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address
PUQfCc6UKzeHisvMRB55YuHo9ia3MNN38aV
total
0
mg
-
conjpub
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current assets
d2024..
/
d573a..
bday:
12334
doc published by
PrGxv..
Param
real
real
:
ι
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Param
Pi
Pi
:
ι
→
(
ι
→
ι
) →
ι
Definition
setexp
setexp
:=
λ x0 x1 .
Pi
x1
(
λ x2 .
x0
)
Param
SNoS_
SNoS_
:
ι
→
ι
Param
omega
omega
:
ι
Param
SNoLt
SNoLt
:
ι
→
ι
→
ο
Param
ap
ap
:
ι
→
ι
→
ι
Param
eps_
eps_
:
ι
→
ι
Param
minus_SNo
minus_SNo
:
ι
→
ι
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Param
SNo
SNo
:
ι
→
ο
Definition
SNoCutP
SNoCutP
:=
λ x0 x1 .
and
(
and
(
∀ x2 .
x2
∈
x0
⟶
SNo
x2
)
(
∀ x2 .
x2
∈
x1
⟶
SNo
x2
)
)
(
∀ x2 .
x2
∈
x0
⟶
∀ x3 .
x3
∈
x1
⟶
SNoLt
x2
x3
)
Param
SNoCut
SNoCut
:
ι
→
ι
→
ι
Known
SNo_approx_real_rep
SNo_approx_real_rep
:
∀ x0 .
x0
∈
real
⟶
∀ x1 : ο .
(
∀ x2 .
x2
∈
setexp
(
SNoS_
omega
)
omega
⟶
∀ x3 .
x3
∈
setexp
(
SNoS_
omega
)
omega
⟶
(
∀ x4 .
x4
∈
omega
⟶
SNoLt
(
ap
x2
x4
)
x0
)
⟶
(
∀ x4 .
x4
∈
omega
⟶
SNoLt
x0
(
add_SNo
(
ap
x2
x4
)
(
eps_
x4
)
)
)
⟶
(
∀ x4 .
x4
∈
omega
⟶
∀ x5 .
x5
∈
x4
⟶
SNoLt
(
ap
x2
x5
)
(
ap
x2
x4
)
)
⟶
(
∀ x4 .
x4
∈
omega
⟶
SNoLt
(
add_SNo
(
ap
x3
x4
)
(
minus_SNo
(
eps_
x4
)
)
)
x0
)
⟶
(
∀ x4 .
x4
∈
omega
⟶
SNoLt
x0
(
ap
x3
x4
)
)
⟶
(
∀ x4 .
x4
∈
omega
⟶
∀ x5 .
x5
∈
x4
⟶
SNoLt
(
ap
x3
x4
)
(
ap
x3
x5
)
)
⟶
SNoCutP
(
prim5
omega
(
ap
x2
)
)
(
prim5
omega
(
ap
x3
)
)
⟶
x0
=
SNoCut
(
prim5
omega
(
ap
x2
)
)
(
prim5
omega
(
ap
x3
)
)
⟶
x1
)
⟶
x1
Param
ordsucc
ordsucc
:
ι
→
ι
Param
lam
Sigma
:
ι
→
(
ι
→
ι
) →
ι
Param
SNoLev
SNoLev
:
ι
→
ι
Param
binunion
binunion
:
ι
→
ι
→
ι
Param
famunion
famunion
:
ι
→
(
ι
→
ι
) →
ι
Param
Subq
Subq
:
ι
→
ι
→
ο
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_approx_real
SNo_approx_real
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
setexp
(
SNoS_
omega
)
omega
⟶
∀ x2 .
x2
∈
setexp
(
SNoS_
omega
)
omega
⟶
(
∀ x3 .
x3
∈
omega
⟶
SNoLt
(
ap
x1
x3
)
x0
)
⟶
(
∀ x3 .
x3
∈
omega
⟶
SNoLt
x0
(
add_SNo
(
ap
x1
x3
)
(
eps_
x3
)
)
)
⟶
(
∀ x3 .
x3
∈
omega
⟶
∀ x4 .
x4
∈
x3
⟶
SNoLt
(
ap
x1
x4
)
(
ap
x1
x3
)
)
⟶
(
∀ x3 .
x3
∈
omega
⟶
SNoLt
x0
(
ap
x2
x3
)
)
⟶
(
∀ x3 .
x3
∈
omega
⟶
∀ x4 .
x4
∈
x3
⟶
SNoLt
(
ap
x2
x3
)
(
ap
x2
x4
)
)
⟶
x0
=
SNoCut
(
prim5
omega
(
ap
x1
)
)
(
prim5
omega
(
ap
x2
)
)
⟶
x0
∈
real
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Known
lam_Pi
lam_Pi
:
∀ x0 .
∀ x1 x2 :
ι → ι
.
(
∀ x3 .
x3
∈
x0
⟶
x2
x3
∈
x1
x3
)
⟶
lam
x0
x2
∈
Pi
x0
x1
Known
add_SNo_SNoS_omega
add_SNo_SNoS_omega
:
∀ x0 .
x0
∈
SNoS_
omega
⟶
∀ x1 .
x1
∈
SNoS_
omega
⟶
add_SNo
x0
x1
∈
SNoS_
omega
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
)
Known
omega_ordsucc
omega_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
ordsucc
x0
∈
omega
Param
nat_p
nat_p
:
ι
→
ο
Known
nat_ordsucc_in_ordsucc
nat_ordsucc_in_ordsucc
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
x0
⟶
ordsucc
x1
∈
ordsucc
x0
Known
omega_nat_p
omega_nat_p
:
∀ x0 .
x0
∈
omega
⟶
nat_p
x0
Known
nat_p_omega
nat_p_omega
:
∀ x0 .
nat_p
x0
⟶
x0
∈
omega
Known
nat_p_trans
nat_p_trans
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
x0
⟶
nat_p
x1
Param
SNoL
SNoL
:
ι
→
ι
Param
SNoR
SNoR
:
ι
→
ι
Known
add_SNo_eq
add_SNo_eq
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
SNo
x1
⟶
add_SNo
x0
x1
=
SNoCut
(
binunion
{
add_SNo
x3
x1
|x3 ∈
SNoL
x0
}
(
prim5
(
SNoL
x1
)
(
add_SNo
x0
)
)
)
(
binunion
{
add_SNo
x3
x1
|x3 ∈
SNoR
x0
}
(
prim5
(
SNoR
x1
)
(
add_SNo
x0
)
)
)
Known
SNoCut_ext
SNoCut_ext
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
(
∀ x4 .
x4
∈
x0
⟶
SNoLt
x4
(
SNoCut
x2
x3
)
)
⟶
(
∀ x4 .
x4
∈
x1
⟶
SNoLt
(
SNoCut
x2
x3
)
x4
)
⟶
(
∀ x4 .
x4
∈
x2
⟶
SNoLt
x4
(
SNoCut
x0
x1
)
)
⟶
(
∀ x4 .
x4
∈
x3
⟶
SNoLt
(
SNoCut
x0
x1
)
x4
)
⟶
SNoCut
x0
x1
=
SNoCut
x2
x3
Known
add_SNo_SNoCutP
add_SNo_SNoCutP
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoCutP
(
binunion
{
add_SNo
x2
x1
|x2 ∈
SNoL
x0
}
(
prim5
(
SNoL
x1
)
(
add_SNo
x0
)
)
)
(
binunion
{
add_SNo
x2
x1
|x2 ∈
SNoR
x0
}
(
prim5
(
SNoR
x1
)
(
add_SNo
x0
)
)
)
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
binunionE
binunionE
:
∀ x0 x1 x2 .
x2
∈
binunion
x0
x1
⟶
or
(
x2
∈
x0
)
(
x2
∈
x1
)
Known
ReplE_impred
ReplE_impred
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
prim5
x0
x1
⟶
∀ x3 : ο .
(
∀ x4 .
x4
∈
x0
⟶
x2
=
x1
x4
⟶
x3
)
⟶
x3
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
SNoLe
SNoLe
:
ι
→
ι
→
ο
Definition
False
False
:=
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Known
dneg
dneg
:
∀ x0 : ο .
not
(
not
x0
)
⟶
x0
Known
SNoLt_irref
SNoLt_irref
:
∀ x0 .
not
(
SNoLt
x0
x0
)
Param
abs_SNo
abs_SNo
:
ι
→
ι
Known
real_SNoS_omega_prop
real_SNoS_omega_prop
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
x1
∈
SNoS_
omega
⟶
(
∀ x2 .
x2
∈
omega
⟶
SNoLt
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
)
(
eps_
x2
)
)
⟶
x1
=
x0
Known
SNoLev_In_real_SNoS_omega
SNoLev_In_real_SNoS_omega
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
SNo
x1
⟶
SNoLev
x1
∈
SNoLev
x0
⟶
x1
∈
SNoS_
omega
Known
abs_SNo_dist_swap
abs_SNo_dist_swap
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
abs_SNo
(
add_SNo
x0
(
minus_SNo
x1
)
)
=
abs_SNo
(
add_SNo
x1
(
minus_SNo
x0
)
)
Known
pos_abs_SNo
pos_abs_SNo
:
∀ x0 .
SNoLt
0
x0
⟶
abs_SNo
x0
=
x0
Known
SNoLt_minus_pos
SNoLt_minus_pos
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
x0
x1
⟶
SNoLt
0
(
add_SNo
x1
(
minus_SNo
x0
)
)
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
SNo_eps_
SNo_eps_
:
∀ x0 .
x0
∈
omega
⟶
SNo
(
eps_
x0
)
Known
SNoLtLe_or
SNoLtLe_or
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
or
(
SNoLt
x0
x1
)
(
SNoLe
x1
x0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
SNoLtLe
SNoLtLe
:
∀ x0 x1 .
SNoLt
x0
x1
⟶
SNoLe
x0
x1
Known
add_SNo_Lt1_cancel
add_SNo_Lt1_cancel
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x2
x1
)
⟶
SNoLt
x0
x2
Known
SNoLeLt_tra
SNoLeLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
x0
x2
Known
add_SNo_com_3b_1_2
add_SNo_com_3b_1_2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
add_SNo
(
add_SNo
x0
x1
)
x2
=
add_SNo
(
add_SNo
x0
x2
)
x1
Known
add_SNo_Le1
add_SNo_Le1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x2
⟶
SNoLe
(
add_SNo
x0
x1
)
(
add_SNo
x2
x1
)
Known
add_SNo_com
add_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
x0
x1
=
add_SNo
x1
x0
Known
ReplI
ReplI
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
x1
x2
∈
prim5
x0
x1
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
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
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
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
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
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
)
)
Known
and3I
and3I
:
∀ x0 x1 x2 : ο .
x0
⟶
x1
⟶
x2
⟶
and
(
and
x0
x1
)
x2
Known
SNoLt_tra
SNoLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
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
minus_add_SNo_distr
minus_add_SNo_distr
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
minus_SNo
(
add_SNo
x0
x1
)
=
add_SNo
(
minus_SNo
x0
)
(
minus_SNo
x1
)
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
beta
beta
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
ap
(
lam
x0
x1
)
x2
=
x1
x2
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
omega_ordinal
omega_ordinal
:
ordinal
omega
Known
ap_Pi
ap_Pi
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 x3 .
x2
∈
Pi
x0
x1
⟶
x3
∈
x0
⟶
ap
x2
x3
∈
x1
x3
Known
real_SNo
real_SNo
:
∀ x0 .
x0
∈
real
⟶
SNo
x0
Theorem
real_add_SNo
real_add_SNo
:
∀ x0 .
x0
∈
real
⟶
∀ x1 .
x1
∈
real
⟶
add_SNo
x0
x1
∈
real
(proof)
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
1a68c..
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
add_SNo
x0
x1
=
add_SNo
x3
x2
⟶
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
=
x3
(proof)
Known
SNoLeE
SNoLeE
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
x0
x1
⟶
or
(
SNoLt
x0
x1
)
(
x0
=
x1
)
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
SNoLe_ref
SNoLe_ref
:
∀ x0 .
SNoLe
x0
x0
Theorem
add_SNo_minus_Le1b3
add_SNo_minus_Le1b3
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLe
(
add_SNo
x0
x1
)
(
add_SNo
x3
x2
)
⟶
SNoLe
(
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
x3
(proof)
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