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Param
int
int
:
ι
Param
omega
omega
:
ι
Param
minus_SNo
minus_SNo
:
ι
→
ι
Param
ordsucc
ordsucc
:
ι
→
ι
Known
int_SNo_cases
int_SNo_cases
:
∀ x0 :
ι → ο
.
(
∀ x1 .
x1
∈
omega
⟶
x0
x1
)
⟶
(
∀ x1 .
x1
∈
omega
⟶
x0
(
minus_SNo
x1
)
)
⟶
∀ x1 .
x1
∈
int
⟶
x0
x1
Param
nat_p
nat_p
:
ι
→
ο
Known
nat_inv_impred
nat_inv_impred
:
∀ x0 :
ι → ο
.
x0
0
⟶
(
∀ x1 .
nat_p
x1
⟶
x0
(
ordsucc
x1
)
)
⟶
∀ x1 .
nat_p
x1
⟶
x0
x1
Known
nat_p_omega
nat_p_omega
:
∀ x0 .
nat_p
x0
⟶
x0
∈
omega
Known
omega_nat_p
omega_nat_p
:
∀ x0 .
x0
∈
omega
⟶
nat_p
x0
Known
minus_SNo_0
minus_SNo_0
:
minus_SNo
0
=
0
Theorem
int_3_cases
int_3_cases
:
∀ x0 .
x0
∈
int
⟶
∀ x1 : ο .
(
∀ x2 .
x2
∈
omega
⟶
x0
=
minus_SNo
(
ordsucc
x2
)
⟶
x1
)
⟶
(
x0
=
0
⟶
x1
)
⟶
(
∀ x2 .
x2
∈
omega
⟶
x0
=
ordsucc
x2
⟶
x1
)
⟶
x1
(proof)
Param
SNo
SNo
:
ι
→
ο
Param
real
real
:
ι
Param
SNoLt
SNoLt
:
ι
→
ι
→
ο
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Param
recip_SNo_pos
recip_SNo_pos
:
ι
→
ι
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Param
ap
ap
:
ι
→
ι
→
ι
Param
SNo_recipaux
SNo_recipaux
:
ι
→
(
ι
→
ι
) →
ι
→
ι
Param
SNoS_
SNoS_
:
ι
→
ι
Param
SNoLev
SNoLev
:
ι
→
ι
Known
SNoLev_ind
SNoLev_ind
:
∀ x0 :
ι → ο
.
(
∀ x1 .
SNo
x1
⟶
(
∀ x2 .
x2
∈
SNoS_
(
SNoLev
x1
)
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
SNo
x1
⟶
x0
x1
Param
abs_SNo
abs_SNo
:
ι
→
ι
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Param
eps_
eps_
:
ι
→
ι
Known
real_E
real_E
:
∀ x0 .
x0
∈
real
⟶
∀ x1 : ο .
(
SNo
x0
⟶
SNoLev
x0
∈
ordsucc
omega
⟶
x0
∈
SNoS_
(
ordsucc
omega
)
⟶
SNoLt
(
minus_SNo
omega
)
x0
⟶
SNoLt
x0
omega
⟶
(
∀ x2 .
x2
∈
SNoS_
omega
⟶
(
∀ x3 .
x3
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x2
(
minus_SNo
x0
)
)
)
(
eps_
x3
)
)
⟶
x2
=
x0
)
⟶
(
∀ x2 .
x2
∈
omega
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
x4
∈
SNoS_
omega
)
(
and
(
SNoLt
x4
x0
)
(
SNoLt
x0
(
add_SNo
x4
(
eps_
x2
)
)
)
)
⟶
x3
)
⟶
x3
)
⟶
x1
)
⟶
x1
Param
mul_SNo
mul_SNo
:
ι
→
ι
→
ι
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Param
famunion
famunion
:
ι
→
(
ι
→
ι
) →
ι
Param
SNoCutP
SNoCutP
:
ι
→
ι
→
ο
Param
SNoCut
SNoCut
:
ι
→
ι
→
ι
Param
binunion
binunion
:
ι
→
ι
→
ι
Param
SNoEq_
SNoEq_
:
ι
→
ι
→
ι
→
ο
Known
SNoCutP_SNoCut_impred
SNoCutP_SNoCut_impred
:
∀ x0 x1 .
SNoCutP
x0
x1
⟶
∀ x2 : ο .
(
SNo
(
SNoCut
x0
x1
)
⟶
SNoLev
(
SNoCut
x0
x1
)
∈
ordsucc
(
binunion
(
famunion
x0
(
λ x3 .
ordsucc
(
SNoLev
x3
)
)
)
(
famunion
x1
(
λ x3 .
ordsucc
(
SNoLev
x3
)
)
)
)
⟶
(
∀ x3 .
x3
∈
x0
⟶
SNoLt
x3
(
SNoCut
x0
x1
)
)
⟶
(
∀ x3 .
x3
∈
x1
⟶
SNoLt
(
SNoCut
x0
x1
)
x3
)
⟶
(
∀ x3 .
SNo
x3
⟶
(
∀ x4 .
x4
∈
x0
⟶
SNoLt
x4
x3
)
⟶
(
∀ x4 .
x4
∈
x1
⟶
SNoLt
x3
x4
)
⟶
and
(
SNoLev
(
SNoCut
x0
x1
)
⊆
SNoLev
x3
)
(
SNoEq_
(
SNoLev
(
SNoCut
x0
x1
)
)
(
SNoCut
x0
x1
)
x3
)
)
⟶
x2
)
⟶
x2
Known
SNo_recipaux_lem2
SNo_recipaux_lem2
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
∀ x1 :
ι → ι
.
(
∀ x2 .
x2
∈
SNoS_
(
SNoLev
x0
)
⟶
SNoLt
0
x2
⟶
and
(
SNo
(
x1
x2
)
)
(
mul_SNo
x2
(
x1
x2
)
=
1
)
)
⟶
SNoCutP
(
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
x1
x2
)
0
)
)
(
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
x1
x2
)
1
)
)
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Known
xm
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Param
exp_SNo_nat
exp_SNo_nat
:
ι
→
ι
→
ι
Known
recip_SNo_pos_eps_
recip_SNo_pos_eps_
:
∀ x0 .
nat_p
x0
⟶
recip_SNo_pos
(
eps_
x0
)
=
exp_SNo_nat
2
x0
Known
SNoS_omega_real
SNoS_omega_real
:
SNoS_
omega
⊆
real
Known
omega_SNoS_omega
omega_SNoS_omega
:
omega
⊆
SNoS_
omega
Known
nat_exp_SNo_nat
nat_exp_SNo_nat
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
nat_p
x1
⟶
nat_p
(
exp_SNo_nat
x0
x1
)
Known
nat_2
nat_2
:
nat_p
2
Known
recip_SNo_pos_2
recip_SNo_pos_2
:
recip_SNo_pos
2
=
eps_
1
Known
SNo_eps_SNoS_omega
SNo_eps_SNoS_omega
:
∀ x0 .
x0
∈
omega
⟶
eps_
x0
∈
SNoS_
omega
Known
nat_1
nat_1
:
nat_p
1
Param
Sep
Sep
:
ι
→
(
ι
→
ο
) →
ι
Param
SNoL
SNoL
:
ι
→
ι
Definition
SNoL_pos
SNoL_pos
:=
λ x0 .
Sep
(
SNoL
x0
)
(
SNoLt
0
)
Known
pos_real_left_approx_double
pos_real_left_approx_double
:
∀ x0 .
x0
∈
real
⟶
SNoLt
0
x0
⟶
(
x0
=
2
⟶
∀ x1 : ο .
x1
)
⟶
(
∀ x1 .
x1
∈
omega
⟶
x0
=
eps_
x1
⟶
∀ x2 : ο .
x2
)
⟶
∀ x1 : ο .
(
∀ x2 .
and
(
x2
∈
SNoL_pos
x0
)
(
SNoLt
x0
(
mul_SNo
2
x2
)
)
⟶
x1
)
⟶
x1
Known
SepE
SepE
:
∀ x0 .
∀ x1 :
ι → ο
.
∀ x2 .
x2
∈
Sep
x0
x1
⟶
and
(
x2
∈
x0
)
(
x1
x2
)
Known
SNoL_E
SNoL_E
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
SNoL
x0
⟶
∀ x2 : ο .
(
SNo
x1
⟶
SNoLev
x1
∈
SNoLev
x0
⟶
SNoLt
x1
x0
⟶
x2
)
⟶
x2
Param
recip_SNo
recip_SNo
:
ι
→
ι
Definition
div_SNo
div_SNo
:=
λ x0 x1 .
mul_SNo
x0
(
recip_SNo
x1
)
Known
recip_SNo_pos_eq
recip_SNo_pos_eq
:
∀ x0 .
SNo
x0
⟶
recip_SNo_pos
x0
=
SNoCut
(
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
0
)
)
(
famunion
omega
(
λ x2 .
ap
(
SNo_recipaux
x0
recip_SNo_pos
x2
)
1
)
)
Known
real_SNoCut
real_SNoCut
:
∀ x0 .
x0
⊆
real
⟶
∀ x1 .
x1
⊆
real
⟶
SNoCutP
x0
x1
⟶
(
x0
=
0
⟶
∀ x2 : ο .
x2
)
⟶
(
x1
=
0
⟶
∀ x2 : ο .
x2
)
⟶
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
x4
∈
x0
)
(
SNoLt
x2
x4
)
⟶
x3
)
⟶
x3
)
⟶
(
∀ x2 .
x2
∈
x1
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
x4
∈
x1
)
(
SNoLt
x4
x2
)
⟶
x3
)
⟶
x3
)
⟶
SNoCut
x0
x1
∈
real
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
EmptyE
EmptyE
:
∀ x0 .
nIn
x0
0
Known
nat_0
nat_0
:
nat_p
0
Param
lam
Sigma
:
ι
→
(
ι
→
ι
) →
ι
Param
If_i
If_i
:
ο
→
ι
→
ι
→
ι
Param
Sing
Sing
:
ι
→
ι
Known
SNo_recipaux_0
SNo_recipaux_0
:
∀ x0 .
∀ x1 :
ι → ι
.
SNo_recipaux
x0
x1
0
=
lam
2
(
λ x3 .
If_i
(
x3
=
0
)
(
Sing
0
)
0
)
Known
tuple_2_0_eq
tuple_2_0_eq
:
∀ x0 x1 .
ap
(
lam
2
(
λ x3 .
If_i
(
x3
=
0
)
x0
x1
)
)
0
=
x0
Known
SingI
SingI
:
∀ x0 .
x0
∈
Sing
x0
Known
div_SNo_pos_LtR
div_SNo_pos_LtR
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
0
x1
⟶
SNoLt
(
mul_SNo
x2
x1
)
x0
⟶
SNoLt
x2
(
div_SNo
x0
x1
)
Known
SNo_add_SNo_4
SNo_add_SNo_4
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
(
add_SNo
x0
(
add_SNo
x1
(
add_SNo
x2
x3
)
)
)
Known
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
x0
)
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Known
add_SNo_0R
add_SNo_0R
:
∀ x0 .
SNo
x0
⟶
add_SNo
x0
0
=
x0
Known
add_SNo_Lt2
add_SNo_Lt2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x1
x2
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x0
x2
)
Known
SNo_0
SNo_0
:
SNo
0
Known
SNo_add_SNo_3
SNo_add_SNo_3
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
(
add_SNo
x0
(
add_SNo
x1
x2
)
)
Known
add_SNo_minus_Lt2b3
add_SNo_minus_Lt2b3
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLt
(
add_SNo
x3
x2
)
(
add_SNo
x0
x1
)
⟶
SNoLt
x3
(
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
Known
add_SNo_0L
add_SNo_0L
:
∀ x0 .
SNo
x0
⟶
add_SNo
0
x0
=
x0
Known
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
Known
add_SNo_com
add_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
x0
x1
=
add_SNo
x1
x0
Known
mul_SNo_oneR
mul_SNo_oneR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
1
=
x0
Known
mul_SNo_Lt
mul_SNo_Lt
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLt
x2
x0
⟶
SNoLt
x3
x1
⟶
SNoLt
(
add_SNo
(
mul_SNo
x2
x1
)
(
mul_SNo
x0
x3
)
)
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x2
x3
)
)
Known
SNo_1
SNo_1
:
SNo
1
Known
mul_SNo_com
mul_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
x0
x1
=
mul_SNo
x1
x0
Known
div_SNo_pos_LtR'
div_SNo_pos_LtR
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
0
x1
⟶
SNoLt
x2
(
div_SNo
x0
x1
)
⟶
SNoLt
(
mul_SNo
x2
x1
)
x0
Known
mul_SNo_oneL
mul_SNo_oneL
:
∀ x0 .
SNo
x0
⟶
mul_SNo
1
x0
=
x0
Known
SNo_recip_SNo
SNo_recip_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
recip_SNo
x0
)
Known
recip_SNo_poscase
recip_SNo_poscase
:
∀ x0 .
SNoLt
0
x0
⟶
recip_SNo
x0
=
recip_SNo_pos
x0
Known
real_SNo
real_SNo
:
∀ x0 .
x0
∈
real
⟶
SNo
x0
Known
div_SNo_pos_LtL
div_SNo_pos_LtL
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
0
x1
⟶
SNoLt
x0
(
mul_SNo
x2
x1
)
⟶
SNoLt
(
div_SNo
x0
x1
)
x2
Known
add_SNo_minus_Lt1b3
add_SNo_minus_Lt1b3
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x3
x2
)
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
x3
Known
div_SNo_pos_LtL'
div_SNo_pos_LtL
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
0
x1
⟶
SNoLt
(
div_SNo
x0
x1
)
x2
⟶
SNoLt
x0
(
mul_SNo
x2
x1
)
Known
mul_div_SNo_nonzero_eq
mul_div_SNo_nonzero_eq
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
(
x1
=
0
⟶
∀ x3 : ο .
x3
)
⟶
x0
=
mul_SNo
x1
x2
⟶
div_SNo
x0
x1
=
x2
Known
SNoLt_irref
SNoLt_irref
:
∀ x0 .
not
(
SNoLt
x0
x0
)
Known
add_SNo_com_3_0_1
add_SNo_com_3_0_1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
add_SNo
x0
(
add_SNo
x1
x2
)
=
add_SNo
x1
(
add_SNo
x0
x2
)
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
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
Known
minus_add_SNo_distr_3
minus_add_SNo_distr_3
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
minus_SNo
(
add_SNo
x0
(
add_SNo
x1
x2
)
)
=
add_SNo
(
minus_SNo
x0
)
(
add_SNo
(
minus_SNo
x1
)
(
minus_SNo
x2
)
)
Known
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
)
Known
add_SNo_assoc_4
add_SNo_assoc_4
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
add_SNo
x0
(
add_SNo
x1
(
add_SNo
x2
x3
)
)
=
add_SNo
(
add_SNo
x0
(
add_SNo
x1
x2
)
)
x3
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
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
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
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
Known
recip_SNo_pos_invR
recip_SNo_pos_invR
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
mul_SNo
x0
(
recip_SNo_pos
x0
)
=
1
Known
SNo_2
SNo_2
:
SNo
2
Known
mul_SNo_pos_pos
mul_SNo_pos_pos
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
0
x0
⟶
SNoLt
0
x1
⟶
SNoLt
0
(
mul_SNo
x0
x1
)
Known
famunionE_impred
famunionE_impred
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
famunion
x0
x1
⟶
∀ x3 : ο .
(
∀ x4 .
x4
∈
x0
⟶
x2
∈
x1
x4
⟶
x3
)
⟶
x3
Known
omega_ordsucc
omega_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
ordsucc
x0
∈
omega
Param
SNo_recipauxset
SNo_recipauxset
:
ι
→
ι
→
ι
→
(
ι
→
ι
) →
ι
Param
SNoR
SNoR
:
ι
→
ι
Known
SNo_recipaux_S
SNo_recipaux_S
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
nat_p
x2
⟶
SNo_recipaux
x0
x1
(
ordsucc
x2
)
=
lam
2
(
λ x4 .
If_i
(
x4
=
0
)
(
binunion
(
binunion
(
ap
(
SNo_recipaux
x0
x1
x2
)
0
)
(
SNo_recipauxset
(
ap
(
SNo_recipaux
x0
x1
x2
)
0
)
x0
(
SNoR
x0
)
x1
)
)
(
SNo_recipauxset
(
ap
(
SNo_recipaux
x0
x1
x2
)
1
)
x0
(
SNoL_pos
x0
)
x1
)
)
(
binunion
(
binunion
(
ap
(
SNo_recipaux
x0
x1
x2
)
1
)
(
SNo_recipauxset
(
ap
(
SNo_recipaux
x0
x1
x2
)
0
)
x0
(
SNoL_pos
x0
)
x1
)
)
(
SNo_recipauxset
(
ap
(
SNo_recipaux
x0
x1
x2
)
1
)
x0
(
SNoR
x0
)
x1
)
)
)
Known
binunionI2
binunionI2
:
∀ x0 x1 x2 .
x2
∈
x1
⟶
x2
∈
binunion
x0
x1
Known
SNo_recipauxset_I
SNo_recipauxset_I
:
∀ x0 x1 x2 .
∀ x3 :
ι → ι
.
∀ x4 .
x4
∈
x0
⟶
∀ x5 .
x5
∈
x2
⟶
mul_SNo
(
add_SNo
1
(
mul_SNo
(
add_SNo
x5
(
minus_SNo
x1
)
)
x4
)
)
(
x3
x5
)
∈
SNo_recipauxset
x0
x1
x2
x3
Known
famunionI
famunionI
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 x3 .
x2
∈
x0
⟶
x3
∈
x1
x2
⟶
x3
∈
famunion
x0
x1
Known
tuple_2_1_eq
tuple_2_1_eq
:
∀ x0 x1 .
ap
(
lam
2
(
λ x3 .
If_i
(
x3
=
0
)
x0
x1
)
)
1
=
x1
Known
binunionI1
binunionI1
:
∀ x0 x1 x2 .
x2
∈
x0
⟶
x2
∈
binunion
x0
x1
Known
SNo_recip_SNo_pos
SNo_recip_SNo_pos
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
SNo
(
recip_SNo_pos
x0
)
Param
ordinal
ordinal
:
ι
→
ο
Param
SNo_
SNo_
:
ι
→
ι
→
ο
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
Known
SNoLev_ordinal
SNoLev_ordinal
:
∀ x0 .
SNo
x0
⟶
ordinal
(
SNoLev
x0
)
Known
recip_SNo_pos_prop1
recip_SNo_pos_prop1
:
∀ x0 .
SNo
x0
⟶
SNoLt
0
x0
⟶
and
(
SNo
(
recip_SNo_pos
x0
)
)
(
mul_SNo
x0
(
recip_SNo_pos
x0
)
=
1
)
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
SingE
SingE
:
∀ x0 x1 .
x1
∈
Sing
x0
⟶
x1
=
x0
Known
real_0
real_0
:
0
∈
real
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
binunionE'
binunionE
:
∀ x0 x1 x2 .
∀ x3 : ο .
(
x2
∈
x0
⟶
x3
)
⟶
(
x2
∈
x1
⟶
x3
)
⟶
x2
∈
binunion
x0
x1
⟶
x3
Known
SNo_recipauxset_E
SNo_recipauxset_E
:
∀ x0 x1 x2 .
∀ x3 :
ι → ι
.
∀ x4 .
x4
∈
SNo_recipauxset
x0
x1
x2
x3
⟶
∀ x5 : ο .
(
∀ x6 .
x6
∈
x0
⟶
∀ x7 .
x7
∈
x2
⟶
x4
=
mul_SNo
(
add_SNo
1
(
mul_SNo
(
add_SNo
x7
(
minus_SNo
x1
)
)
x6
)
)
(
x3
x7
)
⟶
x5
)
⟶
x5
Known
SNoR_E
SNoR_E
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
SNoR
x0
⟶
∀ x2 : ο .
(
SNo
x1
⟶
SNoLev
x1
∈
SNoLev
x0
⟶
SNoLt
x0
x1
⟶
x2
)
⟶
x2
Known
SNoS_I2
SNoS_I2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLev
x0
∈
SNoLev
x1
⟶
x0
∈
SNoS_
(
SNoLev
x1
)
Known
SNoLt_tra
SNoLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
x0
x2
Known
real_mul_SNo
real_mul_SNo
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
x1
∈
real
⟶
mul_SNo
x0
x1
∈
real
Known
real_add_SNo
real_add_SNo
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
x1
∈
real
⟶
add_SNo
x0
x1
∈
real
Known
real_1
real_1
:
1
∈
real
Known
real_minus_SNo
real_minus_SNo
:
∀ x0 .
x0
∈
real
⟶
minus_SNo
x0
∈
real
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
Known
ordsuccE
ordsuccE
:
∀ x0 x1 .
x1
∈
ordsucc
x0
⟶
or
(
x1
∈
x0
)
(
x1
=
x0
)
Known
nat_p_trans
nat_p_trans
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
x0
⟶
nat_p
x1
Theorem
real_recip_SNo_lem1
real_recip_SNo_lem1
:
∀ x0 .
SNo
x0
⟶
x0
∈
real
⟶
SNoLt
0
x0
⟶
and
(
recip_SNo_pos
x0
∈
real
)
(
∀ x1 .
nat_p
x1
⟶
and
(
ap
(
SNo_recipaux
x0
recip_SNo_pos
x1
)
0
⊆
real
)
(
ap
(
SNo_recipaux
x0
recip_SNo_pos
x1
)
1
⊆
real
)
)
(proof)
Theorem
real_recip_SNo_pos
real_recip_SNo_pos
:
∀ x0 .
x0
∈
real
⟶
SNoLt
0
x0
⟶
recip_SNo_pos
x0
∈
real
(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
Known
recip_SNo_negcase
recip_SNo_negcase
:
∀ x0 .
SNo
x0
⟶
SNoLt
x0
0
⟶
recip_SNo
x0
=
minus_SNo
(
recip_SNo_pos
(
minus_SNo
x0
)
)
Known
minus_SNo_Lt_contra2
minus_SNo_Lt_contra2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
x0
(
minus_SNo
x1
)
⟶
SNoLt
x1
(
minus_SNo
x0
)
Known
recip_SNo_0
recip_SNo_0
:
recip_SNo
0
=
0
Theorem
real_recip_SNo
real_recip_SNo
:
∀ x0 .
x0
∈
real
⟶
recip_SNo
x0
∈
real
(proof)
Theorem
real_div_SNo
real_div_SNo
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
x1
∈
real
⟶
div_SNo
x0
x1
∈
real
(proof)