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Proofgold Asset
asset id
a62bd21a0ab14bbcd0127e7278d2034c57eac600c1de5284cc52198f4d942aef
asset hash
f47713b676614d271ceb6bdc7b5f69cfd389e5b357d7b994e5d8409a7f4b838e
bday / block
4981
tx
0d8c8..
preasset
doc published by
Pr6Pc..
Param
famunion
famunion
:
ι
→
(
ι
→
ι
) →
ι
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Known
Empty_Subq_eq
Empty_Subq_eq
:
∀ x0 .
x0
⊆
0
⟶
x0
=
0
Known
famunionE_impred
famunionE_impred
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
famunion
x0
x1
⟶
∀ x3 : ο .
(
∀ x4 .
x4
∈
x0
⟶
x2
∈
x1
x4
⟶
x3
)
⟶
x3
Definition
False
False
:=
∀ x0 : ο .
x0
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
EmptyE
EmptyE
:
∀ x0 .
nIn
x0
0
Theorem
famunion_Empty
famunion_Empty
:
∀ x0 :
ι → ι
.
famunion
0
x0
=
0
(proof)
Param
SNo
SNo
:
ι
→
ο
Param
and
and
:
ο
→
ο
→
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
)
Known
and3I
and3I
:
∀ x0 x1 x2 : ο .
x0
⟶
x1
⟶
x2
⟶
and
(
and
x0
x1
)
x2
Theorem
SNoCutP_L_0
SNoCutP_L_0
:
∀ x0 .
(
∀ x1 .
x1
∈
x0
⟶
SNo
x1
)
⟶
SNoCutP
x0
0
(proof)
Theorem
SNoCutP_0_R
SNoCutP_0_R
:
∀ x0 .
(
∀ x1 .
x1
∈
x0
⟶
SNo
x1
)
⟶
SNoCutP
0
x0
(proof)
Theorem
SNoCutP_0_0
SNoCutP_0_0
:
SNoCutP
0
0
(proof)
Param
SNoCut
SNoCut
:
ι
→
ι
→
ι
Param
SNoLev
SNoLev
:
ι
→
ι
Param
ordsucc
ordsucc
:
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
binunion_idl
binunion_idl
:
∀ x0 .
binunion
0
x0
=
x0
Known
SNo_eq
SNo_eq
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLev
x0
=
SNoLev
x1
⟶
SNoEq_
(
SNoLev
x0
)
x0
x1
⟶
x0
=
x1
Known
SNo_0
SNo_0
:
SNo
0
Known
SNoLev_0
SNoLev_0
:
SNoLev
0
=
0
Param
iff
iff
:
ο
→
ο
→
ο
Known
SNoEq_I
SNoEq_I
:
∀ x0 x1 x2 .
(
∀ x3 .
x3
∈
x0
⟶
iff
(
x3
∈
x1
)
(
x3
∈
x2
)
)
⟶
SNoEq_
x0
x1
x2
Known
cases_1
cases_1
:
∀ x0 .
x0
∈
1
⟶
∀ x1 :
ι → ο
.
x1
0
⟶
x1
x0
Theorem
SNoCut_0_0
SNoCut_0_0
:
SNoCut
0
0
=
0
(proof)
Param
SNoL
SNoL
:
ι
→
ι
Known
set_ext
set_ext
:
∀ x0 x1 .
x0
⊆
x1
⟶
x1
⊆
x0
⟶
x0
=
x1
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
Known
SNo_1
SNo_1
:
SNo
1
Param
ordinal
ordinal
:
ι
→
ο
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
nat_1
nat_1
:
nat_p
1
Known
In_0_1
In_0_1
:
0
∈
1
Known
SNoL_I
SNoL_I
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
SNo
x1
⟶
SNoLev
x1
∈
SNoLev
x0
⟶
SNoLt
x1
x0
⟶
x1
∈
SNoL
x0
Known
ordinal_In_SNoLt
ordinal_In_SNoLt
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
x1
∈
x0
⟶
SNoLt
x1
x0
Theorem
SNoL_1
SNoL_1
:
SNoL
1
=
1
(proof)
Param
SNoR
SNoR
:
ι
→
ι
Known
ordinal_SNoR
ordinal_SNoR
:
∀ x0 .
ordinal
x0
⟶
SNoR
x0
=
0
Theorem
SNoR_1
SNoR_1
:
SNoR
1
=
0
(proof)
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Theorem
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
)
)
(proof)
Param
minus_SNo
minus_SNo
:
ι
→
ι
Known
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
x0
)
Theorem
SNo_add_SNo_3c
SNo_add_SNo_3c
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
(
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
(proof)
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
)
Theorem
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
)
)
(proof)
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_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
)
Theorem
add_SNo_3_3_3_Lt1
add_SNo_3_3_3_Lt1
:
∀ x0 x1 x2 x3 x4 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNoLt
(
add_SNo
x0
x1
)
(
add_SNo
x2
x3
)
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x1
x4
)
)
(
add_SNo
x2
(
add_SNo
x3
x4
)
)
(proof)
Known
add_SNo_rotate_3_1
add_SNo_rotate_3_1
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
add_SNo
x0
(
add_SNo
x1
x2
)
=
add_SNo
x2
(
add_SNo
x0
x1
)
Theorem
add_SNo_3_2_3_Lt1
add_SNo_3_2_3_Lt1
:
∀ x0 x1 x2 x3 x4 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNoLt
(
add_SNo
x1
x0
)
(
add_SNo
x2
x3
)
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x4
x1
)
)
(
add_SNo
x2
(
add_SNo
x3
x4
)
)
(proof)
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
add_SNo_minus_SNo_linv
add_SNo_minus_SNo_linv
:
∀ x0 .
SNo
x0
⟶
add_SNo
(
minus_SNo
x0
)
x0
=
0
Known
add_SNo_0R
add_SNo_0R
:
∀ x0 .
SNo
x0
⟶
add_SNo
x0
0
=
x0
Theorem
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
(proof)
Theorem
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
)
)
(proof)
Known
add_SNo_com
add_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
x0
x1
=
add_SNo
x1
x0
Theorem
add_SNo_minus_Lt12b3
add_SNo_minus_Lt12b3
:
∀ x0 x1 x2 x3 x4 x5 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
x4
⟶
SNo
x5
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x1
x5
)
)
(
add_SNo
x3
(
add_SNo
x4
x2
)
)
⟶
SNoLt
(
add_SNo
x0
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
(
add_SNo
x3
(
add_SNo
x4
(
minus_SNo
x5
)
)
)
(proof)
Param
mul_SNo
mul_SNo
:
ι
→
ι
→
ι
Known
mul_SNo_prop_1
mul_SNo_prop_1
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
SNo
x1
⟶
∀ x2 : ο .
(
SNo
(
mul_SNo
x0
x1
)
⟶
(
∀ x3 .
x3
∈
SNoL
x0
⟶
∀ x4 .
x4
∈
SNoL
x1
⟶
SNoLt
(
add_SNo
(
mul_SNo
x3
x1
)
(
mul_SNo
x0
x4
)
)
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x3
x4
)
)
)
⟶
(
∀ x3 .
x3
∈
SNoR
x0
⟶
∀ x4 .
x4
∈
SNoR
x1
⟶
SNoLt
(
add_SNo
(
mul_SNo
x3
x1
)
(
mul_SNo
x0
x4
)
)
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x3
x4
)
)
)
⟶
(
∀ x3 .
x3
∈
SNoL
x0
⟶
∀ x4 .
x4
∈
SNoR
x1
⟶
SNoLt
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x3
x4
)
)
(
add_SNo
(
mul_SNo
x3
x1
)
(
mul_SNo
x0
x4
)
)
)
⟶
(
∀ x3 .
x3
∈
SNoR
x0
⟶
∀ x4 .
x4
∈
SNoL
x1
⟶
SNoLt
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x3
x4
)
)
(
add_SNo
(
mul_SNo
x3
x1
)
(
mul_SNo
x0
x4
)
)
)
⟶
x2
)
⟶
x2
Theorem
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
(proof)
Theorem
SNo_mul_SNo_lem
SNo_mul_SNo_lem
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNo
(
add_SNo
(
mul_SNo
x2
x1
)
(
add_SNo
(
mul_SNo
x0
x3
)
(
minus_SNo
(
mul_SNo
x2
x3
)
)
)
)
(proof)
Known
mul_SNo_eq_2
mul_SNo_eq_2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
∀ x2 : ο .
(
∀ x3 x4 .
(
∀ x5 .
x5
∈
x3
⟶
∀ x6 : ο .
(
∀ x7 .
x7
∈
SNoL
x0
⟶
∀ x8 .
x8
∈
SNoL
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
(
∀ x7 .
x7
∈
SNoR
x0
⟶
∀ x8 .
x8
∈
SNoR
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
x6
)
⟶
(
∀ x5 .
x5
∈
SNoL
x0
⟶
∀ x6 .
x6
∈
SNoL
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x3
)
⟶
(
∀ x5 .
x5
∈
SNoR
x0
⟶
∀ x6 .
x6
∈
SNoR
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x3
)
⟶
(
∀ x5 .
x5
∈
x4
⟶
∀ x6 : ο .
(
∀ x7 .
x7
∈
SNoL
x0
⟶
∀ x8 .
x8
∈
SNoR
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
(
∀ x7 .
x7
∈
SNoR
x0
⟶
∀ x8 .
x8
∈
SNoL
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
x6
)
⟶
(
∀ x5 .
x5
∈
SNoL
x0
⟶
∀ x6 .
x6
∈
SNoR
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x4
)
⟶
(
∀ x5 .
x5
∈
SNoR
x0
⟶
∀ x6 .
x6
∈
SNoL
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x4
)
⟶
mul_SNo
x0
x1
=
SNoCut
x3
x4
⟶
x2
)
⟶
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
SNoLt_tra
SNoLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
x0
x2
Theorem
mul_SNo_eq_3
mul_SNo_eq_3
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
∀ x2 : ο .
(
∀ x3 x4 .
SNoCutP
x3
x4
⟶
(
∀ x5 .
x5
∈
x3
⟶
∀ x6 : ο .
(
∀ x7 .
x7
∈
SNoL
x0
⟶
∀ x8 .
x8
∈
SNoL
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
(
∀ x7 .
x7
∈
SNoR
x0
⟶
∀ x8 .
x8
∈
SNoR
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
x6
)
⟶
(
∀ x5 .
x5
∈
SNoL
x0
⟶
∀ x6 .
x6
∈
SNoL
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x3
)
⟶
(
∀ x5 .
x5
∈
SNoR
x0
⟶
∀ x6 .
x6
∈
SNoR
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x3
)
⟶
(
∀ x5 .
x5
∈
x4
⟶
∀ x6 : ο .
(
∀ x7 .
x7
∈
SNoL
x0
⟶
∀ x8 .
x8
∈
SNoR
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
(
∀ x7 .
x7
∈
SNoR
x0
⟶
∀ x8 .
x8
∈
SNoL
x1
⟶
x5
=
add_SNo
(
mul_SNo
x7
x1
)
(
add_SNo
(
mul_SNo
x0
x8
)
(
minus_SNo
(
mul_SNo
x7
x8
)
)
)
⟶
x6
)
⟶
x6
)
⟶
(
∀ x5 .
x5
∈
SNoL
x0
⟶
∀ x6 .
x6
∈
SNoR
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x4
)
⟶
(
∀ x5 .
x5
∈
SNoR
x0
⟶
∀ x6 .
x6
∈
SNoL
x1
⟶
add_SNo
(
mul_SNo
x5
x1
)
(
add_SNo
(
mul_SNo
x0
x6
)
(
minus_SNo
(
mul_SNo
x5
x6
)
)
)
∈
x4
)
⟶
mul_SNo
x0
x1
=
SNoCut
x3
x4
⟶
x2
)
⟶
x2
(proof)
Theorem
mul_SNo_Subq_lem
mul_SNo_Subq_lem
:
∀ x0 x1 x2 x3 x4 x5 x6 x7 .
(
∀ x8 .
x8
∈
x6
⟶
∀ x9 : ο .
(
∀ x10 .
x10
∈
x2
⟶
∀ x11 .
x11
∈
x3
⟶
x8
=
add_SNo
(
mul_SNo
x10
x1
)
(
add_SNo
(
mul_SNo
x0
x11
)
(
minus_SNo
(
mul_SNo
x10
x11
)
)
)
⟶
x9
)
⟶
(
∀ x10 .
x10
∈
x4
⟶
∀ x11 .
x11
∈
x5
⟶
x8
=
add_SNo
(
mul_SNo
x10
x1
)
(
add_SNo
(
mul_SNo
x0
x11
)
(
minus_SNo
(
mul_SNo
x10
x11
)
)
)
⟶
x9
)
⟶
x9
)
⟶
(
∀ x8 .
x8
∈
x2
⟶
∀ x9 .
x9
∈
x3
⟶
add_SNo
(
mul_SNo
x8
x1
)
(
add_SNo
(
mul_SNo
x0
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
∈
x7
)
⟶
(
∀ x8 .
x8
∈
x4
⟶
∀ x9 .
x9
∈
x5
⟶
add_SNo
(
mul_SNo
x8
x1
)
(
add_SNo
(
mul_SNo
x0
x9
)
(
minus_SNo
(
mul_SNo
x8
x9
)
)
)
∈
x7
)
⟶
x6
⊆
x7
(proof)
Known
SNoR_0
SNoR_0
:
SNoR
0
=
0
Known
SNoL_0
SNoL_0
:
SNoL
0
=
0
Theorem
mul_SNo_zeroR
mul_SNo_zeroR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
0
=
0
(proof)
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
Known
SNo_eta
SNo_eta
:
∀ x0 .
SNo
x0
⟶
x0
=
SNoCut
(
SNoL
x0
)
(
SNoR
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
SNoCutP_SNoL_SNoR
SNoCutP_SNoL_SNoR
:
∀ x0 .
SNo
x0
⟶
SNoCutP
(
SNoL
x0
)
(
SNoR
x0
)
Known
SNoCutP_SNoCut_L
SNoCutP_SNoCut_L
:
∀ x0 x1 .
SNoCutP
x0
x1
⟶
∀ x2 .
x2
∈
x0
⟶
SNoLt
x2
(
SNoCut
x0
x1
)
Known
minus_SNo_0
minus_SNo_0
:
minus_SNo
0
=
0
Known
SNoL_SNoS
SNoL_SNoS
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
SNoL
x0
⟶
x1
∈
SNoS_
(
SNoLev
x0
)
Known
SNoCutP_SNoCut_R
SNoCutP_SNoCut_R
:
∀ x0 x1 .
SNoCutP
x0
x1
⟶
∀ x2 .
x2
∈
x1
⟶
SNoLt
(
SNoCut
x0
x1
)
x2
Known
SNoR_SNoS
SNoR_SNoS
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
x1
∈
SNoR
x0
⟶
x1
∈
SNoS_
(
SNoLev
x0
)
Theorem
mul_SNo_oneR
mul_SNo_oneR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
1
=
x0
(proof)
Known
SNoLev_ind2
SNoLev_ind2
:
∀ x0 :
ι →
ι → ο
.
(
∀ x1 x2 .
SNo
x1
⟶
SNo
x2
⟶
(
∀ x3 .
x3
∈
SNoS_
(
SNoLev
x1
)
⟶
x0
x3
x2
)
⟶
(
∀ x3 .
x3
∈
SNoS_
(
SNoLev
x2
)
⟶
x0
x1
x3
)
⟶
(
∀ x3 .
x3
∈
SNoS_
(
SNoLev
x1
)
⟶
∀ x4 .
x4
∈
SNoS_
(
SNoLev
x2
)
⟶
x0
x3
x4
)
⟶
x0
x1
x2
)
⟶
∀ x1 x2 .
SNo
x1
⟶
SNo
x2
⟶
x0
x1
x2
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
)
Theorem
mul_SNo_com
mul_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
mul_SNo
x0
x1
=
mul_SNo
x1
x0
(proof)
Known
ReplI
ReplI
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
x1
x2
∈
prim5
x0
x1
Known
minus_SNo_invol
minus_SNo_invol
:
∀ x0 .
SNo
x0
⟶
minus_SNo
(
minus_SNo
x0
)
=
x0
Known
SNoR_I
SNoR_I
:
∀ x0 .
SNo
x0
⟶
∀ x1 .
SNo
x1
⟶
SNoLev
x1
∈
SNoLev
x0
⟶
SNoLt
x0
x1
⟶
x1
∈
SNoR
x0
Known
minus_SNo_Lev
minus_SNo_Lev
:
∀ x0 .
SNo
x0
⟶
SNoLev
(
minus_SNo
x0
)
=
SNoLev
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
minus_SNo_Lt_contra1
minus_SNo_Lt_contra1
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
(
minus_SNo
x0
)
x1
⟶
SNoLt
(
minus_SNo
x1
)
x0
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
minus_SNo_Lt_contra
minus_SNo_Lt_contra
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
x0
x1
⟶
SNoLt
(
minus_SNo
x1
)
(
minus_SNo
x0
)
Known
minus_SNoCut_eq
minus_SNoCut_eq
:
∀ x0 x1 .
SNoCutP
x0
x1
⟶
minus_SNo
(
SNoCut
x0
x1
)
=
SNoCut
(
prim5
x1
minus_SNo
)
(
prim5
x0
minus_SNo
)
Theorem
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
)
(proof)