:: GFACIRC2 semantic presentation
REAL
is
set
NAT
is non
empty
V12
()
V21
()
V22
()
V23
()
V28
()
cardinal
limit_cardinal
Element
of
K27
(
REAL
)
K27
(
REAL
) is
set
NAT
is non
empty
V12
()
V21
()
V22
()
V23
()
V28
()
cardinal
limit_cardinal
set
K27
(
NAT
) is
V12
()
V28
()
set
K27
(
NAT
) is
V12
()
V28
()
set
{}
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
{
{}
,1
}
is non
empty
V28
()
V32
() non
with_pair
set
K228
() is
set
K27
(
K228
()) is
set
K229
() is
Element
of
K27
(
K228
())
K279
() is non
empty
V83
()
L10
()
the
carrier
of
K279
() is non
empty
set
K232
( the
carrier
of
K279
()) is non
empty
M19
( the
carrier
of
K279
())
K278
(
K279
()) is
Element
of
K27
(
K232
( the
carrier
of
K279
()))
K27
(
K232
( the
carrier
of
K279
())) is
set
K28
(
K278
(
K279
()),
NAT
) is
Relation-like
set
K27
(
K28
(
K278
(
K279
()),
NAT
)) is
set
K28
(
NAT
,
K278
(
K279
())) is
Relation-like
set
K27
(
K28
(
NAT
,
K278
(
K279
()))) is
set
BOOLEAN
is non
empty
set
0
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
Element
of
NAT
{
0
,1
}
is non
empty
V28
()
V32
() non
with_pair
set
2 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
2
-tuples_on
BOOLEAN
is
functional
non
empty
FinSequence-membered
FinSequenceSet
of
BOOLEAN
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
) is
Relation-like
set
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
)) is
set
3 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
3
-tuples_on
BOOLEAN
is
functional
non
empty
FinSequence-membered
FinSequenceSet
of
BOOLEAN
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
) is
Relation-like
set
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
)) is
set
'xor'
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
1
-tuples_on
BOOLEAN
is
functional
non
empty
FinSequence-membered
FinSequenceSet
of
BOOLEAN
K28
(
(
1
-tuples_on
BOOLEAN
)
,
BOOLEAN
) is
Relation-like
set
K27
(
K28
(
(
1
-tuples_on
BOOLEAN
)
,
BOOLEAN
)) is
set
Seg
1 is non
empty
V12
()
V28
() 1
-element
Element
of
K27
(
NAT
)
K28
(
NAT
,
NAT
) is
Relation-like
V12
()
V28
()
set
K27
(
K28
(
NAT
,
NAT
)) is
V12
()
V28
()
set
xor2
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
and2
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
xor2c
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
and2c
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
and2a
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
0
-tuples_on
BOOLEAN
is
functional
non
empty
FinSequence-membered
FinSequenceSet
of
BOOLEAN
FALSE
is
boolean
Element
of
BOOLEAN
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
is
Relation-like
Function-like
V18
(
0
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
0
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
K28
(
(
0
-tuples_on
BOOLEAN
)
,
BOOLEAN
) is
Relation-like
set
K27
(
K28
(
(
0
-tuples_on
BOOLEAN
)
,
BOOLEAN
)) is
set
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
]
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
n
is
set
Sn
.
0
is
set
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
0
is
set
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
n
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
h0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
h
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is
set
g
.
(
o0
+
1
)
is
set
An
is
set
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
is non
empty
pair
set
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
non-empty
MSAlgebra
over
h
BitGFA0Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
)
)
)
+*
(
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
Sn
+*
(
BitGFA0Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
MSAlgebra
over
h
+*
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
h
+*
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
n
is
set
h
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
n
is
set
An
.
0
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
n
is
set
Sn
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
n
is
set
g1
.
0
is
set
h1
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
An
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
An
+
1
)
is
set
g
.
(
An
+
1
)
is
set
Sn
is
set
BitGFA0Str
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
]
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
]
,
Sn
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2
]
,
Sn
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
]
is non
empty
pair
set
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
+*
(
BitGFA0Str
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
g1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
g1
+
1
)
is
set
g
.
(
g1
+
1
)
is
set
f1
is
set
BitGFA0Str
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
+*
(
BitGFA0Str
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f2
+
1
)
is
set
g
.
(
f2
+
1
)
is
set
f1
is
set
BitGFA0Str
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
+*
(
BitGFA0Str
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f3
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
n
+
1
)
is
set
g
.
(
n
+
1
)
is
set
f0
is
set
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f3
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
x
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
h2
+
1
)
is
set
g
.
(
h2
+
1
)
is
set
g2
is
set
BitGFA0Str
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
]
,
g2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
]
,
g2
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2
]
,
g2
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
]
is non
empty
pair
set
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
x
+*
(
BitGFA0Str
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is
set
g
.
(
o0
+
1
)
is
set
An
is
set
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
is non
empty
pair
set
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
non-empty
MSAlgebra
over
h
BitGFA0Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2
)
)
)
+*
(
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
Sn
+*
(
BitGFA0Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
MSAlgebra
over
h
+*
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
h
+*
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is
set
g
.
(
f1
+
1
)
is
set
o0
is
set
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
is non
empty
pair
set
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
+*
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
is
strict
non-empty
finitely-generated
V95
(
h
)
gate`2=den
Boolean
MSAlgebra
over
h
BitGFA0Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
GFA0AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
2GatesCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
GFA0CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
1GateCircuit
(
o0
,
(
f
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2
)
)
)
+*
(
1GateCircuit
(
o0
,
(
f
.
(
f1
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
An
+*
(
BitGFA0Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is
strict
non-empty
finitely-generated
V95
(
h
+*
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
h
+*
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
n
is
set
h
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
n
is
set
Sn
.
0
is
set
An
.
0
is
set
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
0
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
A0
is
Element
of
InnerVertices
(
n
,
f
,
g
)
h0
is
Element
of
InnerVertices
(
n
,
f
,
g
)
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
n
is
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
n
is
set
h
.
0
is
set
proj1
N
is
set
proj1
h
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
n
is
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
InnerVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is non
empty
set
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
) is
set
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
h
.
Sn
is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
N
.
(
Sn
+
1
)
is
set
h
.
(
Sn
+
1
)
is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
GFA0CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
(
h
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
(
h
.
Sn
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
(
h
.
Sn
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
the
carrier
of
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
) is
set
An
+*
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
InnerVertices
(
An
+*
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
An
+*
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
)
the
carrier
of
(
An
+*
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
is non
empty
set
K27
( the
carrier
of
(
An
+*
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
) is
set
h
.
n
is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
Sn
is
Element
of
InnerVertices
(
n
,
f
,
g
)
An
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
An
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h
.
(
An
+
1
)
is
set
f
.
(
An
+
1
)
is
set
g
.
(
An
+
1
)
is
set
h
.
An
is
set
GFA0CarryOutput
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
f1
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
o0
is
set
g1
is
set
h
.
o0
is
set
N
.
An
is
set
o0
is non
empty
V56
()
ManySortedSign
n
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g
.
0
is
set
S0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
S0
.
0
is
set
A0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
A0
.
0
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
n
,
f
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
g
.
Sn
is
set
(
Sn
,
n
,
f
) is
strict
non-empty
finitely-generated
V95
((
Sn
,
n
,
f
))
gate`2=den
Boolean
MSAlgebra
over (
Sn
,
n
,
f
)
S0
.
Sn
is
set
(
Sn
,
n
,
f
) is
Element
of
InnerVertices
(
Sn
,
n
,
f
)
the
carrier
of (
Sn
,
n
,
f
) is non
empty
set
InnerVertices
(
Sn
,
n
,
f
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
n
,
f
))
K27
( the
carrier
of (
Sn
,
n
,
f
)) is
set
A0
.
Sn
is
set
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
Sn
is
set
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
Sn
is
set
An
.
0
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
g1
is non
empty
V56
()
ManySortedSign
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
n
.
(
f2
+
1
)
is
set
f
.
(
f2
+
1
)
is
set
f1
is
set
BitGFA0Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
is non
empty
pair
set
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
is
non-empty
MSAlgebra
over
g1
BitGFA0Circ
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
GFA0AdderCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
2GatesCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2
)
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
GFA0CarryCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
GFA0CarryICirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
1GateCircuit
(
f1
,
(
n
.
(
f2
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2
)
)
)
+*
(
1GateCircuit
(
f1
,
(
n
.
(
f2
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
h1
+*
(
BitGFA0Circ
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is
strict
non-empty
MSAlgebra
over
g1
+*
(
BitGFA0Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
g1
+*
(
BitGFA0Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
proj1
A0
is
set
g1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
g1
+
1
)
is
set
n
.
(
g1
+
1
)
is
set
f
.
(
g1
+
1
)
is
set
A0
.
g1
is
set
GFA0CarryOutput
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
) is
set
[
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
Sn
is
set
g1
.
0
is
set
proj1
g1
is
set
h1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g1
.
(
h1
+
1
)
is
set
n
.
(
h1
+
1
)
is
set
f
.
(
h1
+
1
)
is
set
g1
.
h1
is
set
GFA0CarryOutput
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
) is
set
[
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
n
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
0
,
n
,
f
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
0
,
n
,
f
) is
strict
non-empty
finitely-generated
V95
((
0
,
n
,
f
))
gate`2=den
Boolean
MSAlgebra
over (
0
,
n
,
f
)
(
0
,
n
,
f
) is
Element
of
InnerVertices
(
0
,
n
,
f
)
the
carrier
of (
0
,
n
,
f
) is non
empty
set
InnerVertices
(
0
,
n
,
f
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
n
,
f
))
K27
( the
carrier
of (
0
,
n
,
f
)) is
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(1,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
S0
.
1 is
set
A0
.
1 is
set
(1,
S0
,
A0
) is
strict
non-empty
finitely-generated
V95
((1,
S0
,
A0
))
gate`2=den
Boolean
MSAlgebra
over (1,
S0
,
A0
)
(1,
S0
,
A0
) is
Element
of
InnerVertices
(1,
S0
,
A0
)
the
carrier
of (1,
S0
,
A0
) is non
empty
set
InnerVertices
(1,
S0
,
A0
) is non
empty
Element
of
K27
( the
carrier
of (1,
S0
,
A0
))
K27
( the
carrier
of (1,
S0
,
A0
)) is
set
h0
is
set
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
1
)
,
h0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h0
,
(
S0
.
1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
is non
empty
pair
set
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA0Circ
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
GFA0AdderCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
2GatesCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2
)
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
)
1GateCircuit
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
)
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2
]
,
h0
*>
,
xor2
)
)
GFA0CarryCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
GFA0CarryICirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
1GateCircuit
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
1GateCircuit
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2
)
)
+*
(
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
1GateCircuit
(
h0
,
(
S0
.
1
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
1GateCircuit
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
(
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2
)
)
+*
(
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2
)
)
)
+*
(
1GateCircuit
(
h0
,
(
S0
.
1
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA0CarryCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Circ
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
GFA0CarryOutput
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
the
carrier
of
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
) is
set
[
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
1 is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
1 is
set
N
.
0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
Sn
.
(
0
+
1
)
is
set
n
is
set
<*
n
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
f
is
set
<*
f
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
(1,
<*
n
*>
,
<*
f
*>
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(1,
<*
n
*>
,
<*
f
*>
) is
strict
non-empty
finitely-generated
V95
((1,
<*
n
*>
,
<*
f
*>
))
gate`2=den
Boolean
MSAlgebra
over (1,
<*
n
*>
,
<*
f
*>
)
(1,
<*
n
*>
,
<*
f
*>
) is
Element
of
InnerVertices
(1,
<*
n
*>
,
<*
f
*>
)
the
carrier
of (1,
<*
n
*>
,
<*
f
*>
) is non
empty
set
InnerVertices
(1,
<*
n
*>
,
<*
f
*>
) is non
empty
Element
of
K27
( the
carrier
of (1,
<*
n
*>
,
<*
f
*>
))
K27
( the
carrier
of (1,
<*
n
*>
,
<*
f
*>
)) is
set
<*
n
*>
.
1 is
set
<*
f
*>
.
1 is
set
g
is
set
BitGFA0Str
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
n
,
f
,
g
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
n
,
f
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
n
,
f
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
n
,
f
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
n
,
f
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f
,
g
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f
,
g
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
g
,
n
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
g
,
n
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
n
,
f
*>
,
and2
]
is non
empty
pair
set
[
<*
f
,
g
*>
,
and2
]
is non
empty
pair
set
[
<*
g
,
n
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA0CarryStr
(
n
,
f
,
g
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
n
,
f
,
g
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA0Circ
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
n
,
f
,
g
)
GFA0AdderCirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
n
,
f
,
g
)
2GatesCircuit
(
n
,
f
,
g
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
n
,
f
,
g
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
n
,
f
,
g
,
xor2
)
1GateCircuit
(
n
,
f
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
xor2
)
1GateCircuit
(
<*
n
,
f
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
xor2
)
1GateCircuit
(
[
<*
n
,
f
*>
,
xor2
]
,
g
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
)
(
1GateCircuit
(
n
,
f
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
n
,
f
*>
,
xor2
]
,
g
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2
]
,
g
*>
,
xor2
)
)
GFA0CarryCirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
n
,
f
,
g
)
GFA0CarryICirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
n
,
f
,
g
)
1GateCircuit
(
n
,
f
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
1GateCircuit
(
<*
n
,
f
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
1GateCircuit
(
f
,
g
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
1GateCircuit
(
<*
f
,
g
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
(
1GateCircuit
(
n
,
f
,
and2
)
)
+*
(
1GateCircuit
(
f
,
g
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
1GateCircuit
(
g
,
n
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
1GateCircuit
(
<*
g
,
n
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
(
(
1GateCircuit
(
n
,
f
,
and2
)
)
+*
(
1GateCircuit
(
f
,
g
,
and2
)
)
)
+*
(
1GateCircuit
(
g
,
n
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
1GateCircuit
(
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
n
,
f
,
g
)
)
+*
(
1GateCircuit
(
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
n
,
f
,
g
)
)
+*
(
GFA0CarryCirc
(
n
,
f
,
g
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA0CarryStr
(
n
,
f
,
g
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA0CarryStr
(
n
,
f
,
g
)
)
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Circ
(
n
,
f
,
g
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
n
,
f
,
g
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
+*
(
BitGFA0Str
(
n
,
f
,
g
)
)
GFA0CarryOutput
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
n
,
f
,
g
)
)
the
carrier
of
(
GFA0CarryStr
(
n
,
f
,
g
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
n
,
f
,
g
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
n
,
f
,
g
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
n
,
f
,
g
)
)
) is
set
[
<*
[
<*
n
,
f
*>
,
and2
]
,
[
<*
f
,
g
*>
,
and2
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
^
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
^
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
^
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
N
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
^
N
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is
strict
non-empty
finitely-generated
V95
((
n
,
(
f
^
S0
)
,
(
g
^
h0
)
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is
strict
non-empty
finitely-generated
V95
((
n
,
(
f
^
A0
)
,
(
g
^
N
)
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
(
f
^
A0
)
,
(
g
^
N
)
)
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is
Element
of
InnerVertices
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)
the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
set
InnerVertices
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
))
K27
( the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)) is
set
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is
Element
of
InnerVertices
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
)
the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
set
InnerVertices
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
))
K27
( the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
)) is
set
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
n
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
n
is
set
h1
.
0
is
set
f1
.
0
is
set
f2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f2
.
0
is
set
x
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
x
.
n
is
set
g2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g2
.
n
is
set
x
.
0
is
set
g2
.
0
is
set
h2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h2
.
0
is
set
i
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
.
i
is
set
h2
.
i
is
set
h1
.
i
is
set
x
.
i
is
set
f1
.
i
is
set
g2
.
i
is
set
i
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f2
.
(
i
+
1
)
is
set
h2
.
(
i
+
1
)
is
set
h1
.
(
i
+
1
)
is
set
x
.
(
i
+
1
)
is
set
f1
.
(
i
+
1
)
is
set
g2
.
(
i
+
1
)
is
set
len
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
len
g
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
dom
f
is
V28
()
n
-element
Element
of
K27
(
NAT
)
Seg
n
is
V28
()
n
-element
Element
of
K27
(
NAT
)
dom
g
is
V28
()
n
-element
Element
of
K27
(
NAT
)
0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
f
^
S0
)
.
(
i
+
1
)
is
set
f
.
(
i
+
1
)
is
set
(
f
^
A0
)
.
(
i
+
1
)
is
set
(
g
^
h0
)
.
(
i
+
1
)
is
set
g
.
(
i
+
1
)
is
set
(
g
^
N
)
.
(
i
+
1
)
is
set
A
is non
empty
V56
()
ManySortedSign
S
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h1
.
S
is
set
c
26
is
non-empty
MSAlgebra
over
A
f1
.
S
is
set
c
27
is
set
f2
.
S
is
set
S
is non
empty
V56
()
ManySortedSign
c
27
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
c
27
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
f
^
S0
)
.
(
c
27
+
1
)
is
set
(
g
^
h0
)
.
(
c
27
+
1
)
is
set
c
26
is
set
BitGFA0Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
is non
empty
pair
set
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A
is
non-empty
MSAlgebra
over
S
BitGFA0Circ
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
GFA0AdderCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
2GatesCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2
)
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
)
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2
]
,
c
26
*>
,
xor2
)
)
GFA0CarryCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
GFA0CarryICirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
1GateCircuit
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
1GateCircuit
(
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2
)
)
)
+*
(
1GateCircuit
(
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA0CarryCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
A
+*
(
BitGFA0Circ
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is
strict
non-empty
MSAlgebra
over
S
+*
(
BitGFA0Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
S
+*
(
BitGFA0Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
S
is non
empty
V56
()
ManySortedSign
A
is
non-empty
MSAlgebra
over
S
GFA0CarryOutput
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
) is
set
[
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA0Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
S
+*
(
BitGFA0Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
BitGFA0Circ
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
GFA0AdderCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
2GatesCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2
)
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
)
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2
]
,
(
h2
.
i
)
*>
,
xor2
)
)
GFA0CarryCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
GFA0CarryICirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
1GateCircuit
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
1GateCircuit
(
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2
)
)
)
+*
(
1GateCircuit
(
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA0CarryCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA0CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
A
+*
(
BitGFA0Circ
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is
strict
non-empty
MSAlgebra
over
S
+*
(
BitGFA0Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
f2
.
n
is
set
h2
.
n
is
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
S0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
S0
-element
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
S0
-element
FinSequence-like
FinSubsequence-like
set
(
S0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
) is
Element
of
InnerVertices
(
S0
,
A0
,
h0
)
the
carrier
of (
S0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
S0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
A0
,
h0
))
K27
( the
carrier
of (
S0
,
A0
,
h0
)) is
set
(
S0
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
N
is
set
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
A0
^
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
S0
+
1
-element
FinSequence-like
FinSubsequence-like
set
h
is
set
<*
h
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
h0
^
<*
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
S0
+
1
-element
FinSequence-like
FinSubsequence-like
set
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
N
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
N
,
h
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
N
,
h
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
h
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
,
A0
,
h0
),
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
N
,
h
*>
,
and2
]
is non
empty
pair
set
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
strict
non-empty
finitely-generated
V95
((
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
gate`2=den
Boolean
MSAlgebra
over (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
BitGFA0Circ
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
))
GFA0AdderCirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
2GatesCircuit
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2
)
1GateCircuit
(
N
,
h
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
xor2
)
1GateCircuit
(
<*
N
,
h
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
xor2
)
1GateCircuit
(
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
),
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
(
1GateCircuit
(
N
,
h
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
),
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
GFA0CarryCirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
GFA0CarryICirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
1GateCircuit
(
N
,
h
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
1GateCircuit
(
<*
N
,
h
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
1GateCircuit
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
(
1GateCircuit
(
N
,
h
,
and2
)
)
+*
(
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
1GateCircuit
((
S0
,
A0
,
h0
),
N
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
1GateCircuit
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
(
(
1GateCircuit
(
N
,
h
,
and2
)
)
+*
(
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2
)
)
)
+*
(
1GateCircuit
((
S0
,
A0
,
h0
),
N
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
1GateCircuit
(
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircuit
(
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryCirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
(
S0
,
A0
,
h0
)
+*
(
BitGFA0Circ
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
Element
of
InnerVertices
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
set
InnerVertices
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
Element
of
K27
( the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
K27
( the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)) is
set
GFA0CarryOutput
(
N
,
h
,(
S0
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
the
carrier
of
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
N
,
h
*>
,
and2
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
strict
non-empty
finitely-generated
V95
((
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
S0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
S0
is
set
o0
.
0
is
set
f1
.
0
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
0
is
set
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
Element
of
InnerVertices
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
set
InnerVertices
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
K27
( the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)) is
set
g1
.
S0
is
set
o0
.
(
S0
+
1
)
is
set
f1
.
(
S0
+
1
)
is
set
g1
.
(
S0
+
1
)
is
set
len
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
len
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
(
A0
^
<*
N
*>
)
.
(
S0
+
1
)
is
set
(
h0
^
<*
h
*>
)
.
(
S0
+
1
)
is
set
A0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
S0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
S0
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A0
.
(
S0
+
1
)
is
set
h0
.
(
S0
+
1
)
is
set
(
S0
,
A0
,
h0
) is
Element
of
InnerVertices
(
S0
,
A0
,
h0
)
the
carrier
of (
S0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
S0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
A0
,
h0
))
K27
( the
carrier
of (
S0
,
A0
,
h0
)) is
set
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
S0
+
1
)
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
(
S0
+
1
)
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
(
S0
+
1
)
,
A0
,
h0
)
(
S0
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
BitGFA0Circ
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
GFA0AdderCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
2GatesCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2
)
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
),
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
),
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2
]
,(
S0
,
A0
,
h0
)
*>
,
xor2
)
)
GFA0CarryCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
GFA0CarryICirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
1GateCircuit
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
1GateCircuit
((
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2
)
)
)
+*
(
1GateCircuit
((
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
(
S0
,
A0
,
h0
)
+*
(
BitGFA0Circ
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
+*
(
BitGFA0Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
(
(
S0
+
1
)
,
A0
,
h0
) is
Element
of
InnerVertices
(
(
S0
+
1
)
,
A0
,
h0
)
the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
S0
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
)) is
set
GFA0CarryOutput
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
S0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
S0
is
set
N
.
0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
Sn
.
S0
is
set
N
.
(
S0
+
1
)
is
set
h
.
(
S0
+
1
)
is
set
Sn
.
(
S0
+
1
)
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
g
,
S0
))
the
carrier
of (
n
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
n
,
g
,
S0
)) is
set
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
(
n
+
0
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
0
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
0
)
,
g
,
S0
))
the
carrier
of (
(
n
+
0
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
0
)
,
g
,
S0
)) is
set
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
(
n
+
h0
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
h0
)
,
g
,
S0
))
the
carrier
of (
(
n
+
h0
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
h0
)
,
g
,
S0
)) is
set
h0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
n
+
(
h0
+
1
)
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
set
(
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
))
the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
)) is
set
(
n
+
h0
)
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
.
(
(
n
+
h0
)
+
1
)
is
set
S0
.
(
(
n
+
h0
)
+
1
)
is
set
(
(
n
+
h0
)
,
g
,
S0
) is
Element
of
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
)
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
+*
(
GFA0CarryStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
the
carrier
of
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
) is
set
(
InnerVertices
(
n
,
g
,
S0
)
)
\/
(
InnerVertices
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
(
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
)
)
\/
(
InnerVertices
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
)
the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
K27
( the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA0Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
n
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
))
the
carrier
of (
(
n
+
1
)
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
)) is
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
f
.
(
n
+
1
)
is
set
g
.
(
n
+
1
)
is
set
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
n
,
f
,
g
)
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,(
n
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,(
n
,
f
,
g
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,(
n
,
f
,
g
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
+*
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
the
carrier
of
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
) is
set
(
InnerVertices
(
n
,
f
,
g
)
)
\/
(
InnerVertices
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
set
(
n
,
f
,
g
)
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
,
f
,
g
)
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
)
the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
set
K27
( the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
g
.
n
is
set
S0
.
n
is
set
N
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
N
,
g
,
S0
) is
Element
of
InnerVertices
(
N
,
g
,
S0
)
(
N
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
N
,
g
,
S0
) is non
empty
set
InnerVertices
(
N
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
N
,
g
,
S0
))
K27
( the
carrier
of (
N
,
g
,
S0
)) is
set
GFA0AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
N
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
N
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
N
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
N
,
g
,
S0
)
*>
,
xor2
]
is non
empty
pair
set
h
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h0
is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
h
,
g
,
S0
) is
Element
of
InnerVertices
(
h
,
g
,
S0
)
(
h
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
h
,
g
,
S0
) is non
empty
set
InnerVertices
(
h
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
h
,
g
,
S0
))
K27
( the
carrier
of (
h
,
g
,
S0
)) is
set
GFA0AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h
,
g
,
S0
)
*>
,
xor2
]
is non
empty
pair
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
1
+
A0
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
h0
,
g
,
S0
) is
Element
of
InnerVertices
(
h0
,
g
,
S0
)
(
h0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
h0
,
g
,
S0
) is non
empty
set
InnerVertices
(
h0
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
h0
,
g
,
S0
))
K27
( the
carrier
of (
h0
,
g
,
S0
)) is
set
GFA0AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
,
xor2
]
is non
empty
pair
set
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
n
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
n
))
the
carrier
of
H
1
(
n
) is non
empty
set
K27
( the
carrier
of
H
1
(
n
)) is
set
InnerVertices
H
1
(
f
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
f
))
the
carrier
of
H
1
(
f
) is non
empty
set
K27
( the
carrier
of
H
1
(
f
)) is
set
h0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
.
(
h0
+
1
)
is
set
S0
.
(
h0
+
1
)
is
set
BitGFA0Str
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2
]
,(
h0
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
+*
(
GFA0CarryStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
2
(
h0
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
h0
))
the
carrier
of
H
2
(
h0
) is non
empty
set
K27
( the
carrier
of
H
2
(
h0
)) is
set
H
1
(
h0
)
+*
H
2
(
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
1
(
h0
) is non
empty
set
the
carrier
of
H
2
(
h0
)
\/
the
carrier
of
H
1
(
h0
) is non
empty
set
h
is
Element
of the
carrier
of
H
2
(
h0
)
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
f
+
1
)
,
n
,
g
,
S0
) is
Element
of
InnerVertices
(
n
,
g
,
S0
)
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
g
,
S0
) is non
empty
set
InnerVertices
(
n
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
g
,
S0
))
K27
( the
carrier
of (
n
,
g
,
S0
)) is
set
g
.
(
f
+
1
)
is
set
S0
.
(
f
+
1
)
is
set
(
f
,
g
,
S0
) is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
GFA0AdderOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
f
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
f
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
f
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2
)
)
) is
set
[
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
f
,
g
,
S0
)
*>
,
xor2
]
is non
empty
pair
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
A0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
A0
,
g
,
S0
) is
Element
of
InnerVertices
(
A0
,
g
,
S0
)
(
A0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
A0
,
g
,
S0
) is non
empty
set
InnerVertices
(
A0
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
A0
,
g
,
S0
))
K27
( the
carrier
of (
A0
,
g
,
S0
)) is
set
GFA0AdderOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
A0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
A0
,
g
,
S0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
A0
,
g
,
S0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2
)
)
) is
set
[
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2
]
,(
A0
,
g
,
S0
)
*>
,
xor2
]
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
0
))
the
carrier
of
H
1
(
0
) is non
empty
set
K27
( the
carrier
of
H
1
(
0
)) is
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
A0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
A0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
A0
))
the
carrier
of
H
1
(
A0
) is non
empty
set
K27
( the
carrier
of
H
1
(
A0
)) is
set
A0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
A0
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f
.
(
A0
+
1
)
is
set
g
.
(
A0
+
1
)
is
set
(
A0
,
f
,
g
) is
Element
of
InnerVertices
(
A0
,
f
,
g
)
the
carrier
of (
A0
,
f
,
g
) is non
empty
set
InnerVertices
(
A0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
A0
,
f
,
g
))
K27
( the
carrier
of (
A0
,
f
,
g
)) is
set
BitGFA0Str
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
]
,(
A0
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
]
,(
A0
,
f
,
g
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2
]
,(
A0
,
f
,
g
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
+*
(
GFA0CarryStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
H
1
(
A0
)
+*
H
2
(
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
2
(
A0
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
A0
))
the
carrier
of
H
2
(
A0
) is non
empty
set
K27
( the
carrier
of
H
2
(
A0
)) is
set
InnerVertices
H
1
(
A0
+
1) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
A0
+
1))
the
carrier
of
H
1
(
A0
+
1) is non
empty
set
K27
( the
carrier
of
H
1
(
A0
+
1)) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
(
0
,
f
,
g
) is
Element
of
InnerVertices
(
0
,
f
,
g
)
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
f
,
g
) is non
empty
set
InnerVertices
(
0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
f
,
g
))
K27
( the
carrier
of (
0
,
f
,
g
)) is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
f
,
g
) is
Element
of
InnerVertices
(
Sn
,
f
,
g
)
(
Sn
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
Sn
,
f
,
g
) is non
empty
set
InnerVertices
(
Sn
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
f
,
g
))
K27
( the
carrier
of (
Sn
,
f
,
g
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
Sn
+
1
)
,
f
,
g
) is
Element
of
InnerVertices
(
(
Sn
+
1
)
,
f
,
g
)
(
(
Sn
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
Sn
+
1
)
,
f
,
g
) is non
empty
set
InnerVertices
(
(
Sn
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
f
,
g
))
K27
( the
carrier
of (
(
Sn
+
1
)
,
f
,
g
)) is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
GFA0CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
or3
is
Relation-like
Function-like
V18
(3
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
3
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
proj1
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
is
set
(
0
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
0
,
A0
,
h0
)
(
0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
A0
,
h0
))
K27
( the
carrier
of (
0
,
A0
,
h0
)) is
set
(
0
,
A0
,
h0
)
`1
is
set
(
0
,
A0
,
h0
)
`2
is
set
proj1
(
(
0
,
A0
,
h0
)
`2
)
is
set
card
(
(
0
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
N
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
N
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
N
,
A0
,
h0
)
(
N
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
N
,
A0
,
h0
) is non
empty
set
InnerVertices
(
N
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
N
,
A0
,
h0
))
K27
( the
carrier
of (
N
,
A0
,
h0
)) is
set
(
N
,
A0
,
h0
)
`1
is
set
(
N
,
A0
,
h0
)
`2
is
set
proj1
(
(
N
,
A0
,
h0
)
`2
)
is
set
card
(
(
N
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
N
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
(
N
+
1
)
,
A0
,
h0
)
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
N
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
N
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
N
+
1
)
,
A0
,
h0
)) is
set
A0
.
(
N
+
1
)
is
set
h0
.
(
N
+
1
)
is
set
GFA0CarryOutput
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
proj1
or3
is
set
(
(
N
+
1
)
,
A0
,
h0
)
`1
is
set
(
(
N
+
1
)
,
A0
,
h0
)
`2
is
set
proj1
(
(
(
N
+
1
)
,
A0
,
h0
)
`2
)
is
set
card
(
(
(
N
+
1
)
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
S0
is
set
A0
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
S0
,
A0
]
is non
empty
pair
set
proj1
A0
is
set
[
S0
,
A0
]
`2
is
set
proj1
(
[
S0
,
A0
]
`2
)
is
set
(
n
,
f
,
g
)
`2
is
set
proj1
(
(
n
,
f
,
g
)
`2
)
is
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
(
0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
0
))
the
carrier
of
H
1
(
0
) is non
empty
set
K27
( the
carrier
of
H
1
(
0
)) is
set
InputVertices
H
1
(
0
) is
Element
of
K27
( the
carrier
of
H
1
(
0
))
N
.
0
is
set
(
0
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
0
,
A0
,
h0
)
the
carrier
of (
0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
A0
,
h0
))
K27
( the
carrier
of (
0
,
A0
,
h0
)) is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
An
is
set
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
)
the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
) is
set
An
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
H
2
(
An
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
the
carrier
of
H
2
(
An
,
Sn
) is non
empty
set
K27
( the
carrier
of
H
2
(
An
,
Sn
)) is
set
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
}
is
V28
()
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
An
is
set
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
An
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
2
(
An
,
Sn
) is non
empty
set
InputVertices
H
2
(
An
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
K27
( the
carrier
of
H
2
(
An
,
Sn
)) is
set
{
An
}
is non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
H
2
(
An
,
Sn
)
)
\
{
An
}
is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
}
is
V28
()
set
o0
is non
empty
pair
set
An
is non
empty
V56
()
ManySortedSign
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is
set
N
.
Sn
is
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
N
.
(
Sn
+
1
)
is
set
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
(
Sn
+
1
)
,
A0
,
h0
)
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
)) is
set
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
is non
empty
pair
set
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
+*
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
+*
H
2
(
o0
,
Sn
) is non
empty
non
void
V56
()
strict
ManySortedSign
GFA0CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
) is
set
[
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
InputVertices
H
2
(
o0
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
o0
,
Sn
))
the
carrier
of
H
2
(
o0
,
Sn
) is non
empty
set
K27
( the
carrier
of
H
2
(
o0
,
Sn
)) is
set
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
}
is
V28
()
set
InnerVertices
H
2
(
o0
,
Sn
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
o0
,
Sn
))
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
GFA0AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
the
carrier
of
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
) is
set
2GatesCircOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2
)
)
) is
set
[
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
o0
*>
,
xor2
]
is non
empty
pair
set
{
(
GFA0AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V28
()
set
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
is
Relation-like
V28
()
set
(
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
)
\/
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
is non
empty
V28
()
set
{
(
GFA0CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
(
(
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
)
\/
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
)
\/
{
(
GFA0CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V28
()
set
{
H
5
(
o0
,
Sn
)
}
is non
empty
V12
()
V28
() 1
-element
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
(
Sn
+
1
)
,
A0
,
h0
) is
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
))
the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
set
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
)) is
set
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
Sn
,
A0
,
h0
) is
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,(
Sn
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,(
Sn
,
A0
,
h0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2
]
,(
Sn
,
A0
,
h0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
+*
(
GFA0CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is non
empty
set
InputVertices
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
) is
set
{
(
Sn
,
A0
,
h0
)
}
is
Relation-like
non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
\
{
(
Sn
,
A0
,
h0
)
}
is
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
(
InputVertices
(
Sn
,
A0
,
h0
)
)
\/
(
(
InputVertices
(
BitGFA0Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
\
{
(
Sn
,
A0
,
h0
)
}
)
is
set
N
.
Sn
is
set
g
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
0
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
S0
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
0
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
(
0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
0
,
g
,
S0
) is
Element
of
K27
( the
carrier
of (
0
,
g
,
S0
))
the
carrier
of (
0
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
0
,
g
,
S0
)) is
set
proj2
g
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
proj2
S0
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
(
proj2
g
)
\/
(
proj2
S0
)
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
proj2
{}
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
g
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
(
(
g
+
1
)
,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
(
g
+
1
)
,
S0
,
A0
) is
Element
of
K27
( the
carrier
of (
(
g
+
1
)
,
S0
,
A0
))
the
carrier
of (
(
g
+
1
)
,
S0
,
A0
) is non
empty
set
K27
( the
carrier
of (
(
g
+
1
)
,
S0
,
A0
)) is
set
proj2
S0
is
V28
() non
with_pair
set
proj2
A0
is
V28
() non
with_pair
set
(
proj2
S0
)
\/
(
proj2
A0
)
is
V28
() non
with_pair
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
N
is non
pair
set
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h0
^
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
set
h
is
Relation-like
NAT
-defined
Function-like
V28
()
g
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
Sn
is non
pair
set
<*
Sn
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h
^
<*
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
set
dom
<*
N
*>
is non
empty
V12
()
V28
() 1
-element
Element
of
K27
(
NAT
)
len
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
S0
.
(
g
+
1
)
is non
pair
set
<*
N
*>
.
1 is non
pair
set
dom
<*
Sn
*>
is non
empty
V12
()
V28
() 1
-element
Element
of
K27
(
NAT
)
len
h
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
g
+
1
)
is non
pair
set
<*
Sn
*>
.
1 is non
pair
set
(
g
,
S0
,
A0
) is non
empty
pair
Element
of
InnerVertices
(
g
,
S0
,
A0
)
(
g
,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
g
,
S0
,
A0
) is non
empty
set
InnerVertices
(
g
,
S0
,
A0
) is non
empty
Element
of
K27
( the
carrier
of (
g
,
S0
,
A0
))
K27
( the
carrier
of (
g
,
S0
,
A0
)) is
set
{
N
,
Sn
,
H
1
(
g
)
}
is
V28
()
set
{
H
1
(
g
),
N
,
Sn
}
is
V28
()
set
proj2
h0
is
V28
() non
with_pair
set
proj2
<*
N
*>
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h0
)
\/
(
proj2
<*
N
*>
)
is non
empty
V28
() non
with_pair
set
{
N
}
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h0
)
\/
{
N
}
is non
empty
V28
() non
with_pair
set
proj2
h
is
V28
() non
with_pair
set
proj2
<*
Sn
*>
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h
)
\/
(
proj2
<*
Sn
*>
)
is non
empty
V28
() non
with_pair
set
{
Sn
}
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
<*
N
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
[
<*
N
,
Sn
*>
,
and2
]
is non
empty
pair
set
[
<*
N
,
Sn
*>
,
xor2
]
is non
empty
pair
set
h0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
g
,
h0
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
g
,
h0
,
h
) is
Element
of
K27
( the
carrier
of (
g
,
h0
,
h
))
the
carrier
of (
g
,
h0
,
h
) is non
empty
set
K27
( the
carrier
of (
g
,
h0
,
h
)) is
set
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
N
,
Sn
,(
g
,
S0
,
A0
),
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
Sn
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
N
,
Sn
*>
,
xor2
]
,(
g
,
S0
,
A0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
xor2
]
,(
g
,
S0
,
A0
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
Sn
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
xor2
]
,(
g
,
S0
,
A0
)
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
Sn
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
Sn
,(
g
,
S0
,
A0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
Sn
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
,
S0
,
A0
),
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
N
,
Sn
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
N
,
Sn
*>
,
and2
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
and2
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
and2
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
+*
(
GFA0CarryStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
is non
empty
set
InputVertices
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
is
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
)
K27
( the
carrier
of
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
) is
set
{
H
1
(
g
)
}
is
Relation-like
non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
is
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
)
(
InputVertices
(
g
,
h0
,
h
)
)
\/
(
(
InputVertices
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
)
is
set
(
proj2
h0
)
\/
(
proj2
h
)
is
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
(
InputVertices
(
BitGFA0Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
)
is
set
{
N
,
Sn
,
H
1
(
g
)
}
\
{
H
1
(
g
)
}
is
V28
()
Element
of
K27
(
{
N
,
Sn
,
H
1
(
g
)
}
)
K27
(
{
N
,
Sn
,
H
1
(
g
)
}
) is
V28
()
V32
()
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
{
N
,
Sn
,
H
1
(
g
)
}
\
{
H
1
(
g
)
}
)
is
V28
()
set
{
N
,
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
,
Sn
}
is non
empty
V28
() non
with_pair
set
{
N
}
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
{
N
}
\/
{
Sn
}
)
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
}
is non
empty
V28
() non
with_pair
set
(
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
}
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
{
N
}
)
\/
(
proj2
h
)
is non
empty
V28
() non
with_pair
set
(
(
(
proj2
h0
)
\/
{
N
}
)
\/
(
proj2
h
)
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
2
*
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
1
+
(
2
*
n
)
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
the
Sorts
of (
n
,
f
,
g
) is
Relation-like
the
carrier
of (
n
,
f
,
g
)
-defined
Function-like
V14
( the
carrier
of (
n
,
f
,
g
))
set
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K106
( the
Sorts
of (
n
,
f
,
g
)) is
set
N
is
Relation-like
Function-like
V18
(
NAT
,
NAT
)
Element
of
K27
(
K28
(
NAT
,
NAT
))
N
.
0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
N
.
1 is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
N
.
2 is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
Sn
+
1
)
is non
pair
set
g
.
(
Sn
+
1
)
is non
pair
set
h
is
set
BitGFA0Circ
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
BitGFA0Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
is non
empty
pair
set
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
2GatesCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2
)
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2
]
,
h
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
GFA0CarryICirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
1GateCircuit
(
h
,
(
f
.
(
Sn
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2
)
)
)
+*
(
1GateCircuit
(
h
,
(
f
.
(
Sn
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is
Relation-like
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
-defined
Function-like
V14
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
set
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is non
empty
set
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
) is
set
h
is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
Following
(
h
,1) is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
Following
h
is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
Following
(
h
,
H
4
(
0
)) is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is non
pair
set
g
.
(
o0
+
1
)
is non
pair
set
h
.
o0
is
set
f1
is
set
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA0Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2
]
,
f1
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
1GateCircuit
(
f1
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2
)
)
)
+*
(
1GateCircuit
(
f1
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
g1
is
non-empty
finitely-generated
V95
(
H
1
(
f1
,
o0
))
MSAlgebra
over
H
1
(
f1
,
o0
)
the
Sorts
of
g1
is
Relation-like
the
carrier
of
H
1
(
f1
,
o0
)
-defined
Function-like
V14
( the
carrier
of
H
1
(
f1
,
o0
))
set
the
carrier
of
H
1
(
f1
,
o0
) is non
empty
set
K106
( the
Sorts
of
g1
) is
set
(
o0
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
o0
,
f
,
g
)
(
o0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
o0
,
f
,
g
) is non
empty
set
InnerVertices
(
o0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
o0
,
f
,
g
))
K27
( the
carrier
of (
o0
,
f
,
g
)) is
set
h1
is
Element
of
K106
( the
Sorts
of
g1
)
Following
(
h1
,
(
N
.
1
)
) is
Element
of
K106
( the
Sorts
of
g1
)
the
Sorts
of
g1
is
Relation-like
the
carrier
of
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
-defined
Function-like
V14
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
)
set
the
carrier
of
(
BitGFA0Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is non
empty
set
K106
( the
Sorts
of
g1
) is
set
(
0
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
0
,
f
,
g
)
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
f
,
g
) is non
empty
set
InnerVertices
(
0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
f
,
g
))
K27
( the
carrier
of (
0
,
f
,
g
)) is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
n
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
n
is
set
f1
.
0
is
set
g1
.
0
is
set
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
0
is
set
f1
is
set
h1
.
f1
is
set
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
f2
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
f2
,
f
,
g
)
(
f2
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f2
,
f
,
g
) is non
empty
set
InnerVertices
(
f2
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
f2
,
f
,
g
))
K27
( the
carrier
of (
f2
,
f
,
g
)) is
set
h
.
f1
is
set
h
.
0
is
set
f1
.
H
4
(2) is
set
g1
.
H
4
(2) is
set
f2
is non
empty
V56
()
ManySortedSign
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is non
pair
set
g
.
(
f1
+
1
)
is non
pair
set
f0
is
set
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f1
.
f1
is
set
f3
is
non-empty
MSAlgebra
over
f2
g1
.
f1
is
set
h
.
f1
is
set
n
is
non-empty
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
BitGFA0Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
GFA0AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
2GatesCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
,
xor2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
)
GFA0CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
GFA0CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
1GateCircuit
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
1GateCircuit
(
f0
,
(
f
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2
)
)
)
+*
(
1GateCircuit
(
f0
,
(
f
.
(
f1
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA0CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA0AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
f1
.
(
f1
+
1
)
is
set
f2
+*
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
g1
.
(
f1
+
1
)
is
set
f3
+*
n
is
strict
non-empty
MSAlgebra
over
f2
+*
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
h
.
(
f1
+
1
)
is
set
GFA0CarryOutput
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
) is
set
[
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
InnerVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
) is
set
InputVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
is
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
)
)
)
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
FALSE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is non
pair
set
g
.
(
f1
+
1
)
is non
pair
set
f2
is
set
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f2
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2
]
,
f2
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
]
is non
empty
pair
set
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
)
the
carrier
of
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA0Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
) is
set
f3
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
.
f3
is
set
f3
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f3
+
1
)
is non
pair
set
g
.
(
f3
+
1
)
is non
pair
set
f0
is
set
BitGFA0Str
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2
]
,
f0
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
1
(
f0
,
f3
) is non
empty
set
InputVertices
H
1
(
f0
,
f3
) is
Element
of
K27
( the
carrier
of
H
1
(
f0
,
f3
))
K27
( the
carrier
of
H
1
(
f0
,
f3
)) is
set
{
f0
}
is non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
H
1
(
f0
,
f3
)
)
\
{
f0
}
is
Element
of
K27
( the
carrier
of
H
1
(
f0
,
f3
))
(
f3
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
f3
,
f
,
g
)
(
f3
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f3
,
f
,
g
) is non
empty
set
InnerVertices
(
f3
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
f3
,
f
,
g
))
K27
( the
carrier
of (
f3
,
f
,
g
)) is
set
{
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
}
is
V28
()
set
n
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
.
n
is
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h
.
(
n
+
1
)
is
set
f
.
(
n
+
1
)
is non
pair
set
g
.
(
n
+
1
)
is non
pair
set
x
is
set
GFA0CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is
Element
of
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,
x
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
x
,
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
)
)
)
+*
(
1GateCircStr
(
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
is non
empty
pair
set
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
set
InnerVertices
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
)
K27
( the
carrier
of
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
) is
set
[
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA0Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
x
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
x
*>
,
xor2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
x
*>
,
xor2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
+*
(
GFA0CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
H
1
(
x
,
n
) is
Element
of
K27
( the
carrier
of
H
1
(
x
,
n
))
the
carrier
of
H
1
(
x
,
n
) is non
empty
set
K27
( the
carrier
of
H
1
(
x
,
n
)) is
set
InnerVertices
H
1
(
x
,
n
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
x
,
n
))
(
n
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
(
(
n
+
1
)
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
(
n
+
1
)
,
f
,
g
)
(
(
n
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
n
+
1
)
,
f
,
g
) is non
empty
set
InnerVertices
(
(
n
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
))
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
)) is
set
{
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
}
is
V28
()
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
GFA0AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is
Element
of
InnerVertices
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
the
carrier
of
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
set
InnerVertices
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
)
K27
( the
carrier
of
(
GFA0AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
) is
set
2GatesCircOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
)
)
the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2
)
)
) is
set
[
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
,
x
*>
,
xor2
]
is non
empty
pair
set
{
(
GFA0AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V28
()
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
is
Relation-like
V28
()
set
(
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
)
\/
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
is non
empty
V28
()
set
{
(
GFA0CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
(
(
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2
]
}
\/
{
(
GFA0AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
)
\/
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
)
\/
{
(
GFA0CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V28
()
set
{
H
3
(
x
,
n
)
}
is non
empty
V12
()
V28
() 1
-element
set
H
4
(2)
*
H
4
(1) is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
H
4
(
0
)
+
(
H
4
(2)
*
H
4
(1)
)
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
f1
is
Element
of
K106
( the
Sorts
of (
n
,
f
,
g
))
Following
(
f1
,
(
1
+
(
2
*
n
)
)
) is
Element
of
K106
( the
Sorts
of (
n
,
f
,
g
))
TRUE
is
boolean
Element
of
BOOLEAN
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
is
Relation-like
Function-like
V18
(
0
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
0
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
]
is non
empty
pair
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
n
is
set
Sn
.
0
is
set
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
0
is
set
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
n
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
S0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
h
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is
set
g
.
(
o0
+
1
)
is
set
An
is
set
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
is non
empty
pair
set
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
non-empty
MSAlgebra
over
h
BitGFA1Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
)
)
)
+*
(
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
Sn
+*
(
BitGFA1Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
MSAlgebra
over
h
+*
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
h
+*
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
n
is
set
h
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
n
is
set
An
.
0
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
n
is
set
Sn
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
n
is
set
g1
.
0
is
set
h1
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
An
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
An
+
1
)
is
set
g
.
(
An
+
1
)
is
set
Sn
is
set
BitGFA1Str
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
]
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
]
,
Sn
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
xor2c
]
,
Sn
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
]
is non
empty
pair
set
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
Sn
*>
,
and2a
]
,
[
<*
Sn
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
+*
(
BitGFA1Str
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
Sn
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
g1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
g1
+
1
)
is
set
g
.
(
g1
+
1
)
is
set
f1
is
set
BitGFA1Str
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
g1
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
+*
(
BitGFA1Str
(
(
f
.
(
g1
+
1
)
)
,
(
g
.
(
g1
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f2
+
1
)
is
set
g
.
(
f2
+
1
)
is
set
f1
is
set
BitGFA1Str
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
+*
(
BitGFA1Str
(
(
f
.
(
f2
+
1
)
)
,
(
g
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f3
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
n
+
1
)
is
set
g
.
(
n
+
1
)
is
set
f0
is
set
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f3
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
x
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
h2
+
1
)
is
set
g
.
(
h2
+
1
)
is
set
g2
is
set
BitGFA1Str
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
]
,
g2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
]
,
g2
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
xor2c
]
,
g2
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
]
is non
empty
pair
set
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
h2
+
1
)
)
,
g2
*>
,
and2a
]
,
[
<*
g2
,
(
f
.
(
h2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
x
+*
(
BitGFA1Str
(
(
f
.
(
h2
+
1
)
)
,
(
g
.
(
h2
+
1
)
)
,
g2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is
set
g
.
(
o0
+
1
)
is
set
An
is
set
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
is non
empty
pair
set
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is
non-empty
MSAlgebra
over
h
BitGFA1Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
,
xor2c
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
An
,
and2a
)
)
)
+*
(
1GateCircuit
(
An
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
Sn
+*
(
BitGFA1Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is
strict
non-empty
MSAlgebra
over
h
+*
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
h
+*
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
h
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
Sn
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is
set
g
.
(
f1
+
1
)
is
set
o0
is
set
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
is non
empty
pair
set
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h
+*
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
is
strict
non-empty
finitely-generated
V95
(
h
)
gate`2=den
Boolean
MSAlgebra
over
h
BitGFA1Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
GFA1AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
2GatesCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
,
xor2c
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
GFA1CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
1GateCircuit
(
o0
,
(
f
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
o0
,
and2a
)
)
)
+*
(
1GateCircuit
(
o0
,
(
f
.
(
f1
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
An
+*
(
BitGFA1Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
is
strict
non-empty
finitely-generated
V95
(
h
+*
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
h
+*
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
o0
)
)
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
n
is
set
h
is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
n
is
set
Sn
.
0
is
set
An
.
0
is
set
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
0
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
A0
is
Element
of
InnerVertices
(
n
,
f
,
g
)
h0
is
Element
of
InnerVertices
(
n
,
f
,
g
)
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
n
is
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
n
is
set
h
.
0
is
set
proj1
N
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
N
.
(
Sn
+
1
)
is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
N
.
Sn
is
set
GFA1CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
]
,
[
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
]
,
[
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
]
,
[
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
N
.
Sn
)
*>
,
and2a
]
,
[
<*
(
N
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
proj1
h
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h
.
(
Sn
+
1
)
is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
h
.
Sn
is
set
GFA1CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
n
is
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
InnerVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is non
empty
set
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
) is
set
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
h
.
Sn
is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
N
.
(
Sn
+
1
)
is
set
h
.
(
Sn
+
1
)
is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
GFA1CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
*>
,
and2a
]
,
[
<*
(
h
.
Sn
)
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
(
h
.
Sn
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
(
h
.
Sn
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
(
h
.
Sn
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
the
carrier
of
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
) is
set
An
+*
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
InnerVertices
(
An
+*
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
An
+*
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
)
the
carrier
of
(
An
+*
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
is non
empty
set
K27
( the
carrier
of
(
An
+*
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
(
h
.
Sn
)
)
)
)
) is
set
h
.
n
is
set
An
is non
empty
V56
()
ManySortedSign
InnerVertices
An
is
Element
of
K27
( the
carrier
of
An
)
the
carrier
of
An
is non
empty
set
K27
( the
carrier
of
An
) is
set
Sn
is
Element
of
InnerVertices
(
n
,
f
,
g
)
An
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
An
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h
.
(
An
+
1
)
is
set
f
.
(
An
+
1
)
is
set
g
.
(
An
+
1
)
is
set
h
.
An
is
set
GFA1CarryOutput
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
)
)
) is
set
[
<*
[
<*
(
f
.
(
An
+
1
)
)
,
(
g
.
(
An
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
An
+
1
)
)
,
(
h
.
An
)
*>
,
and2a
]
,
[
<*
(
h
.
An
)
,
(
f
.
(
An
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
f1
is non
empty
V56
()
ManySortedSign
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
o0
is
set
g1
is
set
h
.
o0
is
set
N
.
An
is
set
o0
is non
empty
V56
()
ManySortedSign
n
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g
.
0
is
set
S0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
S0
.
0
is
set
A0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
A0
.
0
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
n
,
f
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
g
.
Sn
is
set
(
Sn
,
n
,
f
) is
strict
non-empty
finitely-generated
V95
((
Sn
,
n
,
f
))
gate`2=den
Boolean
MSAlgebra
over (
Sn
,
n
,
f
)
S0
.
Sn
is
set
(
Sn
,
n
,
f
) is
Element
of
InnerVertices
(
Sn
,
n
,
f
)
the
carrier
of (
Sn
,
n
,
f
) is non
empty
set
InnerVertices
(
Sn
,
n
,
f
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
n
,
f
))
K27
( the
carrier
of (
Sn
,
n
,
f
)) is
set
A0
.
Sn
is
set
An
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
An
.
Sn
is
set
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
Sn
is
set
An
.
0
is
set
o0
.
0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
0
is
set
g1
is non
empty
V56
()
ManySortedSign
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
n
.
(
f2
+
1
)
is
set
f
.
(
f2
+
1
)
is
set
f1
is
set
BitGFA1Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
is non
empty
pair
set
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
h1
is
non-empty
MSAlgebra
over
g1
BitGFA1Circ
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
GFA1AdderCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
2GatesCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
,
xor2c
)
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
GFA1CarryICirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
1GateCircuit
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
1GateCircuit
(
f1
,
(
n
.
(
f2
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
f
.
(
f2
+
1
)
)
,
f1
,
and2a
)
)
)
+*
(
1GateCircuit
(
f1
,
(
n
.
(
f2
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircuit
(
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
f2
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
n
.
(
f2
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryCirc
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
h1
+*
(
BitGFA1Circ
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is
strict
non-empty
MSAlgebra
over
g1
+*
(
BitGFA1Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
g1
+*
(
BitGFA1Str
(
(
n
.
(
f2
+
1
)
)
,
(
f
.
(
f2
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
proj1
A0
is
set
g1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
g1
+
1
)
is
set
n
.
(
g1
+
1
)
is
set
f
.
(
g1
+
1
)
is
set
A0
.
g1
is
set
GFA1CarryOutput
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
)
)
) is
set
[
<*
[
<*
(
n
.
(
g1
+
1
)
)
,
(
f
.
(
g1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
g1
+
1
)
)
,
(
A0
.
g1
)
*>
,
and2a
]
,
[
<*
(
A0
.
g1
)
,
(
n
.
(
g1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
Sn
is
set
g1
.
0
is
set
proj1
g1
is
set
h1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g1
.
(
h1
+
1
)
is
set
n
.
(
h1
+
1
)
is
set
f
.
(
h1
+
1
)
is
set
g1
.
h1
is
set
GFA1CarryOutput
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
)
)
) is
set
[
<*
[
<*
(
n
.
(
h1
+
1
)
)
,
(
f
.
(
h1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
f
.
(
h1
+
1
)
)
,
(
g1
.
h1
)
*>
,
and2a
]
,
[
<*
(
g1
.
h1
)
,
(
n
.
(
h1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
n
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
0
,
n
,
f
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
0
,
n
,
f
) is
strict
non-empty
finitely-generated
V95
((
0
,
n
,
f
))
gate`2=den
Boolean
MSAlgebra
over (
0
,
n
,
f
)
(
0
,
n
,
f
) is
Element
of
InnerVertices
(
0
,
n
,
f
)
the
carrier
of (
0
,
n
,
f
) is non
empty
set
InnerVertices
(
0
,
n
,
f
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
n
,
f
))
K27
( the
carrier
of (
0
,
n
,
f
)) is
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
h0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h0
.
0
is
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
0
is
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(1,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
S0
.
1 is
set
A0
.
1 is
set
(1,
S0
,
A0
) is
strict
non-empty
finitely-generated
V95
((1,
S0
,
A0
))
gate`2=den
Boolean
MSAlgebra
over (1,
S0
,
A0
)
(1,
S0
,
A0
) is
Element
of
InnerVertices
(1,
S0
,
A0
)
the
carrier
of (1,
S0
,
A0
) is non
empty
set
InnerVertices
(1,
S0
,
A0
) is non
empty
Element
of
K27
( the
carrier
of (1,
S0
,
A0
))
K27
( the
carrier
of (1,
S0
,
A0
)) is
set
h0
is
set
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
1
)
,
h0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h0
,
(
S0
.
1
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
is non
empty
pair
set
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA1Circ
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
GFA1AdderCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
2GatesCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
,
xor2c
)
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
)
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
xor2c
]
,
h0
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
GFA1CarryICirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
1GateCircuit
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
1GateCircuit
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
1GateCircuit
(
h0
,
(
S0
.
1
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
1GateCircuit
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
(
(
1GateCircuit
(
(
S0
.
1
)
,
(
A0
.
1
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
A0
.
1
)
,
h0
,
and2a
)
)
)
+*
(
1GateCircuit
(
h0
,
(
S0
.
1
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
A0
.
1
)
,
h0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h0
,
(
S0
.
1
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircuit
(
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA1CarryCirc
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
+*
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Circ
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
GFA1CarryOutput
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
the
carrier
of
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
S0
.
1
)
,
(
A0
.
1
)
,
h0
)
)
) is
set
[
<*
[
<*
(
S0
.
1
)
,
(
A0
.
1
)
*>
,
and2c
]
,
[
<*
(
A0
.
1
)
,
h0
*>
,
and2a
]
,
[
<*
h0
,
(
S0
.
1
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
1 is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
1 is
set
N
.
0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
Sn
.
(
0
+
1
)
is
set
n
is
set
<*
n
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
f
is
set
<*
f
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
(1,
<*
n
*>
,
<*
f
*>
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(1,
<*
n
*>
,
<*
f
*>
) is
strict
non-empty
finitely-generated
V95
((1,
<*
n
*>
,
<*
f
*>
))
gate`2=den
Boolean
MSAlgebra
over (1,
<*
n
*>
,
<*
f
*>
)
(1,
<*
n
*>
,
<*
f
*>
) is
Element
of
InnerVertices
(1,
<*
n
*>
,
<*
f
*>
)
the
carrier
of (1,
<*
n
*>
,
<*
f
*>
) is non
empty
set
InnerVertices
(1,
<*
n
*>
,
<*
f
*>
) is non
empty
Element
of
K27
( the
carrier
of (1,
<*
n
*>
,
<*
f
*>
))
K27
( the
carrier
of (1,
<*
n
*>
,
<*
f
*>
)) is
set
<*
n
*>
.
1 is
set
<*
f
*>
.
1 is
set
g
is
set
BitGFA1Str
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
n
,
f
,
g
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
n
,
f
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
n
,
f
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
n
,
f
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f
,
g
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f
,
g
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
g
,
n
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
g
,
n
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
n
,
f
*>
,
and2c
]
is non
empty
pair
set
[
<*
f
,
g
*>
,
and2a
]
is non
empty
pair
set
[
<*
g
,
n
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA1CarryStr
(
n
,
f
,
g
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
n
,
f
,
g
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA1Circ
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
n
,
f
,
g
)
GFA1AdderCirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
n
,
f
,
g
)
2GatesCircuit
(
n
,
f
,
g
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
n
,
f
,
g
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
n
,
f
,
g
,
xor2c
)
1GateCircuit
(
n
,
f
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
)
1GateCircuit
(
<*
n
,
f
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
)
1GateCircuit
(
[
<*
n
,
f
*>
,
xor2c
]
,
g
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
)
(
1GateCircuit
(
n
,
f
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
n
,
f
*>
,
xor2c
]
,
g
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
n
,
f
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
xor2c
]
,
g
*>
,
xor2c
)
)
GFA1CarryCirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
n
,
f
,
g
)
GFA1CarryICirc
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
n
,
f
,
g
)
1GateCircuit
(
n
,
f
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
1GateCircuit
(
<*
n
,
f
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
1GateCircuit
(
f
,
g
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
1GateCircuit
(
<*
f
,
g
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
(
1GateCircuit
(
n
,
f
,
and2c
)
)
+*
(
1GateCircuit
(
f
,
g
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
1GateCircuit
(
g
,
n
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
1GateCircuit
(
<*
g
,
n
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
(
(
1GateCircuit
(
n
,
f
,
and2c
)
)
+*
(
1GateCircuit
(
f
,
g
,
and2a
)
)
)
+*
(
1GateCircuit
(
g
,
n
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
n
,
f
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
f
,
g
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
g
,
n
*>
,
and2
)
)
1GateCircuit
(
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
n
,
f
,
g
)
)
+*
(
1GateCircuit
(
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
n
,
f
,
g
)
)
+*
(
1GateCircStr
(
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
n
,
f
,
g
)
)
+*
(
GFA1CarryCirc
(
n
,
f
,
g
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA1CarryStr
(
n
,
f
,
g
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
n
,
f
,
g
)
)
+*
(
GFA1CarryStr
(
n
,
f
,
g
)
)
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Circ
(
n
,
f
,
g
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
n
,
f
,
g
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
+*
(
BitGFA1Str
(
n
,
f
,
g
)
)
GFA1CarryOutput
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
n
,
f
,
g
)
)
the
carrier
of
(
GFA1CarryStr
(
n
,
f
,
g
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
n
,
f
,
g
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
n
,
f
,
g
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
n
,
f
,
g
)
)
) is
set
[
<*
[
<*
n
,
f
*>
,
and2c
]
,
[
<*
f
,
g
*>
,
and2a
]
,
[
<*
g
,
n
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
^
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
^
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
f
^
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
N
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
^
N
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is
strict
non-empty
finitely-generated
V95
((
n
,
(
f
^
S0
)
,
(
g
^
h0
)
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is
strict
non-empty
finitely-generated
V95
((
n
,
(
f
^
A0
)
,
(
g
^
N
)
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
(
f
^
A0
)
,
(
g
^
N
)
)
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is
Element
of
InnerVertices
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)
the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
set
InnerVertices
(
n
,
(
f
^
S0
)
,
(
g
^
h0
)
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
))
K27
( the
carrier
of (
n
,
(
f
^
S0
)
,
(
g
^
h0
)
)) is
set
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is
Element
of
InnerVertices
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
)
the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
set
InnerVertices
(
n
,
(
f
^
A0
)
,
(
g
^
N
)
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
))
K27
( the
carrier
of (
n
,
(
f
^
A0
)
,
(
g
^
N
)
)) is
set
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
n
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
n
is
set
h1
.
0
is
set
f1
.
0
is
set
f2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f2
.
0
is
set
x
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
x
.
n
is
set
g2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g2
.
n
is
set
x
.
0
is
set
g2
.
0
is
set
h2
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h2
.
0
is
set
i
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f2
.
i
is
set
h2
.
i
is
set
h1
.
i
is
set
x
.
i
is
set
f1
.
i
is
set
g2
.
i
is
set
i
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f2
.
(
i
+
1
)
is
set
h2
.
(
i
+
1
)
is
set
h1
.
(
i
+
1
)
is
set
x
.
(
i
+
1
)
is
set
f1
.
(
i
+
1
)
is
set
g2
.
(
i
+
1
)
is
set
len
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
len
g
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
dom
f
is
V28
()
n
-element
Element
of
K27
(
NAT
)
Seg
n
is
V28
()
n
-element
Element
of
K27
(
NAT
)
dom
g
is
V28
()
n
-element
Element
of
K27
(
NAT
)
0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
f
^
S0
)
.
(
i
+
1
)
is
set
f
.
(
i
+
1
)
is
set
(
f
^
A0
)
.
(
i
+
1
)
is
set
(
g
^
h0
)
.
(
i
+
1
)
is
set
g
.
(
i
+
1
)
is
set
(
g
^
N
)
.
(
i
+
1
)
is
set
A
is non
empty
V56
()
ManySortedSign
S
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h1
.
S
is
set
c
26
is
non-empty
MSAlgebra
over
A
f1
.
S
is
set
c
27
is
set
f2
.
S
is
set
S
is non
empty
V56
()
ManySortedSign
c
27
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
c
27
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
f
^
S0
)
.
(
c
27
+
1
)
is
set
(
g
^
h0
)
.
(
c
27
+
1
)
is
set
c
26
is
set
BitGFA1Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
is non
empty
pair
set
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A
is
non-empty
MSAlgebra
over
S
BitGFA1Circ
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
GFA1AdderCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
2GatesCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
xor2c
)
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
)
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
xor2c
]
,
c
26
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
GFA1CarryICirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
1GateCircuit
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
1GateCircuit
(
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
,
and2a
)
)
)
+*
(
1GateCircuit
(
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
*>
,
and2a
]
,
[
<*
c
26
,
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA1CarryCirc
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
A
+*
(
BitGFA1Circ
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is
strict
non-empty
MSAlgebra
over
S
+*
(
BitGFA1Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
S
+*
(
BitGFA1Str
(
(
(
f
^
S0
)
.
(
c
27
+
1
)
)
,
(
(
g
^
h0
)
.
(
c
27
+
1
)
)
,
c
26
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
S
is non
empty
V56
()
ManySortedSign
A
is
non-empty
MSAlgebra
over
S
GFA1CarryOutput
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
) is
set
[
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA1Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
S
+*
(
BitGFA1Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
BitGFA1Circ
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
GFA1AdderCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
2GatesCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
xor2c
)
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
)
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
xor2c
]
,
(
h2
.
i
)
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
GFA1CarryICirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
1GateCircuit
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
1GateCircuit
(
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
,
and2a
)
)
)
+*
(
1GateCircuit
(
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircuit
(
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
*>
,
and2c
]
,
[
<*
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
*>
,
and2a
]
,
[
<*
(
h2
.
i
)
,
(
(
f
^
A0
)
.
(
i
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA1CarryCirc
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
+*
(
GFA1CarryStr
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
A
+*
(
BitGFA1Circ
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
is
strict
non-empty
MSAlgebra
over
S
+*
(
BitGFA1Str
(
(
(
f
^
A0
)
.
(
i
+
1
)
)
,
(
(
g
^
N
)
.
(
i
+
1
)
)
,
(
h2
.
i
)
)
)
f2
.
n
is
set
h2
.
n
is
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
S0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
S0
-element
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
S0
-element
FinSequence-like
FinSubsequence-like
set
(
S0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
) is
Element
of
InnerVertices
(
S0
,
A0
,
h0
)
the
carrier
of (
S0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
S0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
A0
,
h0
))
K27
( the
carrier
of (
S0
,
A0
,
h0
)) is
set
(
S0
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
N
is
set
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
A0
^
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
S0
+
1
-element
FinSequence-like
FinSubsequence-like
set
h
is
set
<*
h
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
set
h0
^
<*
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
S0
+
1
-element
FinSequence-like
FinSubsequence-like
set
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
N
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
N
,
h
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
h
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
,
A0
,
h0
),
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
N
,
h
*>
,
and2c
]
is non
empty
pair
set
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
strict
non-empty
finitely-generated
V95
((
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
gate`2=den
Boolean
MSAlgebra
over (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
BitGFA1Circ
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
))
GFA1AdderCirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
2GatesCircuit
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
N
,
h
,(
S0
,
A0
,
h0
),
xor2c
)
1GateCircuit
(
N
,
h
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
)
1GateCircuit
(
<*
N
,
h
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
)
1GateCircuit
(
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
),
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
(
1GateCircuit
(
N
,
h
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
),
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
N
,
h
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
GFA1CarryCirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
GFA1CarryICirc
(
N
,
h
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
1GateCircuit
(
N
,
h
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
1GateCircuit
(
<*
N
,
h
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
1GateCircuit
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
(
1GateCircuit
(
N
,
h
,
and2c
)
)
+*
(
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
1GateCircuit
((
S0
,
A0
,
h0
),
N
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
1GateCircuit
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
(
(
1GateCircuit
(
N
,
h
,
and2c
)
)
+*
(
1GateCircuit
(
h
,(
S0
,
A0
,
h0
),
and2a
)
)
)
+*
(
1GateCircuit
((
S0
,
A0
,
h0
),
N
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
N
,
h
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
)
)
1GateCircuit
(
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircuit
(
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryCirc
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
(
S0
,
A0
,
h0
)
+*
(
BitGFA1Circ
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
N
,
h
,(
S0
,
A0
,
h0
))
)
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
Element
of
InnerVertices
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
set
InnerVertices
(
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
Element
of
K27
( the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
K27
( the
carrier
of (
(
S0
+
1
)
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)) is
set
GFA1CarryOutput
(
N
,
h
,(
S0
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
the
carrier
of
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
N
,
h
,(
S0
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
N
,
h
*>
,
and2c
]
,
[
<*
h
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
N
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
strict
non-empty
finitely-generated
V95
((
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
o0
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
.
S0
is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
S0
is
set
o0
.
0
is
set
f1
.
0
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
0
is
set
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is
Element
of
InnerVertices
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)
the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
set
InnerVertices
(
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
))
K27
( the
carrier
of (
S0
,
(
A0
^
<*
N
*>
)
,
(
h0
^
<*
h
*>
)
)) is
set
g1
.
S0
is
set
o0
.
(
S0
+
1
)
is
set
f1
.
(
S0
+
1
)
is
set
g1
.
(
S0
+
1
)
is
set
len
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
len
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
(
A0
^
<*
N
*>
)
.
(
S0
+
1
)
is
set
(
h0
^
<*
h
*>
)
.
(
S0
+
1
)
is
set
A0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
S0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
S0
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
A0
.
(
S0
+
1
)
is
set
h0
.
(
S0
+
1
)
is
set
(
S0
,
A0
,
h0
) is
Element
of
InnerVertices
(
S0
,
A0
,
h0
)
the
carrier
of (
S0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
S0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
S0
,
A0
,
h0
))
K27
( the
carrier
of (
S0
,
A0
,
h0
)) is
set
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
S0
+
1
)
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
(
S0
+
1
)
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
(
S0
+
1
)
,
A0
,
h0
)
(
S0
,
A0
,
h0
) is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
))
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
BitGFA1Circ
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
GFA1AdderCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
2GatesCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
xor2c
)
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
),
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
),
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
xor2c
]
,(
S0
,
A0
,
h0
)
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
GFA1CarryICirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
1GateCircuit
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
1GateCircuit
((
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
),
and2a
)
)
)
+*
(
1GateCircuit
((
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircuit
(
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryCirc
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
(
S0
,
A0
,
h0
)
+*
(
BitGFA1Circ
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is
strict
non-empty
finitely-generated
V95
((
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
gate`2=den
Boolean
MSAlgebra
over (
S0
,
A0
,
h0
)
+*
(
BitGFA1Str
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
(
(
S0
+
1
)
,
A0
,
h0
) is
Element
of
InnerVertices
(
(
S0
+
1
)
,
A0
,
h0
)
the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
S0
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
S0
+
1
)
,
A0
,
h0
)) is
set
GFA1CarryOutput
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
(
A0
.
(
S0
+
1
)
)
,
(
h0
.
(
S0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
S0
+
1
)
)
,(
S0
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
S0
,
A0
,
h0
),
(
A0
.
(
S0
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
N
.
S0
is
set
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h
.
S0
is
set
N
.
0
is
set
h
.
0
is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
Sn
.
S0
is
set
N
.
(
S0
+
1
)
is
set
h
.
(
S0
+
1
)
is
set
Sn
.
(
S0
+
1
)
is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
g
,
S0
))
the
carrier
of (
n
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
n
,
g
,
S0
)) is
set
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
(
n
+
0
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
0
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
0
)
,
g
,
S0
))
the
carrier
of (
(
n
+
0
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
0
)
,
g
,
S0
)) is
set
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
(
n
+
h0
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
h0
)
,
g
,
S0
))
the
carrier
of (
(
n
+
h0
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
h0
)
,
g
,
S0
)) is
set
h0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
n
+
(
h0
+
1
)
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
set
(
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
))
the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
(
n
+
(
h0
+
1
)
)
,
g
,
S0
)) is
set
(
n
+
h0
)
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
.
(
(
n
+
h0
)
+
1
)
is
set
S0
.
(
(
n
+
h0
)
+
1
)
is
set
(
(
n
+
h0
)
,
g
,
S0
) is
Element
of
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
)
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
xor2c
]
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
(
n
+
h0
)
,
g
,
S0
),
(
g
.
(
(
n
+
h0
)
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
+*
(
GFA1CarryStr
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
the
carrier
of
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
) is
set
(
InnerVertices
(
n
,
g
,
S0
)
)
\/
(
InnerVertices
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
(
InnerVertices
(
(
n
+
h0
)
,
g
,
S0
)
)
\/
(
InnerVertices
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
)
the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
is non
empty
set
K27
( the
carrier
of
(
(
(
n
+
h0
)
,
g
,
S0
)
+*
(
BitGFA1Str
(
(
g
.
(
(
n
+
h0
)
+
1
)
)
,
(
S0
.
(
(
n
+
h0
)
+
1
)
)
,(
(
n
+
h0
)
,
g
,
S0
))
)
)
) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
n
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
))
the
carrier
of (
(
n
+
1
)
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
)) is
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
f
.
(
n
+
1
)
is
set
g
.
(
n
+
1
)
is
set
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
n
,
f
,
g
)
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,(
n
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,(
n
,
f
,
g
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,(
n
,
f
,
g
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
n
,
f
,
g
),
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
+*
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
the
carrier
of
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
) is
set
(
InnerVertices
(
n
,
f
,
g
)
)
\/
(
InnerVertices
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
set
(
n
,
f
,
g
)
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
(
n
,
f
,
g
)
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
)
the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
is non
empty
set
K27
( the
carrier
of
(
(
n
,
f
,
g
)
+*
(
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,(
n
,
f
,
g
))
)
)
) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
g
.
n
is
set
S0
.
n
is
set
N
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
N
,
g
,
S0
) is
Element
of
InnerVertices
(
N
,
g
,
S0
)
(
N
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
N
,
g
,
S0
) is non
empty
set
InnerVertices
(
N
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
N
,
g
,
S0
))
K27
( the
carrier
of (
N
,
g
,
S0
)) is
set
GFA1AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
N
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
N
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
N
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
N
,
g
,
S0
),
xor2c
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
N
,
g
,
S0
)
*>
,
xor2c
]
is non
empty
pair
set
h
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h0
is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
h
,
g
,
S0
) is
Element
of
InnerVertices
(
h
,
g
,
S0
)
(
h
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
h
,
g
,
S0
) is non
empty
set
InnerVertices
(
h
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
h
,
g
,
S0
))
K27
( the
carrier
of (
h
,
g
,
S0
)) is
set
GFA1AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h
,
g
,
S0
),
xor2c
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h
,
g
,
S0
)
*>
,
xor2c
]
is non
empty
pair
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
1
+
A0
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
h0
,
g
,
S0
) is
Element
of
InnerVertices
(
h0
,
g
,
S0
)
(
h0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
h0
,
g
,
S0
) is non
empty
set
InnerVertices
(
h0
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
h0
,
g
,
S0
))
K27
( the
carrier
of (
h0
,
g
,
S0
)) is
set
GFA1AdderOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
n
)
,
(
S0
.
n
)
,(
h0
,
g
,
S0
),
xor2c
)
)
) is
set
[
<*
[
<*
(
g
.
n
)
,
(
S0
.
n
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
,
xor2c
]
is non
empty
pair
set
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
n
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
n
))
the
carrier
of
H
1
(
n
) is non
empty
set
K27
( the
carrier
of
H
1
(
n
)) is
set
InnerVertices
H
1
(
f
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
f
))
the
carrier
of
H
1
(
f
) is non
empty
set
K27
( the
carrier
of
H
1
(
f
)) is
set
h0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
.
(
h0
+
1
)
is
set
S0
.
(
h0
+
1
)
is
set
BitGFA1Str
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
xor2c
]
,(
h0
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
)
*>
,
and2a
]
,
[
<*
(
h0
,
g
,
S0
),
(
g
.
(
h0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
+*
(
GFA1CarryStr
(
(
g
.
(
h0
+
1
)
)
,
(
S0
.
(
h0
+
1
)
)
,(
h0
,
g
,
S0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
2
(
h0
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
h0
))
the
carrier
of
H
2
(
h0
) is non
empty
set
K27
( the
carrier
of
H
2
(
h0
)) is
set
H
1
(
h0
)
+*
H
2
(
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
1
(
h0
) is non
empty
set
the
carrier
of
H
2
(
h0
)
\/
the
carrier
of
H
1
(
h0
) is non
empty
set
h
is
Element
of the
carrier
of
H
2
(
h0
)
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
(
f
+
1
)
,
n
,
g
,
S0
) is
Element
of
InnerVertices
(
n
,
g
,
S0
)
(
n
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
g
,
S0
) is non
empty
set
InnerVertices
(
n
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
g
,
S0
))
K27
( the
carrier
of (
n
,
g
,
S0
)) is
set
g
.
(
f
+
1
)
is
set
S0
.
(
f
+
1
)
is
set
(
f
,
g
,
S0
) is
Element
of
InnerVertices
(
f
,
g
,
S0
)
(
f
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f
,
g
,
S0
) is non
empty
set
InnerVertices
(
f
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
f
,
g
,
S0
))
K27
( the
carrier
of (
f
,
g
,
S0
)) is
set
GFA1AdderOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
f
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
f
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
f
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
f
,
g
,
S0
),
xor2c
)
)
) is
set
[
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
f
,
g
,
S0
)
*>
,
xor2c
]
is non
empty
pair
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
A0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
A0
,
g
,
S0
) is
Element
of
InnerVertices
(
A0
,
g
,
S0
)
(
A0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
A0
,
g
,
S0
) is non
empty
set
InnerVertices
(
A0
,
g
,
S0
) is non
empty
Element
of
K27
( the
carrier
of (
A0
,
g
,
S0
))
K27
( the
carrier
of (
A0
,
g
,
S0
)) is
set
GFA1AdderOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
)) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
A0
,
g
,
S0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
A0
,
g
,
S0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
A0
,
g
,
S0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
))
)
) is
set
2GatesCircOutput
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
,(
A0
,
g
,
S0
),
xor2c
)
)
) is
set
[
<*
[
<*
(
g
.
(
f
+
1
)
)
,
(
S0
.
(
f
+
1
)
)
*>
,
xor2c
]
,(
A0
,
g
,
S0
)
*>
,
xor2c
]
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
0
))
the
carrier
of
H
1
(
0
) is non
empty
set
K27
( the
carrier
of
H
1
(
0
)) is
set
A0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
A0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
A0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
A0
))
the
carrier
of
H
1
(
A0
) is non
empty
set
K27
( the
carrier
of
H
1
(
A0
)) is
set
A0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
A0
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f
.
(
A0
+
1
)
is
set
g
.
(
A0
+
1
)
is
set
(
A0
,
f
,
g
) is
Element
of
InnerVertices
(
A0
,
f
,
g
)
the
carrier
of (
A0
,
f
,
g
) is non
empty
set
InnerVertices
(
A0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
A0
,
f
,
g
))
K27
( the
carrier
of (
A0
,
f
,
g
)) is
set
BitGFA1Str
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
]
,(
A0
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
]
,(
A0
,
f
,
g
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
xor2c
]
,(
A0
,
f
,
g
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
A0
,
f
,
g
),
(
f
.
(
A0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
+*
(
GFA1CarryStr
(
(
f
.
(
A0
+
1
)
)
,
(
g
.
(
A0
+
1
)
)
,(
A0
,
f
,
g
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
H
1
(
A0
)
+*
H
2
(
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
2
(
A0
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
A0
))
the
carrier
of
H
2
(
A0
) is non
empty
set
K27
( the
carrier
of
H
2
(
A0
)) is
set
InnerVertices
H
1
(
A0
+
1) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
A0
+
1))
the
carrier
of
H
1
(
A0
+
1) is non
empty
set
K27
( the
carrier
of
H
1
(
A0
+
1)) is
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
(
0
,
f
,
g
) is
Element
of
InnerVertices
(
0
,
f
,
g
)
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
f
,
g
) is non
empty
set
InnerVertices
(
0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
f
,
g
))
K27
( the
carrier
of (
0
,
f
,
g
)) is
set
Sn
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
Sn
.
0
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
f
,
g
) is
Element
of
InnerVertices
(
Sn
,
f
,
g
)
(
Sn
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
Sn
,
f
,
g
) is non
empty
set
InnerVertices
(
Sn
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
f
,
g
))
K27
( the
carrier
of (
Sn
,
f
,
g
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
Sn
+
1
)
,
f
,
g
) is
Element
of
InnerVertices
(
(
Sn
+
1
)
,
f
,
g
)
(
(
Sn
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
Sn
+
1
)
,
f
,
g
) is non
empty
set
InnerVertices
(
(
Sn
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
f
,
g
))
K27
( the
carrier
of (
(
Sn
+
1
)
,
f
,
g
)) is
set
f
.
(
Sn
+
1
)
is
set
g
.
(
Sn
+
1
)
is
set
GFA1CarryOutput
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
))
)
) is
set
[
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,(
Sn
,
f
,
g
)
*>
,
and2a
]
,
[
<*
(
Sn
,
f
,
g
),
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
proj1
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
is
set
(
0
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
0
,
A0
,
h0
)
(
0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
A0
,
h0
))
K27
( the
carrier
of (
0
,
A0
,
h0
)) is
set
(
0
,
A0
,
h0
)
`1
is
set
(
0
,
A0
,
h0
)
`2
is
set
proj1
(
(
0
,
A0
,
h0
)
`2
)
is
set
card
(
(
0
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
N
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
N
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
N
,
A0
,
h0
)
(
N
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
N
,
A0
,
h0
) is non
empty
set
InnerVertices
(
N
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
N
,
A0
,
h0
))
K27
( the
carrier
of (
N
,
A0
,
h0
)) is
set
(
N
,
A0
,
h0
)
`1
is
set
(
N
,
A0
,
h0
)
`2
is
set
proj1
(
(
N
,
A0
,
h0
)
`2
)
is
set
card
(
(
N
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
N
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
(
N
+
1
)
,
A0
,
h0
)
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
N
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
N
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
N
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
N
+
1
)
,
A0
,
h0
)) is
set
A0
.
(
N
+
1
)
is
set
h0
.
(
N
+
1
)
is
set
GFA1CarryOutput
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
))
)
) is
set
[
<*
[
<*
(
A0
.
(
N
+
1
)
)
,
(
h0
.
(
N
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
N
+
1
)
)
,(
N
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
N
,
A0
,
h0
),
(
A0
.
(
N
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
proj1
or3
is
set
(
(
N
+
1
)
,
A0
,
h0
)
`1
is
set
(
(
N
+
1
)
,
A0
,
h0
)
`2
is
set
proj1
(
(
(
N
+
1
)
,
A0
,
h0
)
`2
)
is
set
card
(
(
(
N
+
1
)
,
A0
,
h0
)
`1
)
is
V21
()
V22
()
V23
()
cardinal
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
n
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
S0
is
set
A0
is
Relation-like
Function-like
V18
(2
-tuples_on
BOOLEAN
,
BOOLEAN
)
boolean-valued
Element
of
K27
(
K28
(
(
2
-tuples_on
BOOLEAN
)
,
BOOLEAN
))
[
S0
,
A0
]
is non
empty
pair
set
proj1
A0
is
set
[
S0
,
A0
]
`2
is
set
proj1
(
[
S0
,
A0
]
`2
)
is
set
(
n
,
f
,
g
)
`2
is
set
proj1
(
(
n
,
f
,
g
)
`2
)
is
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
N
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
(
0
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
H
1
(
0
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
0
))
the
carrier
of
H
1
(
0
) is non
empty
set
K27
( the
carrier
of
H
1
(
0
)) is
set
InputVertices
H
1
(
0
) is
Element
of
K27
( the
carrier
of
H
1
(
0
))
N
.
0
is
set
(
0
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
0
,
A0
,
h0
)
the
carrier
of (
0
,
A0
,
h0
) is non
empty
set
InnerVertices
(
0
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
A0
,
h0
))
K27
( the
carrier
of (
0
,
A0
,
h0
)) is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
An
is
set
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
)
the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
) is
set
An
is
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
H
2
(
An
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
the
carrier
of
H
2
(
An
,
Sn
) is non
empty
set
K27
( the
carrier
of
H
2
(
An
,
Sn
)) is
set
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
}
is
V28
()
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
N
.
Sn
is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
An
is
set
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
An
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
is non
empty
pair
set
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
An
*>
,
and2a
]
,
[
<*
An
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
+*
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
2
(
An
,
Sn
) is non
empty
set
InputVertices
H
2
(
An
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
K27
( the
carrier
of
H
2
(
An
,
Sn
)) is
set
{
An
}
is non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
H
2
(
An
,
Sn
)
)
\
{
An
}
is
Element
of
K27
( the
carrier
of
H
2
(
An
,
Sn
))
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
An
}
is
V28
()
set
o0
is non
empty
pair
set
An
is non
empty
V56
()
ManySortedSign
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
o0
is
set
N
.
Sn
is
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
N
.
(
Sn
+
1
)
is
set
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
(
Sn
+
1
)
,
A0
,
h0
)
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
set
InnerVertices
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
))
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
)) is
set
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
is non
empty
pair
set
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
+*
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
An
+*
H
2
(
o0
,
Sn
) is non
empty
non
void
V56
()
strict
ManySortedSign
GFA1CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
) is
set
[
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
InputVertices
H
2
(
o0
,
Sn
) is
Element
of
K27
( the
carrier
of
H
2
(
o0
,
Sn
))
the
carrier
of
H
2
(
o0
,
Sn
) is non
empty
set
K27
( the
carrier
of
H
2
(
o0
,
Sn
)) is
set
{
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
}
is
V28
()
set
InnerVertices
H
2
(
o0
,
Sn
) is non
empty
Element
of
K27
( the
carrier
of
H
2
(
o0
,
Sn
))
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
GFA1AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
the
carrier
of
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
) is
set
2GatesCircOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
,
xor2c
)
)
) is
set
[
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
o0
*>
,
xor2c
]
is non
empty
pair
set
{
(
GFA1AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V28
()
set
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
is
Relation-like
V28
()
set
(
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
)
\/
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
is non
empty
V28
()
set
{
(
GFA1CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
(
(
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
)
\/
{
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,
o0
*>
,
and2a
]
,
[
<*
o0
,
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
}
)
\/
{
(
GFA1CarryOutput
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,
o0
)
)
}
is non
empty
V28
()
set
{
H
5
(
o0
,
Sn
)
}
is non
empty
V12
()
V28
() 1
-element
set
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
(
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
(
Sn
+
1
)
,
A0
,
h0
) is
Element
of
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
))
the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
) is non
empty
set
K27
( the
carrier
of (
(
Sn
+
1
)
,
A0
,
h0
)) is
set
(
Sn
,
A0
,
h0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
Sn
,
A0
,
h0
) is
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
the
carrier
of (
Sn
,
A0
,
h0
) is non
empty
set
K27
( the
carrier
of (
Sn
,
A0
,
h0
)) is
set
A0
.
(
Sn
+
1
)
is non
pair
set
h0
.
(
Sn
+
1
)
is non
pair
set
(
Sn
,
A0
,
h0
) is non
empty
pair
Element
of
InnerVertices
(
Sn
,
A0
,
h0
)
InnerVertices
(
Sn
,
A0
,
h0
) is non
empty
Element
of
K27
( the
carrier
of (
Sn
,
A0
,
h0
))
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,(
Sn
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,(
Sn
,
A0
,
h0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,(
Sn
,
A0
,
h0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
+*
(
1GateCircStr
(
<*
[
<*
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
)
*>
,
and2a
]
,
[
<*
(
Sn
,
A0
,
h0
),
(
A0
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
+*
(
GFA1CarryStr
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is non
empty
set
InputVertices
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
is
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
K27
( the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
) is
set
{
(
Sn
,
A0
,
h0
)
}
is
Relation-like
non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
\
{
(
Sn
,
A0
,
h0
)
}
is
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
(
InputVertices
(
Sn
,
A0
,
h0
)
)
\/
(
(
InputVertices
(
BitGFA1Str
(
(
A0
.
(
Sn
+
1
)
)
,
(
h0
.
(
Sn
+
1
)
)
,(
Sn
,
A0
,
h0
))
)
)
\
{
(
Sn
,
A0
,
h0
)
}
)
is
set
N
.
Sn
is
set
g
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
0
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
S0
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
0
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
(
0
,
g
,
S0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
0
,
g
,
S0
) is
Element
of
K27
( the
carrier
of (
0
,
g
,
S0
))
the
carrier
of (
0
,
g
,
S0
) is non
empty
set
K27
( the
carrier
of (
0
,
g
,
S0
)) is
set
proj2
g
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
proj2
S0
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
(
proj2
g
)
\/
(
proj2
S0
)
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
proj2
{}
is
Relation-like
non-empty
empty-yielding
NAT
-defined
Function-like
one-to-one
constant
functional
empty
V12
() non
pair
V21
()
V22
()
V23
()
V25
()
V26
()
V27
()
V28
()
V29
()
V32
()
cardinal
{}
-element
V37
()
FinSequence-like
FinSubsequence-like
FinSequence-membered
non
with_pair
nonpair-yielding
ext-real
non
positive
non
negative
V115
()
V126
()
set
g
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
g
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
S0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
A0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
(
(
g
+
1
)
,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
(
g
+
1
)
,
S0
,
A0
) is
Element
of
K27
( the
carrier
of (
(
g
+
1
)
,
S0
,
A0
))
the
carrier
of (
(
g
+
1
)
,
S0
,
A0
) is non
empty
set
K27
( the
carrier
of (
(
g
+
1
)
,
S0
,
A0
)) is
set
proj2
S0
is
V28
() non
with_pair
set
proj2
A0
is
V28
() non
with_pair
set
(
proj2
S0
)
\/
(
proj2
A0
)
is
V28
() non
with_pair
set
h0
is
Relation-like
NAT
-defined
Function-like
V28
()
g
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
N
is non
pair
set
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h0
^
<*
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
set
h
is
Relation-like
NAT
-defined
Function-like
V28
()
g
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
Sn
is non
pair
set
<*
Sn
*>
is
Relation-like
NAT
-defined
Function-like
constant
non
empty
V12
()
V28
() 1
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
h
^
<*
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
()
g
+
1
-element
FinSequence-like
FinSubsequence-like
set
dom
<*
N
*>
is non
empty
V12
()
V28
() 1
-element
Element
of
K27
(
NAT
)
len
h0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
S0
.
(
g
+
1
)
is non
pair
set
<*
N
*>
.
1 is non
pair
set
dom
<*
Sn
*>
is non
empty
V12
()
V28
() 1
-element
Element
of
K27
(
NAT
)
len
h
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
A0
.
(
g
+
1
)
is non
pair
set
<*
Sn
*>
.
1 is non
pair
set
(
g
,
S0
,
A0
) is non
empty
pair
Element
of
InnerVertices
(
g
,
S0
,
A0
)
(
g
,
S0
,
A0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
g
,
S0
,
A0
) is non
empty
set
InnerVertices
(
g
,
S0
,
A0
) is non
empty
Element
of
K27
( the
carrier
of (
g
,
S0
,
A0
))
K27
( the
carrier
of (
g
,
S0
,
A0
)) is
set
{
N
,
Sn
,
H
1
(
g
)
}
is
V28
()
set
{
H
1
(
g
),
N
,
Sn
}
is
V28
()
set
proj2
h0
is
V28
() non
with_pair
set
proj2
<*
N
*>
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h0
)
\/
(
proj2
<*
N
*>
)
is non
empty
V28
() non
with_pair
set
{
N
}
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h0
)
\/
{
N
}
is non
empty
V28
() non
with_pair
set
proj2
h
is
V28
() non
with_pair
set
proj2
<*
Sn
*>
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h
)
\/
(
proj2
<*
Sn
*>
)
is non
empty
V28
() non
with_pair
set
{
Sn
}
is non
empty
V12
()
V28
() 1
-element
non
with_pair
set
(
proj2
h
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
<*
N
,
Sn
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
[
<*
N
,
Sn
*>
,
and2c
]
is non
empty
pair
set
[
<*
N
,
Sn
*>
,
xor2c
]
is non
empty
pair
set
h0
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
h
^
{}
is
Relation-like
NAT
-defined
Function-like
V28
()
FinSequence-like
FinSubsequence-like
set
(
g
,
h0
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
(
g
,
h0
,
h
) is
Element
of
K27
( the
carrier
of (
g
,
h0
,
h
))
the
carrier
of (
g
,
h0
,
h
) is non
empty
set
K27
( the
carrier
of (
g
,
h0
,
h
)) is
set
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
N
,
Sn
,(
g
,
S0
,
A0
),
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
Sn
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
[
<*
N
,
Sn
*>
,
xor2c
]
,(
g
,
S0
,
A0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
xor2c
]
,(
g
,
S0
,
A0
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
Sn
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
xor2c
]
,(
g
,
S0
,
A0
)
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
N
,
Sn
,(
g
,
S0
,
A0
)) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
N
,
Sn
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
Sn
,(
g
,
S0
,
A0
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
N
,
Sn
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
,
S0
,
A0
),
N
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
N
,
Sn
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
]
is non
empty
pair
set
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
N
,
Sn
*>
,
and2c
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
and2c
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
+*
(
1GateCircStr
(
<*
[
<*
N
,
Sn
*>
,
and2c
]
,
[
<*
Sn
,(
g
,
S0
,
A0
)
*>
,
and2a
]
,
[
<*
(
g
,
S0
,
A0
),
N
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
+*
(
GFA1CarryStr
(
N
,
Sn
,(
g
,
S0
,
A0
))
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
is non
empty
set
InputVertices
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
is
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
)
K27
( the
carrier
of
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
) is
set
{
H
1
(
g
)
}
is
Relation-like
non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
is
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
)
(
InputVertices
(
g
,
h0
,
h
)
)
\/
(
(
InputVertices
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
)
is
set
(
proj2
h0
)
\/
(
proj2
h
)
is
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
(
InputVertices
(
BitGFA1Str
(
N
,
Sn
,
H
1
(
g
))
)
)
\
{
H
1
(
g
)
}
)
is
set
{
N
,
Sn
,
H
1
(
g
)
}
\
{
H
1
(
g
)
}
is
V28
()
Element
of
K27
(
{
N
,
Sn
,
H
1
(
g
)
}
)
K27
(
{
N
,
Sn
,
H
1
(
g
)
}
) is
V28
()
V32
()
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
{
N
,
Sn
,
H
1
(
g
)
}
\
{
H
1
(
g
)
}
)
is
V28
()
set
{
N
,
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
,
Sn
}
is non
empty
V28
() non
with_pair
set
{
N
}
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
(
{
N
}
\/
{
Sn
}
)
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
}
is non
empty
V28
() non
with_pair
set
(
(
(
proj2
h0
)
\/
(
proj2
h
)
)
\/
{
N
}
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
(
(
proj2
h0
)
\/
{
N
}
)
\/
(
proj2
h
)
is non
empty
V28
() non
with_pair
set
(
(
(
proj2
h0
)
\/
{
N
}
)
\/
(
proj2
h
)
)
\/
{
Sn
}
is non
empty
V28
() non
with_pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
2
*
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
1
+
(
2
*
n
)
is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
g
is
Relation-like
NAT
-defined
Function-like
V28
()
n
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
n
,
f
,
g
) is
strict
non-empty
finitely-generated
V95
((
n
,
f
,
g
))
gate`2=den
Boolean
MSAlgebra
over (
n
,
f
,
g
)
the
Sorts
of (
n
,
f
,
g
) is
Relation-like
the
carrier
of (
n
,
f
,
g
)
-defined
Function-like
V14
( the
carrier
of (
n
,
f
,
g
))
set
the
carrier
of (
n
,
f
,
g
) is non
empty
set
K106
( the
Sorts
of (
n
,
f
,
g
)) is
set
N
is
Relation-like
Function-like
V18
(
NAT
,
NAT
)
Element
of
K27
(
K28
(
NAT
,
NAT
))
N
.
0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
N
.
1 is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
N
.
2 is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
Sn
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
Sn
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
Sn
+
1
)
is non
pair
set
g
.
(
Sn
+
1
)
is non
pair
set
h
is
set
BitGFA1Circ
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
BitGFA1Str
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
is non
empty
pair
set
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
2GatesCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
,
xor2c
)
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
xor2c
]
,
h
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
GFA1CarryICirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
1GateCircuit
(
h
,
(
f
.
(
Sn
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
Sn
+
1
)
)
,
h
,
and2a
)
)
)
+*
(
1GateCircuit
(
h
,
(
f
.
(
Sn
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
Sn
+
1
)
)
,
h
*>
,
and2a
]
,
[
<*
h
,
(
f
.
(
Sn
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
Sn
+
1
)
)
,
(
g
.
(
Sn
+
1
)
)
,
h
)
)
the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is
Relation-like
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
-defined
Function-like
V14
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
set
the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is non
empty
set
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
) is
set
h
is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
Following
(
h
,1) is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
Following
h
is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
Following
(
h
,
H
4
(
0
)) is
Element
of
K106
( the
Sorts
of
(
1GateCircuit
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
h
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
o0
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
o0
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
o0
+
1
)
is non
pair
set
g
.
(
o0
+
1
)
is non
pair
set
h
.
o0
is
set
f1
is
set
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
is non
empty
pair
set
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
BitGFA1Circ
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
2GatesCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
,
xor2c
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
xor2c
]
,
f1
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
1GateCircuit
(
f1
,
(
f
.
(
o0
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
o0
+
1
)
)
,
f1
,
and2a
)
)
)
+*
(
1GateCircuit
(
f1
,
(
f
.
(
o0
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
o0
+
1
)
)
,
f1
*>
,
and2a
]
,
[
<*
f1
,
(
f
.
(
o0
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
g1
is
non-empty
finitely-generated
V95
(
H
1
(
f1
,
o0
))
MSAlgebra
over
H
1
(
f1
,
o0
)
the
Sorts
of
g1
is
Relation-like
the
carrier
of
H
1
(
f1
,
o0
)
-defined
Function-like
V14
( the
carrier
of
H
1
(
f1
,
o0
))
set
the
carrier
of
H
1
(
f1
,
o0
) is non
empty
set
K106
( the
Sorts
of
g1
) is
set
(
o0
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
o0
,
f
,
g
)
(
o0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
o0
,
f
,
g
) is non
empty
set
InnerVertices
(
o0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
o0
,
f
,
g
))
K27
( the
carrier
of (
o0
,
f
,
g
)) is
set
h1
is
Element
of
K106
( the
Sorts
of
g1
)
Following
(
h1
,
(
N
.
1
)
) is
Element
of
K106
( the
Sorts
of
g1
)
the
Sorts
of
g1
is
Relation-like
the
carrier
of
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
-defined
Function-like
V14
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
)
set
the
carrier
of
(
BitGFA1Str
(
(
f
.
(
o0
+
1
)
)
,
(
g
.
(
o0
+
1
)
)
,
f1
)
)
is non
empty
set
K106
( the
Sorts
of
g1
) is
set
(
0
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
0
,
f
,
g
)
(
0
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
0
,
f
,
g
) is non
empty
set
InnerVertices
(
0
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
0
,
f
,
g
))
K27
( the
carrier
of (
0
,
f
,
g
)) is
set
f1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
f1
.
n
is
set
g1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
g1
.
n
is
set
f1
.
0
is
set
g1
.
0
is
set
h1
is
Relation-like
NAT
-defined
Function-like
V14
(
NAT
)
set
h1
.
0
is
set
f1
is
set
h1
.
f1
is
set
f2
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
(
f2
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
f2
,
f
,
g
)
(
f2
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f2
,
f
,
g
) is non
empty
set
InnerVertices
(
f2
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
f2
,
f
,
g
))
K27
( the
carrier
of (
f2
,
f
,
g
)) is
set
h
.
f1
is
set
h
.
0
is
set
f1
.
H
4
(2) is
set
g1
.
H
4
(2) is
set
f2
is non
empty
V56
()
ManySortedSign
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is non
pair
set
g
.
(
f1
+
1
)
is non
pair
set
f0
is
set
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
f1
.
f1
is
set
f3
is
non-empty
MSAlgebra
over
f2
g1
.
f1
is
set
h
.
f1
is
set
n
is
non-empty
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
BitGFA1Circ
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
GFA1AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
2GatesCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
,
xor2c
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
xor2c
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
,
xor2c
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
)
GFA1CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
GFA1CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
strict
non-empty
finitely-generated
V95
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
))
gate`2=den
Boolean
MSAlgebra
over
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
1GateCircuit
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2a
)
)
is
strict
non-empty
finitely-generated
V95
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
1GateCircuit
(
f0
,
(
f
.
(
f1
+
1
)
)
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
1GateCircuit
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
(
(
1GateCircuit
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
and2c
)
)
+*
(
1GateCircuit
(
(
g
.
(
f1
+
1
)
)
,
f0
,
and2a
)
)
)
+*
(
1GateCircuit
(
f0
,
(
f
.
(
f1
+
1
)
)
,
and2
)
)
is
strict
non-empty
finitely-generated
V95
(
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
1GateCircuit
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is
strict
non-empty
finitely-generated
V95
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
))
gate`2=den
Boolean
MSAlgebra
over
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
(
GFA1CarryICirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircuit
(
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
,
or3
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
(
GFA1AdderCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryCirc
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is
strict
non-empty
finitely-generated
V95
(
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
)
gate`2=den
Boolean
MSAlgebra
over
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
f1
.
(
f1
+
1
)
is
set
f2
+*
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
ManySortedSign
g1
.
(
f1
+
1
)
is
set
f3
+*
n
is
strict
non-empty
MSAlgebra
over
f2
+*
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
h
.
(
f1
+
1
)
is
set
GFA1CarryOutput
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f0
)
)
) is
set
[
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
InnerVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
) is
set
InputVertices
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
is
Element
of
K27
( the
carrier
of
(
1GateCircStr
(
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
)
)
)
{
[
{}
,
(
(
0
-tuples_on
BOOLEAN
)
-->
TRUE
)
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
f1
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
f1
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f1
+
1
)
is non
pair
set
g
.
(
f1
+
1
)
is non
pair
set
f2
is
set
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f2
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
xor2c
]
,
f2
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
]
is non
empty
pair
set
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f1
+
1
)
)
,
f2
*>
,
and2a
]
,
[
<*
f2
,
(
f
.
(
f1
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InnerVertices
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
)
the
carrier
of
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
is non
empty
set
K27
( the
carrier
of
(
BitGFA1Str
(
(
f
.
(
f1
+
1
)
)
,
(
g
.
(
f1
+
1
)
)
,
f2
)
)
) is
set
f3
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
.
f3
is
set
f3
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
f
.
(
f3
+
1
)
is non
pair
set
g
.
(
f3
+
1
)
is non
pair
set
f0
is
set
BitGFA1Str
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
]
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
xor2c
]
,
f0
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
]
is non
empty
pair
set
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
f3
+
1
)
)
,
f0
*>
,
and2a
]
,
[
<*
f0
,
(
f
.
(
f3
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
H
1
(
f0
,
f3
) is non
empty
set
InputVertices
H
1
(
f0
,
f3
) is
Element
of
K27
( the
carrier
of
H
1
(
f0
,
f3
))
K27
( the
carrier
of
H
1
(
f0
,
f3
)) is
set
{
f0
}
is non
empty
V12
()
V28
() 1
-element
set
(
InputVertices
H
1
(
f0
,
f3
)
)
\
{
f0
}
is
Element
of
K27
( the
carrier
of
H
1
(
f0
,
f3
))
(
f3
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
f3
,
f
,
g
)
(
f3
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
f3
,
f
,
g
) is non
empty
set
InnerVertices
(
f3
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
f3
,
f
,
g
))
K27
( the
carrier
of (
f3
,
f
,
g
)) is
set
{
(
f
.
(
f3
+
1
)
)
,
(
g
.
(
f3
+
1
)
)
,
f0
}
is
V28
()
set
n
is non
empty
pair
set
n
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
set
h
.
n
is
set
n
+
1 is non
empty
non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
positive
non
negative
V115
()
V126
()
Element
of
NAT
h
.
(
n
+
1
)
is
set
f
.
(
n
+
1
)
is non
pair
set
g
.
(
n
+
1
)
is non
pair
set
x
is
set
GFA1CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is
Element
of
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
nonpair-yielding
set
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
(
g
.
(
n
+
1
)
)
,
x
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
<*
x
,
(
f
.
(
n
+
1
)
)
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
)
)
+*
(
1GateCircStr
(
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
)
)
)
+*
(
1GateCircStr
(
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
is non
empty
pair
set
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
is non
empty
pair
set
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 3
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1CarryIStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
set
InnerVertices
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
)
K27
( the
carrier
of
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
) is
set
[
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
*>
,
or3
]
is non
empty
pair
set
BitGFA1Str
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
is non
empty
pair
set
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
x
*>
is
Relation-like
NAT
-defined
Function-like
non
empty
V28
() 2
-element
FinSequence-like
FinSubsequence-like
set
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
x
*>
,
xor2c
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
1GateCircStr
(
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
)
)
+*
(
1GateCircStr
(
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
x
*>
,
xor2c
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
+*
(
GFA1CarryStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
InputVertices
H
1
(
x
,
n
) is
Element
of
K27
( the
carrier
of
H
1
(
x
,
n
))
the
carrier
of
H
1
(
x
,
n
) is non
empty
set
K27
( the
carrier
of
H
1
(
x
,
n
)) is
set
InnerVertices
H
1
(
x
,
n
) is non
empty
Element
of
K27
( the
carrier
of
H
1
(
x
,
n
))
(
n
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
n
,
f
,
g
)
(
n
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
n
,
f
,
g
) is non
empty
set
InnerVertices
(
n
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
n
,
f
,
g
))
K27
( the
carrier
of (
n
,
f
,
g
)) is
set
(
(
n
+
1
)
,
f
,
g
) is non
empty
pair
Element
of
InnerVertices
(
(
n
+
1
)
,
f
,
g
)
(
(
n
+
1
)
,
f
,
g
) is non
empty
non
void
V56
()
strict
Circuit-like
unsplit
gate`1=arity
gate`2isBoolean
gate`2=den
ManySortedSign
the
carrier
of (
(
n
+
1
)
,
f
,
g
) is non
empty
set
InnerVertices
(
(
n
+
1
)
,
f
,
g
) is non
empty
Element
of
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
))
K27
( the
carrier
of (
(
n
+
1
)
,
f
,
g
)) is
set
{
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
}
is
V28
()
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
}
is
Relation-like
Function-like
constant
non
empty
V12
()
V28
() 1
-element
set
GFA1AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
) is
Element
of
InnerVertices
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
the
carrier
of
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
set
InnerVertices
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
)
K27
( the
carrier
of
(
GFA1AdderStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
) is
set
2GatesCircOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
) is non
empty
pair
Element
of
InnerVertices
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
)
)
the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
)
)
is non
empty
set
InnerVertices
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
)
)
is non
empty
Element
of
K27
( the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
)
)
)
K27
( the
carrier
of
(
2GatesCircStr
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
,
xor2c
)
)
) is
set
[
<*
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
,
x
*>
,
xor2c
]
is non
empty
pair
set
{
(
GFA1AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V28
()
set
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
is
Relation-like
V28
()
set
(
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
)
\/
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
is non
empty
V28
()
set
{
(
GFA1CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V12
()
V28
() 1
-element
set
(
(
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
xor2c
]
}
\/
{
(
GFA1AdderOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
)
\/
{
[
<*
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
*>
,
and2c
]
,
[
<*
(
g
.
(
n
+
1
)
)
,
x
*>
,
and2a
]
,
[
<*
x
,
(
f
.
(
n
+
1
)
)
*>
,
and2
]
}
)
\/
{
(
GFA1CarryOutput
(
(
f
.
(
n
+
1
)
)
,
(
g
.
(
n
+
1
)
)
,
x
)
)
}
is non
empty
V28
()
set
{
H
3
(
x
,
n
)
}
is non
empty
V12
()
V28
() 1
-element
set
H
4
(2)
*
H
4
(1) is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
H
4
(
0
)
+
(
H
4
(2)
*
H
4
(1)
)
is non
pair
V21
()
V22
()
V23
()
V27
()
V28
()
cardinal
ext-real
non
negative
V115
()
V126
()
Element
of
NAT
f1
is
Element
of
K106
( the
Sorts
of (
n
,
f
,
g
))
Following
(
f1
,
(
1
+
(
2
*
n
)
)
) is
Element
of
K106
( the
Sorts
of (
n
,
f
,
g
))