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Proofgold Asset
asset id
134916c5ccc7f5983b5dd0b6d0ed11f4c064cf52ad112b09050e1f212b4c240b
asset hash
04d203f6fa5f0cccc39657525d0683369db4ddc87da8e44dc2176f46777a029c
bday / block
27765
tx
9a4a7..
preasset
doc published by
PrQUS..
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Param
SNo
SNo
:
ι
→
ο
Param
SNoLt
SNoLt
:
ι
→
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
Param
SNoL
SNoL
:
ι
→
ι
Param
mul_SNo
mul_SNo
:
ι
→
ι
→
ι
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Param
SNoLe
SNoLe
:
ι
→
ι
→
ο
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Param
SNoLev
SNoLev
:
ι
→
ι
Param
ordsucc
ordsucc
:
ι
→
ι
Param
binunion
binunion
:
ι
→
ι
→
ι
Param
famunion
famunion
:
ι
→
(
ι
→
ι
) →
ι
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
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
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
SNoS_
SNoS_
:
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
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
)
Known
SNoLtLe_tra
SNoLtLe_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLe
x1
x2
⟶
SNoLt
x0
x2
Param
minus_SNo
minus_SNo
:
ι
→
ι
Known
mul_SNoCut_abs
mul_SNoCut_abs
:
∀ x0 x1 x2 x3 x4 x5 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
x4
=
SNoCut
x0
x1
⟶
x5
=
SNoCut
x2
x3
⟶
∀ x6 : ο .
(
∀ x7 x8 x9 x10 .
(
∀ x11 .
x11
∈
x7
⟶
∀ x12 : ο .
(
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x2
⟶
SNo
x13
⟶
SNo
x14
⟶
SNoLt
x13
x4
⟶
SNoLt
x14
x5
⟶
x11
=
add_SNo
(
mul_SNo
x13
x5
)
(
add_SNo
(
mul_SNo
x4
x14
)
(
minus_SNo
(
mul_SNo
x13
x14
)
)
)
⟶
x12
)
⟶
x12
)
⟶
(
∀ x11 .
x11
∈
x0
⟶
∀ x12 .
x12
∈
x2
⟶
add_SNo
(
mul_SNo
x11
x5
)
(
add_SNo
(
mul_SNo
x4
x12
)
(
minus_SNo
(
mul_SNo
x11
x12
)
)
)
∈
x7
)
⟶
(
∀ x11 .
x11
∈
x8
⟶
∀ x12 : ο .
(
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x3
⟶
SNo
x13
⟶
SNo
x14
⟶
SNoLt
x4
x13
⟶
SNoLt
x5
x14
⟶
x11
=
add_SNo
(
mul_SNo
x13
x5
)
(
add_SNo
(
mul_SNo
x4
x14
)
(
minus_SNo
(
mul_SNo
x13
x14
)
)
)
⟶
x12
)
⟶
x12
)
⟶
(
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x3
⟶
add_SNo
(
mul_SNo
x11
x5
)
(
add_SNo
(
mul_SNo
x4
x12
)
(
minus_SNo
(
mul_SNo
x11
x12
)
)
)
∈
x8
)
⟶
(
∀ x11 .
x11
∈
x9
⟶
∀ x12 : ο .
(
∀ x13 .
x13
∈
x0
⟶
∀ x14 .
x14
∈
x3
⟶
SNo
x13
⟶
SNo
x14
⟶
SNoLt
x13
x4
⟶
SNoLt
x5
x14
⟶
x11
=
add_SNo
(
mul_SNo
x13
x5
)
(
add_SNo
(
mul_SNo
x4
x14
)
(
minus_SNo
(
mul_SNo
x13
x14
)
)
)
⟶
x12
)
⟶
x12
)
⟶
(
∀ x11 .
x11
∈
x0
⟶
∀ x12 .
x12
∈
x3
⟶
add_SNo
(
mul_SNo
x11
x5
)
(
add_SNo
(
mul_SNo
x4
x12
)
(
minus_SNo
(
mul_SNo
x11
x12
)
)
)
∈
x9
)
⟶
(
∀ x11 .
x11
∈
x10
⟶
∀ x12 : ο .
(
∀ x13 .
x13
∈
x1
⟶
∀ x14 .
x14
∈
x2
⟶
SNo
x13
⟶
SNo
x14
⟶
SNoLt
x4
x13
⟶
SNoLt
x14
x5
⟶
x11
=
add_SNo
(
mul_SNo
x13
x5
)
(
add_SNo
(
mul_SNo
x4
x14
)
(
minus_SNo
(
mul_SNo
x13
x14
)
)
)
⟶
x12
)
⟶
x12
)
⟶
(
∀ x11 .
x11
∈
x1
⟶
∀ x12 .
x12
∈
x2
⟶
add_SNo
(
mul_SNo
x11
x5
)
(
add_SNo
(
mul_SNo
x4
x12
)
(
minus_SNo
(
mul_SNo
x11
x12
)
)
)
∈
x10
)
⟶
SNoCutP
(
binunion
x7
x8
)
(
binunion
x9
x10
)
⟶
mul_SNo
x4
x5
=
SNoCut
(
binunion
x7
x8
)
(
binunion
x9
x10
)
⟶
x6
)
⟶
x6
Param
SNoR
SNoR
:
ι
→
ι
Known
SNo_eta
SNo_eta
:
∀ x0 .
SNo
x0
⟶
x0
=
SNoCut
(
SNoL
x0
)
(
SNoR
x0
)
Known
SNoCut_Le
SNoCut_Le
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
(
∀ x4 .
x4
∈
x0
⟶
SNoLt
x4
(
SNoCut
x2
x3
)
)
⟶
(
∀ x4 .
x4
∈
x3
⟶
SNoLt
(
SNoCut
x0
x1
)
x4
)
⟶
SNoLe
(
SNoCut
x0
x1
)
(
SNoCut
x2
x3
)
Known
SNoCutP_SNoL_SNoR
SNoCutP_SNoL_SNoR
:
∀ x0 .
SNo
x0
⟶
SNoCutP
(
SNoL
x0
)
(
SNoR
x0
)
Known
binunionE'
binunionE
:
∀ x0 x1 x2 .
∀ x3 : ο .
(
x2
∈
x0
⟶
x3
)
⟶
(
x2
∈
x1
⟶
x3
)
⟶
x2
∈
binunion
x0
x1
⟶
x3
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
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
Known
SNoLtLe_or
SNoLtLe_or
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
or
(
SNoLt
x0
x1
)
(
SNoLe
x1
x0
)
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Known
orIL
orIL
:
∀ x0 x1 : ο .
x0
⟶
or
x0
x1
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Known
orIR
orIR
:
∀ x0 x1 : ο .
x1
⟶
or
x0
x1
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
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
SNoS_I2
SNoS_I2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLev
x0
∈
SNoLev
x1
⟶
x0
∈
SNoS_
(
SNoLev
x1
)
Param
ordinal
ordinal
:
ι
→
ο
Definition
TransSet
TransSet
:=
λ x0 .
∀ x1 .
x1
∈
x0
⟶
x1
⊆
x0
Known
ordinal_TransSet
ordinal_TransSet
:
∀ x0 .
ordinal
x0
⟶
TransSet
x0
Known
SNoLev_ordinal
SNoLev_ordinal
:
∀ x0 .
SNo
x0
⟶
ordinal
(
SNoLev
x0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
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
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
In_no2cycle
In_no2cycle
:
∀ x0 x1 .
x0
∈
x1
⟶
x1
∈
x0
⟶
False
Theorem
mul_SNo_SNoCut_SNoL_interpolate
mul_SNo_SNoCut_SNoL_interpolate
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
∀ x4 x5 .
x4
=
SNoCut
x0
x1
⟶
x5
=
SNoCut
x2
x3
⟶
∀ x6 .
x6
∈
SNoL
(
mul_SNo
x4
x5
)
⟶
or
(
∀ x7 : ο .
(
∀ x8 .
and
(
x8
∈
x0
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
x10
∈
x2
)
(
SNoLe
(
add_SNo
x6
(
mul_SNo
x8
x10
)
)
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x10
)
)
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
x8
∈
x1
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
x10
∈
x3
)
(
SNoLe
(
add_SNo
x6
(
mul_SNo
x8
x10
)
)
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x10
)
)
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
(proof)
Theorem
mul_SNo_SNoCut_SNoL_interpolate_impred
mul_SNo_SNoCut_SNoL_interpolate_impred
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
∀ x4 x5 .
x4
=
SNoCut
x0
x1
⟶
x5
=
SNoCut
x2
x3
⟶
∀ x6 .
x6
∈
SNoL
(
mul_SNo
x4
x5
)
⟶
∀ x7 : ο .
(
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x2
⟶
SNoLe
(
add_SNo
x6
(
mul_SNo
x8
x9
)
)
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x9
)
)
⟶
x7
)
⟶
(
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x3
⟶
SNoLe
(
add_SNo
x6
(
mul_SNo
x8
x9
)
)
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x9
)
)
⟶
x7
)
⟶
x7
(proof)
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
)
)
)
Theorem
mul_SNo_SNoCut_SNoR_interpolate
mul_SNo_SNoCut_SNoR_interpolate
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
∀ x4 x5 .
x4
=
SNoCut
x0
x1
⟶
x5
=
SNoCut
x2
x3
⟶
∀ x6 .
x6
∈
SNoR
(
mul_SNo
x4
x5
)
⟶
or
(
∀ x7 : ο .
(
∀ x8 .
and
(
x8
∈
x0
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
x10
∈
x3
)
(
SNoLe
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x10
)
)
(
add_SNo
x6
(
mul_SNo
x8
x10
)
)
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
(
∀ x7 : ο .
(
∀ x8 .
and
(
x8
∈
x1
)
(
∀ x9 : ο .
(
∀ x10 .
and
(
x10
∈
x2
)
(
SNoLe
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x10
)
)
(
add_SNo
x6
(
mul_SNo
x8
x10
)
)
)
⟶
x9
)
⟶
x9
)
⟶
x7
)
⟶
x7
)
(proof)
Theorem
mul_SNo_SNoCut_SNoR_interpolate_impred
mul_SNo_SNoCut_SNoR_interpolate_impred
:
∀ x0 x1 x2 x3 .
SNoCutP
x0
x1
⟶
SNoCutP
x2
x3
⟶
∀ x4 x5 .
x4
=
SNoCut
x0
x1
⟶
x5
=
SNoCut
x2
x3
⟶
∀ x6 .
x6
∈
SNoR
(
mul_SNo
x4
x5
)
⟶
∀ x7 : ο .
(
∀ x8 .
x8
∈
x0
⟶
∀ x9 .
x9
∈
x3
⟶
SNoLe
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x9
)
)
(
add_SNo
x6
(
mul_SNo
x8
x9
)
)
⟶
x7
)
⟶
(
∀ x8 .
x8
∈
x1
⟶
∀ x9 .
x9
∈
x2
⟶
SNoLe
(
add_SNo
(
mul_SNo
x8
x5
)
(
mul_SNo
x4
x9
)
)
(
add_SNo
x6
(
mul_SNo
x8
x9
)
)
⟶
x7
)
⟶
x7
(proof)