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Proofgold Asset
asset id
cbca024233ac9f773eabe6ae017a39bbfe350a00cd76bc9c4eddfcd02a2e45db
asset hash
d048ab0823a5e859865eff6c0778a3e68bdb05dc87e853394ef9f27512827151
bday / block
6761
tx
26642..
preasset
doc published by
Pr6Pc..
Definition
False
False
:=
∀ x0 : ο .
x0
Known
In_ind
In_ind
:
∀ x0 :
ι → ο
.
(
∀ x1 .
(
∀ x2 .
x2
∈
x1
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
x0
x1
Theorem
In_no4cycle
In_no4cycle
:
∀ x0 x1 x2 x3 .
x0
∈
x1
⟶
x1
∈
x2
⟶
x2
∈
x3
⟶
x3
∈
x0
⟶
False
(proof)
Param
SNo
SNo
:
ι
→
ο
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
SNo_0
SNo_0
:
SNo
0
Known
mul_SNo_zeroR
mul_SNo_zeroR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
0
=
0
Theorem
mul_SNo_zeroL
mul_SNo_zeroL
:
∀ x0 .
SNo
x0
⟶
mul_SNo
0
x0
=
0
(proof)
Param
ordsucc
ordsucc
:
ι
→
ι
Known
SNo_1
SNo_1
:
SNo
1
Known
mul_SNo_oneR
mul_SNo_oneR
:
∀ x0 .
SNo
x0
⟶
mul_SNo
x0
1
=
x0
Theorem
mul_SNo_oneL
mul_SNo_oneL
:
∀ x0 .
SNo
x0
⟶
mul_SNo
1
x0
=
x0
(proof)
Param
SNoLt
SNoLt
:
ι
→
ι
→
ο
Param
add_SNo
add_SNo
:
ι
→
ι
→
ι
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
add_SNo_0R
add_SNo_0R
:
∀ x0 .
SNo
x0
⟶
add_SNo
x0
0
=
x0
Known
SNo_mul_SNo
SNo_mul_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
mul_SNo
x0
x1
)
Known
add_SNo_0L
add_SNo_0L
:
∀ x0 .
SNo
x0
⟶
add_SNo
0
x0
=
x0
Theorem
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
)
(proof)
Param
SNoLe
SNoLe
:
ι
→
ι
→
ο
Known
mul_SNo_Le
mul_SNo_Le
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLe
x2
x0
⟶
SNoLe
x3
x1
⟶
SNoLe
(
add_SNo
(
mul_SNo
x2
x1
)
(
mul_SNo
x0
x3
)
)
(
add_SNo
(
mul_SNo
x0
x1
)
(
mul_SNo
x2
x3
)
)
Theorem
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
)
(proof)
Theorem
neg_mul_SNo_Lt
neg_mul_SNo_Lt
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNoLt
x0
0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x2
x1
⟶
SNoLt
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
(proof)
Theorem
nonpos_mul_SNo_Le
nonpos_mul_SNo_Le
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNoLe
x0
0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x2
x1
⟶
SNoLe
(
mul_SNo
x0
x1
)
(
mul_SNo
x0
x2
)
(proof)
Param
If_i
If_i
:
ο
→
ι
→
ι
→
ι
Param
minus_SNo
minus_SNo
:
ι
→
ι
Definition
abs_SNo
abs_SNo
:=
λ x0 .
If_i
(
SNoLe
0
x0
)
x0
(
minus_SNo
x0
)
Known
If_i_1
If_i_1
:
∀ x0 : ο .
∀ x1 x2 .
x0
⟶
If_i
x0
x1
x2
=
x1
Theorem
nonneg_abs_SNo
nonneg_abs_SNo
:
∀ x0 .
SNoLe
0
x0
⟶
abs_SNo
x0
=
x0
(proof)
Definition
not
not
:=
λ x0 : ο .
x0
⟶
False
Known
If_i_0
If_i_0
:
∀ x0 : ο .
∀ x1 x2 .
not
x0
⟶
If_i
x0
x1
x2
=
x2
Theorem
not_nonneg_abs_SNo
not_nonneg_abs_SNo
:
∀ x0 .
not
(
SNoLe
0
x0
)
⟶
abs_SNo
x0
=
minus_SNo
x0
(proof)
Known
SNoLe_ref
SNoLe_ref
:
∀ x0 .
SNoLe
x0
x0
Theorem
abs_SNo_0
abs_SNo_0
:
abs_SNo
0
=
0
(proof)
Known
SNoLtLe
SNoLtLe
:
∀ x0 x1 .
SNoLt
x0
x1
⟶
SNoLe
x0
x1
Theorem
pos_abs_SNo
pos_abs_SNo
:
∀ x0 .
SNoLt
0
x0
⟶
abs_SNo
x0
=
x0
(proof)
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
Theorem
neg_abs_SNo
neg_abs_SNo
:
∀ x0 .
SNo
x0
⟶
SNoLt
x0
0
⟶
abs_SNo
x0
=
minus_SNo
x0
(proof)
Definition
or
or
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x2
)
⟶
(
x1
⟶
x2
)
⟶
x2
Known
xm
xm
:
∀ x0 : ο .
or
x0
(
not
x0
)
Known
SNo_minus_SNo
SNo_minus_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
minus_SNo
x0
)
Theorem
SNo_abs_SNo
SNo_abs_SNo
:
∀ x0 .
SNo
x0
⟶
SNo
(
abs_SNo
x0
)
(proof)
Param
SNoLev
SNoLev
:
ι
→
ι
Known
minus_SNo_Lev
minus_SNo_Lev
:
∀ x0 .
SNo
x0
⟶
SNoLev
(
minus_SNo
x0
)
=
SNoLev
x0
Theorem
abs_SNo_Lev
abs_SNo_Lev
:
∀ x0 .
SNo
x0
⟶
SNoLev
(
abs_SNo
x0
)
=
SNoLev
x0
(proof)
Known
SNoLtLe_or
SNoLtLe_or
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
or
(
SNoLt
x0
x1
)
(
SNoLe
x1
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_0
minus_SNo_0
:
minus_SNo
0
=
0
Known
minus_SNo_invol
minus_SNo_invol
:
∀ x0 .
SNo
x0
⟶
minus_SNo
(
minus_SNo
x0
)
=
x0
Known
SNoLe_antisym
SNoLe_antisym
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
x0
x1
⟶
SNoLe
x1
x0
⟶
x0
=
x1
Known
minus_SNo_Le_contra
minus_SNo_Le_contra
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
x0
x1
⟶
SNoLe
(
minus_SNo
x1
)
(
minus_SNo
x0
)
Theorem
abs_SNo_minus
abs_SNo_minus
:
∀ x0 .
SNo
x0
⟶
abs_SNo
(
minus_SNo
x0
)
=
abs_SNo
x0
(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
)
Known
add_SNo_com
add_SNo_com
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
x0
x1
=
add_SNo
x1
x0
Known
SNo_add_SNo
SNo_add_SNo
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNo
(
add_SNo
x0
x1
)
Theorem
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
)
)
(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
)
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
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_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
SNoLe_tra
SNoLe_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x1
⟶
SNoLe
x1
x2
⟶
SNoLe
x0
x2
Known
add_SNo_Le3
add_SNo_Le3
:
∀ x0 x1 x2 x3 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNo
x3
⟶
SNoLe
x0
x2
⟶
SNoLe
x1
x3
⟶
SNoLe
(
add_SNo
x0
x1
)
(
add_SNo
x2
x3
)
Theorem
SNo_triangle
SNo_triangle
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLe
(
abs_SNo
(
add_SNo
x0
x1
)
)
(
add_SNo
(
abs_SNo
x0
)
(
abs_SNo
x1
)
)
(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_minus_L2
add_SNo_minus_L2
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
add_SNo
(
minus_SNo
x0
)
(
add_SNo
x0
x1
)
=
x1
Theorem
SNo_triangle2
SNo_triangle2
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
(
abs_SNo
(
add_SNo
x0
(
minus_SNo
x2
)
)
)
(
add_SNo
(
abs_SNo
(
add_SNo
x0
(
minus_SNo
x1
)
)
)
(
abs_SNo
(
add_SNo
x1
(
minus_SNo
x2
)
)
)
)
(proof)
Param
nat_p
nat_p
:
ι
→
ο
Param
SNoS_
SNoS_
:
ι
→
ι
Param
binunion
binunion
:
ι
→
ι
→
ι
Param
Sing
Sing
:
ι
→
ι
Param
SetAdjoin
SetAdjoin
:
ι
→
ι
→
ι
Definition
eps_
eps_
:=
λ x0 .
binunion
(
Sing
0
)
{
SetAdjoin
(
ordsucc
x1
)
(
Sing
1
)
|x1 ∈
x0
}
Param
omega
omega
:
ι
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
nat_p_ordinal
nat_p_ordinal
:
∀ x0 .
nat_p
x0
⟶
ordinal
x0
Known
nat_ordsucc
nat_ordsucc
:
∀ x0 .
nat_p
x0
⟶
nat_p
(
ordsucc
x0
)
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
SNo_eps_
SNo_eps_
:
∀ x0 .
x0
∈
omega
⟶
SNo
(
eps_
x0
)
Known
FalseE
FalseE
:
False
⟶
∀ x0 : ο .
x0
Definition
nIn
nIn
:=
λ x0 x1 .
not
(
x0
∈
x1
)
Known
In_irref
In_irref
:
∀ x0 .
nIn
x0
x0
Known
SNoLev_eps_
SNoLev_eps_
:
∀ x0 .
x0
∈
omega
⟶
SNoLev
(
eps_
x0
)
=
ordsucc
x0
Param
binintersect
binintersect
:
ι
→
ι
→
ι
Param
SNoEq_
SNoEq_
:
ι
→
ι
→
ι
→
ο
Known
SNoLtE
SNoLtE
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLt
x0
x1
⟶
∀ x2 : ο .
(
∀ x3 .
SNo
x3
⟶
SNoLev
x3
∈
binintersect
(
SNoLev
x0
)
(
SNoLev
x1
)
⟶
SNoEq_
(
SNoLev
x3
)
x3
x0
⟶
SNoEq_
(
SNoLev
x3
)
x3
x1
⟶
SNoLt
x0
x3
⟶
SNoLt
x3
x1
⟶
nIn
(
SNoLev
x3
)
x0
⟶
SNoLev
x3
∈
x1
⟶
x2
)
⟶
(
SNoLev
x0
∈
SNoLev
x1
⟶
SNoEq_
(
SNoLev
x0
)
x0
x1
⟶
SNoLev
x0
∈
x1
⟶
x2
)
⟶
(
SNoLev
x1
∈
SNoLev
x0
⟶
SNoEq_
(
SNoLev
x1
)
x0
x1
⟶
nIn
(
SNoLev
x1
)
x0
⟶
x2
)
⟶
x2
Known
SNoLev_0_eq_0
SNoLev_0_eq_0
:
∀ x0 .
SNo
x0
⟶
SNoLev
x0
=
0
⟶
x0
=
0
Known
eps_ordinal_In_eq_0
eps_ordinal_In_eq_0
:
∀ x0 x1 .
ordinal
x1
⟶
x1
∈
eps_
x0
⟶
x1
=
0
Known
SNoLev_ordinal
SNoLev_ordinal
:
∀ x0 .
SNo
x0
⟶
ordinal
(
SNoLev
x0
)
Known
In_no2cycle
In_no2cycle
:
∀ x0 x1 .
x0
∈
x1
⟶
x1
∈
x0
⟶
False
Known
nat_p_omega
nat_p_omega
:
∀ x0 .
nat_p
x0
⟶
x0
∈
omega
Theorem
SNo_pos_eps_Lt
SNo_pos_eps_Lt
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
SNoS_
(
ordsucc
x0
)
⟶
SNoLt
0
x1
⟶
SNoLt
(
eps_
x0
)
x1
(proof)
Known
ordsuccE
ordsuccE
:
∀ x0 x1 .
x1
∈
ordsucc
x0
⟶
or
(
x1
∈
x0
)
(
x1
=
x0
)
Theorem
SNo_pos_eps_Le
SNo_pos_eps_Le
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
SNoS_
(
ordsucc
(
ordsucc
x0
)
)
⟶
SNoLt
0
x1
⟶
SNoLe
(
eps_
x0
)
x1
(proof)
Definition
and
and
:=
λ x0 x1 : ο .
∀ x2 : ο .
(
x0
⟶
x1
⟶
x2
)
⟶
x2
Known
nat_inv
nat_inv
:
∀ x0 .
nat_p
x0
⟶
or
(
x0
=
0
)
(
∀ x1 : ο .
(
∀ x2 .
and
(
nat_p
x2
)
(
x0
=
ordsucc
x2
)
⟶
x1
)
⟶
x1
)
Known
andI
andI
:
∀ x0 x1 : ο .
x0
⟶
x1
⟶
and
x0
x1
Definition
Subq
Subq
:=
λ x0 x1 .
∀ x2 .
x2
∈
x0
⟶
x2
∈
x1
Known
nat_ordsucc_trans
nat_ordsucc_trans
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
ordsucc
x0
⟶
x1
⊆
x0
Known
ordsuccI2
ordsuccI2
:
∀ x0 .
x0
∈
ordsucc
x0
Known
SNo_eq
SNo_eq
:
∀ x0 x1 .
SNo
x0
⟶
SNo
x1
⟶
SNoLev
x0
=
SNoLev
x1
⟶
SNoEq_
(
SNoLev
x0
)
x0
x1
⟶
x0
=
x1
Known
SNoEq_tra_
SNoEq_tra_
:
∀ x0 x1 x2 x3 .
SNoEq_
x0
x1
x2
⟶
SNoEq_
x0
x2
x3
⟶
SNoEq_
x0
x1
x3
Known
SNoEq_sym_
SNoEq_sym_
:
∀ x0 x1 x2 .
SNoEq_
x0
x1
x2
⟶
SNoEq_
x0
x2
x1
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
iffI
iffI
:
∀ x0 x1 : ο .
(
x0
⟶
x1
)
⟶
(
x1
⟶
x0
)
⟶
iff
x0
x1
Known
binunionI1
binunionI1
:
∀ x0 x1 x2 .
x2
∈
x0
⟶
x2
∈
binunion
x0
x1
Known
SingI
SingI
:
∀ x0 .
x0
∈
Sing
x0
Known
nat_p_trans
nat_p_trans
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
x0
⟶
nat_p
x1
Theorem
eps_SNo_eq
eps_SNo_eq
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
SNoS_
(
ordsucc
x0
)
⟶
SNoLt
0
x1
⟶
SNoEq_
(
SNoLev
x1
)
(
eps_
x0
)
x1
⟶
∀ x2 : ο .
(
∀ x3 .
and
(
x3
∈
x0
)
(
x1
=
eps_
x3
)
⟶
x2
)
⟶
x2
(proof)
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
Known
SingE
SingE
:
∀ x0 x1 .
x1
∈
Sing
x0
⟶
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
SNo_eps_pos
SNo_eps_pos
:
∀ x0 .
x0
∈
omega
⟶
SNoLt
0
(
eps_
x0
)
Known
omega_nat_p
omega_nat_p
:
∀ x0 .
x0
∈
omega
⟶
nat_p
x0
Theorem
eps_SNoCutP
eps_SNoCutP
:
∀ x0 .
x0
∈
omega
⟶
SNoCutP
(
Sing
0
)
(
prim5
x0
eps_
)
(proof)
Param
SNoCut
SNoCut
:
ι
→
ι
→
ι
Param
famunion
famunion
:
ι
→
(
ι
→
ι
) →
ι
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
binintersectE2
binintersectE2
:
∀ x0 x1 x2 .
x2
∈
binintersect
x0
x1
⟶
x2
∈
x1
Known
ReplI
ReplI
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
x1
x2
∈
prim5
x0
x1
Known
SNoS_I
SNoS_I
:
∀ x0 .
ordinal
x0
⟶
∀ x1 x2 .
x2
∈
x0
⟶
SNo_
x2
x1
⟶
x1
∈
SNoS_
x0
Known
SNoLev_
SNoLev_
:
∀ x0 .
SNo
x0
⟶
SNo_
(
SNoLev
x0
)
x0
Known
SNo_eps_decr
SNo_eps_decr
:
∀ x0 .
x0
∈
omega
⟶
∀ x1 .
x1
∈
x0
⟶
SNoLt
(
eps_
x0
)
(
eps_
x1
)
Known
famunion_Empty
famunion_Empty
:
∀ x0 :
ι → ι
.
famunion
0
x0
=
0
Known
binunion_idr
binunion_idr
:
∀ x0 .
binunion
x0
0
=
x0
Known
Empty_eq
Empty_eq
:
∀ x0 .
(
∀ x1 .
nIn
x1
x0
)
⟶
x0
=
0
Known
EmptyE
EmptyE
:
∀ x0 .
nIn
x0
0
Known
Subq_binunion_eq
Subq_binunion_eq
:
∀ x0 x1 .
x0
⊆
x1
=
(
binunion
x0
x1
=
x1
)
Known
cases_1
cases_1
:
∀ x0 .
x0
∈
1
⟶
∀ x1 :
ι → ο
.
x1
0
⟶
x1
x0
Known
nat_0_in_ordsucc
nat_0_in_ordsucc
:
∀ x0 .
nat_p
x0
⟶
0
∈
ordsucc
x0
Known
set_ext
set_ext
:
∀ x0 x1 .
x0
⊆
x1
⟶
x1
⊆
x0
⟶
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
nat_trans
nat_trans
:
∀ x0 .
nat_p
x0
⟶
∀ x1 .
x1
∈
x0
⟶
x1
⊆
x0
Known
ordinal_ordsucc_In
ordinal_ordsucc_In
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
x1
∈
x0
⟶
ordsucc
x1
∈
ordsucc
x0
Known
famunionI
famunionI
:
∀ x0 .
∀ x1 :
ι → ι
.
∀ x2 x3 .
x2
∈
x0
⟶
x3
∈
x1
x2
⟶
x3
∈
famunion
x0
x1
Known
SNoLev_0
SNoLev_0
:
SNoLev
0
=
0
Known
In_0_1
In_0_1
:
0
∈
1
Theorem
eps_SNoCut
eps_SNoCut
:
∀ x0 .
x0
∈
omega
⟶
eps_
x0
=
SNoCut
(
Sing
0
)
(
prim5
x0
eps_
)
(proof)
Known
nat_complete_ind
nat_complete_ind
:
∀ x0 :
ι → ο
.
(
∀ x1 .
nat_p
x1
⟶
(
∀ x2 .
x2
∈
x1
⟶
x0
x2
)
⟶
x0
x1
)
⟶
∀ x1 .
nat_p
x1
⟶
x0
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
)
)
)
Known
binunionE
binunionE
:
∀ x0 x1 x2 .
x2
∈
binunion
x0
x1
⟶
or
(
x2
∈
x0
)
(
x2
∈
x1
)
Known
andEL
andEL
:
∀ x0 x1 : ο .
and
x0
x1
⟶
x0
Known
andER
andER
:
∀ x0 x1 : ο .
and
x0
x1
⟶
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
omega_ordsucc
omega_ordsucc
:
∀ x0 .
x0
∈
omega
⟶
ordsucc
x0
∈
omega
Known
SNoLeLt_tra
SNoLeLt_tra
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x0
x1
⟶
SNoLt
x1
x2
⟶
SNoLt
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
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
)
Theorem
eps_ordsucc_half_add
eps_ordsucc_half_add
:
∀ x0 .
nat_p
x0
⟶
add_SNo
(
eps_
(
ordsucc
x0
)
)
(
eps_
(
ordsucc
x0
)
)
=
eps_
x0
(proof)
Known
nat_1
nat_1
:
nat_p
1
Theorem
SNo_eps_1
SNo_eps_1
:
SNo
(
eps_
1
)
(proof)
Known
nat_0
nat_0
:
nat_p
0
Known
eps_0_1
eps_0_1
:
eps_
0
=
1
Theorem
eps_1_half_eq1
eps_1_half_eq1
:
add_SNo
(
eps_
1
)
(
eps_
1
)
=
1
(proof)
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
add_nat_1_1_2
add_nat_1_1_2
:
add_nat
1
1
=
2
Theorem
add_SNo_1_1_2
add_SNo_1_1_2
:
add_SNo
1
1
=
2
(proof)
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
)
Theorem
eps_1_half_eq2
eps_1_half_eq2
:
mul_SNo
2
(
eps_
1
)
=
1
(proof)
Known
SNo_2
SNo_2
:
SNo
2
Theorem
eps_1_half_eq3
eps_1_half_eq3
:
mul_SNo
(
eps_
1
)
2
=
1
(proof)
Theorem
eps_1_split_eq
eps_1_split_eq
:
∀ x0 .
SNo
x0
⟶
add_SNo
(
mul_SNo
(
eps_
1
)
x0
)
(
mul_SNo
(
eps_
1
)
x0
)
=
x0
(proof)
Theorem
eps_1_split_Le_bd
eps_1_split_Le_bd
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x1
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLe
x2
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLe
(
add_SNo
x1
x2
)
x0
(proof)
Theorem
eps_1_split_LeLt_bd
eps_1_split_LeLt_bd
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLe
x1
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLt
x2
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLt
(
add_SNo
x1
x2
)
x0
(proof)
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
eps_1_split_LtLe_bd
eps_1_split_LtLe_bd
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x1
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLe
x2
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLt
(
add_SNo
x1
x2
)
x0
(proof)
Theorem
eps_1_split_Lt_bd
eps_1_split_Lt_bd
:
∀ x0 x1 x2 .
SNo
x0
⟶
SNo
x1
⟶
SNo
x2
⟶
SNoLt
x1
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLt
x2
(
mul_SNo
(
eps_
1
)
x0
)
⟶
SNoLt
(
add_SNo
x1
x2
)
x0
(proof)
Definition
SNo_ord_seq
SNo_ord_seq
:=
λ x0 .
λ x1 :
ι → ι
.
∀ x2 .
x2
∈
x0
⟶
SNo
(
x1
x2
)
Param
setminus
setminus
:
ι
→
ι
→
ι
Definition
2be79..
:=
λ x0 .
λ x1 :
ι → ι
.
λ x2 .
∀ x3 .
SNo
x3
⟶
SNoLt
0
x3
⟶
∀ x4 : ο .
(
∀ x5 .
and
(
x5
∈
x0
)
(
∀ x6 .
x6
∈
setminus
x0
x5
⟶
SNoLt
(
abs_SNo
(
add_SNo
(
x1
x6
)
(
minus_SNo
x2
)
)
)
x3
)
⟶
x4
)
⟶
x4
Definition
c59b5..
:=
λ x0 .
λ x1 :
ι → ι
.
∀ x2 .
SNo
x2
⟶
SNoLt
0
x2
⟶
∀ x3 : ο .
(
∀ x4 .
and
(
x4
∈
x0
)
(
∀ x5 .
x5
∈
setminus
x0
x4
⟶
∀ x6 .
x6
∈
setminus
x0
x4
⟶
SNoLt
(
abs_SNo
(
add_SNo
(
x1
x5
)
(
minus_SNo
(
x1
x6
)
)
)
)
x2
)
⟶
x3
)
⟶
x3
Known
ordinal_linear
ordinal_linear
:
∀ x0 x1 .
ordinal
x0
⟶
ordinal
x1
⟶
or
(
x0
⊆
x1
)
(
x1
⊆
x0
)
Known
setminusI
setminusI
:
∀ x0 x1 x2 .
x2
∈
x0
⟶
nIn
x2
x1
⟶
x2
∈
setminus
x0
x1
Known
ordinal_Hered
ordinal_Hered
:
∀ x0 .
ordinal
x0
⟶
∀ x1 .
x1
∈
x0
⟶
ordinal
x1
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
add_SNo_minus_SNo_rinv
add_SNo_minus_SNo_rinv
:
∀ x0 .
SNo
x0
⟶
add_SNo
x0
(
minus_SNo
x0
)
=
0
Theorem
5bedf..
:
∀ x0 .
ordinal
x0
⟶
∀ x1 :
ι → ι
.
SNo_ord_seq
x0
x1
⟶
∀ x2 .
SNo
x2
⟶
2be79..
x0
x1
x2
⟶
∀ x3 .
SNo
x3
⟶
2be79..
x0
x1
x3
⟶
not
(
SNoLt
x2
x3
)
(proof)
Theorem
5eef1..
:
∀ x0 .
ordinal
x0
⟶
∀ x1 :
ι → ι
.
SNo_ord_seq
x0
x1
⟶
∀ x2 .
SNo
x2
⟶
2be79..
x0
x1
x2
⟶
∀ x3 .
SNo
x3
⟶
2be79..
x0
x1
x3
⟶
x2
=
x3
(proof)