4 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
mul_SNo
x0
(
mul_SNo
x1
(
mul_SNo
x2
(
mul_SNo
x3
x4
)
)
)
=
mul_SNo
x3
(
mul_SNo
x4
(
mul_SNo
x0
(
mul_SNo
x1
x2
)
)
)
(proof)