:: GFACIRC1 semantic presentation
:: deftheorem Def1 defines inv1 GFACIRC1:def 1 :
theorem Th1: :: GFACIRC1:1
:: deftheorem Def2 defines buf1 GFACIRC1:def 2 :
theorem Th2: :: GFACIRC1:2
definition
func and2c -> Function of 2
-tuples_on BOOLEAN ,
BOOLEAN means :
Def3:
:: GFACIRC1:def 3
for
b1,
b2 being
Element of
BOOLEAN holds
a1 . <*b1,b2*> = b1 '&' ('not' b2);
existence
ex b1 being Function of 2 -tuples_on BOOLEAN , BOOLEAN st
for b2, b3 being Element of BOOLEAN holds b1 . <*b2,b3*> = b2 '&' ('not' b3)
uniqueness
for b1, b2 being Function of 2 -tuples_on BOOLEAN , BOOLEAN st ( for b3, b4 being Element of BOOLEAN holds b1 . <*b3,b4*> = b3 '&' ('not' b4) ) & ( for b3, b4 being Element of BOOLEAN holds b2 . <*b3,b4*> = b3 '&' ('not' b4) ) holds
b1 = b2
end;
:: deftheorem Def3 defines and2c GFACIRC1:def 3 :
theorem Th3: :: GFACIRC1:3
for
b1,
b2 being
Element of
BOOLEAN holds
(
and2c . <*b1,b2*> = b1 '&' ('not' b2) &
and2c . <*b1,b2*> = and2a . <*b2,b1*> &
and2c . <*b1,b2*> = nor2a . <*b1,b2*> &
and2c . <*0,0*> = 0 &
and2c . <*0,1*> = 0 &
and2c . <*1,0*> = 1 &
and2c . <*1,1*> = 0 )
definition
func xor2c -> Function of 2
-tuples_on BOOLEAN ,
BOOLEAN means :
Def4:
:: GFACIRC1:def 4
for
b1,
b2 being
Element of
BOOLEAN holds
a1 . <*b1,b2*> = b1 'xor' ('not' b2);
existence
ex b1 being Function of 2 -tuples_on BOOLEAN , BOOLEAN st
for b2, b3 being Element of BOOLEAN holds b1 . <*b2,b3*> = b2 'xor' ('not' b3)
uniqueness
for b1, b2 being Function of 2 -tuples_on BOOLEAN , BOOLEAN st ( for b3, b4 being Element of BOOLEAN holds b1 . <*b3,b4*> = b3 'xor' ('not' b4) ) & ( for b3, b4 being Element of BOOLEAN holds b2 . <*b3,b4*> = b3 'xor' ('not' b4) ) holds
b1 = b2
end;
:: deftheorem Def4 defines xor2c GFACIRC1:def 4 :
theorem Th4: :: GFACIRC1:4
for
b1,
b2 being
Element of
BOOLEAN holds
(
xor2c . <*b1,b2*> = b1 'xor' ('not' b2) &
xor2c . <*b1,b2*> = xor2a . <*b1,b2*> &
xor2c . <*b1,b2*> = or2 . <*(and2b . <*b1,b2*>),(and2 . <*b1,b2*>)*> &
xor2c . <*0,0*> = 1 &
xor2c . <*0,1*> = 0 &
xor2c . <*1,0*> = 0 &
xor2c . <*1,1*> = 1 )
theorem Th5: :: GFACIRC1:5
theorem Th6: :: GFACIRC1:6
theorem Th7: :: GFACIRC1:7
theorem Th8: :: GFACIRC1:8
theorem Th9: :: GFACIRC1:9
theorem Th10: :: GFACIRC1:10
theorem Th11: :: GFACIRC1:11
theorem Th12: :: GFACIRC1:12
Lemma10:
for b1, b2, b3 being Function of 2 -tuples_on BOOLEAN , BOOLEAN
for b4, b5, b6 being set st b4 <> [<*b5,b6*>,b2] & b5 <> [<*b6,b4*>,b3] & b6 <> [<*b4,b5*>,b1] holds
( not [<*b4,b5*>,b1] in {b5,b6} & not b6 in {[<*b4,b5*>,b1],[<*b5,b6*>,b2]} & not b4 in {[<*b4,b5*>,b1],[<*b5,b6*>,b2]} & not [<*b6,b4*>,b3] in {b4,b5,b6} )
Lemma11:
for b1, b2, b3 being Function of 2 -tuples_on BOOLEAN , BOOLEAN
for b4 being Function of 3 -tuples_on BOOLEAN , BOOLEAN
for b5, b6, b7 being set holds {b5,b6,b7} \ {[<*[<*b5,b6*>,b1],[<*b6,b7*>,b2],[<*b7,b5*>,b3]*>,b4]} = {b5,b6,b7}
Lemma12:
for b1 being Function of 2 -tuples_on BOOLEAN , BOOLEAN
for b2, b3, b4 being set st b4 <> [<*b2,b3*>,b1] holds
for b5 being State of (2GatesCircuit b2,b3,b4,b1) holds
( (Following b5) . (2GatesCircOutput b2,b3,b4,b1) = b1 . <*(b5 . [<*b2,b3*>,b1]),(b5 . b4)*> & (Following b5) . [<*b2,b3*>,b1] = b1 . <*(b5 . b2),(b5 . b3)*> & (Following b5) . b2 = b5 . b2 & (Following b5) . b3 = b5 . b3 & (Following b5) . b4 = b5 . b4 )
definition
let c1,
c2,
c3 be
set ;
func GFA0CarryIStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 5
((1GateCircStr <*a1,a2*>,and2 ) +* (1GateCircStr <*a2,a3*>,and2 )) +* (1GateCircStr <*a3,a1*>,and2 );
coherence
((1GateCircStr <*c1,c2*>,and2 ) +* (1GateCircStr <*c2,c3*>,and2 )) +* (1GateCircStr <*c3,c1*>,and2 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def5 defines GFA0CarryIStr GFACIRC1:def 5 :
for
b1,
b2,
b3 being
set holds
GFA0CarryIStr b1,
b2,
b3 = ((1GateCircStr <*b1,b2*>,and2 ) +* (1GateCircStr <*b2,b3*>,and2 )) +* (1GateCircStr <*b3,b1*>,and2 );
definition
let c1,
c2,
c3 be
set ;
func GFA0CarryICirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA0CarryIStr a1,
a2,
a3 equals :: GFACIRC1:def 6
((1GateCircuit a1,a2,and2 ) +* (1GateCircuit a2,a3,and2 )) +* (1GateCircuit a3,a1,and2 );
coherence
((1GateCircuit c1,c2,and2 ) +* (1GateCircuit c2,c3,and2 )) +* (1GateCircuit c3,c1,and2 ) is strict gate`2=den Boolean Circuit of GFA0CarryIStr c1,c2,c3
;
end;
:: deftheorem Def6 defines GFA0CarryICirc GFACIRC1:def 6 :
for
b1,
b2,
b3 being
set holds
GFA0CarryICirc b1,
b2,
b3 = ((1GateCircuit b1,b2,and2 ) +* (1GateCircuit b2,b3,and2 )) +* (1GateCircuit b3,b1,and2 );
definition
let c1,
c2,
c3 be
set ;
func GFA0CarryStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 7
(GFA0CarryIStr a1,a2,a3) +* (1GateCircStr <*[<*a1,a2*>,and2 ],[<*a2,a3*>,and2 ],[<*a3,a1*>,and2 ]*>,or3 );
coherence
(GFA0CarryIStr c1,c2,c3) +* (1GateCircStr <*[<*c1,c2*>,and2 ],[<*c2,c3*>,and2 ],[<*c3,c1*>,and2 ]*>,or3 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def7 defines GFA0CarryStr GFACIRC1:def 7 :
for
b1,
b2,
b3 being
set holds
GFA0CarryStr b1,
b2,
b3 = (GFA0CarryIStr b1,b2,b3) +* (1GateCircStr <*[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]*>,or3 );
definition
let c1,
c2,
c3 be
set ;
func GFA0CarryCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA0CarryStr a1,
a2,
a3 equals :: GFACIRC1:def 8
(GFA0CarryICirc a1,a2,a3) +* (1GateCircuit [<*a1,a2*>,and2 ],[<*a2,a3*>,and2 ],[<*a3,a1*>,and2 ],or3 );
coherence
(GFA0CarryICirc c1,c2,c3) +* (1GateCircuit [<*c1,c2*>,and2 ],[<*c2,c3*>,and2 ],[<*c3,c1*>,and2 ],or3 ) is strict gate`2=den Boolean Circuit of GFA0CarryStr c1,c2,c3
;
end;
:: deftheorem Def8 defines GFA0CarryCirc GFACIRC1:def 8 :
for
b1,
b2,
b3 being
set holds
GFA0CarryCirc b1,
b2,
b3 = (GFA0CarryICirc b1,b2,b3) +* (1GateCircuit [<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ],or3 );
definition
let c1,
c2,
c3 be
set ;
func GFA0CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA0CarryStr a1,a2,a3) equals :: GFACIRC1:def 9
[<*[<*a1,a2*>,and2 ],[<*a2,a3*>,and2 ],[<*a3,a1*>,and2 ]*>,or3 ];
coherence
[<*[<*c1,c2*>,and2 ],[<*c2,c3*>,and2 ],[<*c3,c1*>,and2 ]*>,or3 ] is Element of InnerVertices (GFA0CarryStr c1,c2,c3)
end;
:: deftheorem Def9 defines GFA0CarryOutput GFACIRC1:def 9 :
for
b1,
b2,
b3 being
set holds
GFA0CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]*>,or3 ];
theorem Th13: :: GFACIRC1:13
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA0CarryIStr b1,b2,b3) = {[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]}
theorem Th14: :: GFACIRC1:14
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA0CarryStr b1,b2,b3) = {[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]} \/ {(GFA0CarryOutput b1,b2,b3)}
theorem Th15: :: GFACIRC1:15
theorem Th16: :: GFACIRC1:16
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
InputVertices (GFA0CarryIStr b1,b2,b3) = {b1,b2,b3}
theorem Th17: :: GFACIRC1:17
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
InputVertices (GFA0CarryStr b1,b2,b3) = {b1,b2,b3}
theorem Th18: :: GFACIRC1:18
theorem Th19: :: GFACIRC1:19
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA0CarryStr b1,b2,b3) &
b2 in the
carrier of
(GFA0CarryStr b1,b2,b3) &
b3 in the
carrier of
(GFA0CarryStr b1,b2,b3) &
[<*b1,b2*>,and2 ] in the
carrier of
(GFA0CarryStr b1,b2,b3) &
[<*b2,b3*>,and2 ] in the
carrier of
(GFA0CarryStr b1,b2,b3) &
[<*b3,b1*>,and2 ] in the
carrier of
(GFA0CarryStr b1,b2,b3) &
[<*[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]*>,or3 ] in the
carrier of
(GFA0CarryStr b1,b2,b3) )
theorem Th20: :: GFACIRC1:20
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,and2 ] in InnerVertices (GFA0CarryStr b1,b2,b3) &
[<*b2,b3*>,and2 ] in InnerVertices (GFA0CarryStr b1,b2,b3) &
[<*b3,b1*>,and2 ] in InnerVertices (GFA0CarryStr b1,b2,b3) &
GFA0CarryOutput b1,
b2,
b3 in InnerVertices (GFA0CarryStr b1,b2,b3) )
theorem Th21: :: GFACIRC1:21
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
(
b1 in InputVertices (GFA0CarryStr b1,b2,b3) &
b2 in InputVertices (GFA0CarryStr b1,b2,b3) &
b3 in InputVertices (GFA0CarryStr b1,b2,b3) )
theorem Th22: :: GFACIRC1:22
theorem Th23: :: GFACIRC1:23
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA0CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,and2 ] = b5 '&' b6 &
(Following b4) . [<*b2,b3*>,and2 ] = b6 '&' b7 &
(Following b4) . [<*b3,b1*>,and2 ] = b7 '&' b5 )
theorem Th24: :: GFACIRC1:24
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA0CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,and2 ] &
b6 = b4 . [<*b2,b3*>,and2 ] &
b7 = b4 . [<*b3,b1*>,and2 ] holds
(Following b4) . (GFA0CarryOutput b1,b2,b3) = (b5 'or' b6) 'or' b7
theorem Th25: :: GFACIRC1:25
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
for
b4 being
State of
(GFA0CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA0CarryOutput b1,b2,b3) = ((b5 '&' b6) 'or' (b6 '&' b7)) 'or' (b7 '&' b5) &
(Following b4,2) . [<*b1,b2*>,and2 ] = b5 '&' b6 &
(Following b4,2) . [<*b2,b3*>,and2 ] = b6 '&' b7 &
(Following b4,2) . [<*b3,b1*>,and2 ] = b7 '&' b5 )
theorem Th26: :: GFACIRC1:26
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
for
b4 being
State of
(GFA0CarryCirc b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA0AdderStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 10
2GatesCircStr a1,
a2,
a3,
xor2 ;
coherence
2GatesCircStr c1,c2,c3,xor2 is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def10 defines GFA0AdderStr GFACIRC1:def 10 :
definition
let c1,
c2,
c3 be
set ;
func GFA0AdderCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA0AdderStr a1,
a2,
a3 equals :: GFACIRC1:def 11
2GatesCircuit a1,
a2,
a3,
xor2 ;
coherence
2GatesCircuit c1,c2,c3,xor2 is strict gate`2=den Boolean Circuit of GFA0AdderStr c1,c2,c3
;
end;
:: deftheorem Def11 defines GFA0AdderCirc GFACIRC1:def 11 :
definition
let c1,
c2,
c3 be
set ;
func GFA0AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA0AdderStr a1,a2,a3) equals :: GFACIRC1:def 12
2GatesCircOutput a1,
a2,
a3,
xor2 ;
coherence
2GatesCircOutput c1,c2,c3,xor2 is Element of InnerVertices (GFA0AdderStr c1,c2,c3)
;
end;
:: deftheorem Def12 defines GFA0AdderOutput GFACIRC1:def 12 :
theorem Th27: :: GFACIRC1:27
theorem Th28: :: GFACIRC1:28
theorem Th29: :: GFACIRC1:29
theorem Th30: :: GFACIRC1:30
theorem Th31: :: GFACIRC1:31
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA0AdderStr b1,b2,b3) &
b2 in the
carrier of
(GFA0AdderStr b1,b2,b3) &
b3 in the
carrier of
(GFA0AdderStr b1,b2,b3) &
[<*b1,b2*>,xor2 ] in the
carrier of
(GFA0AdderStr b1,b2,b3) &
[<*[<*b1,b2*>,xor2 ],b3*>,xor2 ] in the
carrier of
(GFA0AdderStr b1,b2,b3) )
by FACIRC_1:60, FACIRC_1:61;
theorem Th32: :: GFACIRC1:32
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2 ] in InnerVertices (GFA0AdderStr b1,b2,b3) &
GFA0AdderOutput b1,
b2,
b3 in InnerVertices (GFA0AdderStr b1,b2,b3) )
theorem Th33: :: GFACIRC1:33
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
(
b1 in InputVertices (GFA0AdderStr b1,b2,b3) &
b2 in InputVertices (GFA0AdderStr b1,b2,b3) &
b3 in InputVertices (GFA0AdderStr b1,b2,b3) )
theorem Th34: :: GFACIRC1:34
theorem Th35: :: GFACIRC1:35
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA0AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,xor2 ] = b5 'xor' b6 &
(Following b4) . b1 = b5 &
(Following b4) . b2 = b6 &
(Following b4) . b3 = b7 )
theorem Th36: :: GFACIRC1:36
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA0AdderCirc b1,b2,b3)for
b5,
b6,
b7,
b8 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,xor2 ] &
b6 = b4 . b1 &
b7 = b4 . b2 &
b8 = b4 . b3 holds
(Following b4) . (GFA0AdderOutput b1,b2,b3) = b5 'xor' b8
theorem Th37: :: GFACIRC1:37
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA0AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA0AdderOutput b1,b2,b3) = (b5 'xor' b6) 'xor' b7 &
(Following b4,2) . [<*b1,b2*>,xor2 ] = b5 'xor' b6 &
(Following b4,2) . b1 = b5 &
(Following b4,2) . b2 = b6 &
(Following b4,2) . b3 = b7 )
theorem Th38: :: GFACIRC1:38
definition
let c1,
c2,
c3 be
set ;
func BitGFA0Str c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 13
(GFA0AdderStr a1,a2,a3) +* (GFA0CarryStr a1,a2,a3);
coherence
(GFA0AdderStr c1,c2,c3) +* (GFA0CarryStr c1,c2,c3) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def13 defines BitGFA0Str GFACIRC1:def 13 :
definition
let c1,
c2,
c3 be
set ;
func BitGFA0Circ c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
BitGFA0Str a1,
a2,
a3 equals :: GFACIRC1:def 14
(GFA0AdderCirc a1,a2,a3) +* (GFA0CarryCirc a1,a2,a3);
coherence
(GFA0AdderCirc c1,c2,c3) +* (GFA0CarryCirc c1,c2,c3) is strict gate`2=den Boolean Circuit of BitGFA0Str c1,c2,c3
;
end;
:: deftheorem Def14 defines BitGFA0Circ GFACIRC1:def 14 :
theorem Th39: :: GFACIRC1:39
for
b1,
b2,
b3 being
set holds
InnerVertices (BitGFA0Str b1,b2,b3) = (({[<*b1,b2*>,xor2 ]} \/ {(GFA0AdderOutput b1,b2,b3)}) \/ {[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]}) \/ {(GFA0CarryOutput b1,b2,b3)}
theorem Th40: :: GFACIRC1:40
theorem Th41: :: GFACIRC1:41
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
InputVertices (BitGFA0Str b1,b2,b3) = {b1,b2,b3}
theorem Th42: :: GFACIRC1:42
theorem Th43: :: GFACIRC1:43
theorem Th44: :: GFACIRC1:44
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(BitGFA0Str b1,b2,b3) &
b2 in the
carrier of
(BitGFA0Str b1,b2,b3) &
b3 in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*b1,b2*>,xor2 ] in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*[<*b1,b2*>,xor2 ],b3*>,xor2 ] in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*b1,b2*>,and2 ] in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*b2,b3*>,and2 ] in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*b3,b1*>,and2 ] in the
carrier of
(BitGFA0Str b1,b2,b3) &
[<*[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]*>,or3 ] in the
carrier of
(BitGFA0Str b1,b2,b3) )
theorem Th45: :: GFACIRC1:45
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2 ] in InnerVertices (BitGFA0Str b1,b2,b3) &
GFA0AdderOutput b1,
b2,
b3 in InnerVertices (BitGFA0Str b1,b2,b3) &
[<*b1,b2*>,and2 ] in InnerVertices (BitGFA0Str b1,b2,b3) &
[<*b2,b3*>,and2 ] in InnerVertices (BitGFA0Str b1,b2,b3) &
[<*b3,b1*>,and2 ] in InnerVertices (BitGFA0Str b1,b2,b3) &
GFA0CarryOutput b1,
b2,
b3 in InnerVertices (BitGFA0Str b1,b2,b3) )
theorem Th46: :: GFACIRC1:46
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
(
b1 in InputVertices (BitGFA0Str b1,b2,b3) &
b2 in InputVertices (BitGFA0Str b1,b2,b3) &
b3 in InputVertices (BitGFA0Str b1,b2,b3) )
definition
let c1,
c2,
c3 be
set ;
func BitGFA0CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA0Str a1,a2,a3) equals :: GFACIRC1:def 15
[<*[<*a1,a2*>,and2 ],[<*a2,a3*>,and2 ],[<*a3,a1*>,and2 ]*>,or3 ];
coherence
[<*[<*c1,c2*>,and2 ],[<*c2,c3*>,and2 ],[<*c3,c1*>,and2 ]*>,or3 ] is Element of InnerVertices (BitGFA0Str c1,c2,c3)
end;
:: deftheorem Def15 defines BitGFA0CarryOutput GFACIRC1:def 15 :
for
b1,
b2,
b3 being
set holds
BitGFA0CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2 ],[<*b2,b3*>,and2 ],[<*b3,b1*>,and2 ]*>,or3 ];
definition
let c1,
c2,
c3 be
set ;
func BitGFA0AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA0Str a1,a2,a3) equals :: GFACIRC1:def 16
2GatesCircOutput a1,
a2,
a3,
xor2 ;
coherence
2GatesCircOutput c1,c2,c3,xor2 is Element of InnerVertices (BitGFA0Str c1,c2,c3)
end;
:: deftheorem Def16 defines BitGFA0AdderOutput GFACIRC1:def 16 :
theorem Th47: :: GFACIRC1:47
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
for
b4 being
State of
(BitGFA0Circ b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA0AdderOutput b1,b2,b3) = (b5 'xor' b6) 'xor' b7 &
(Following b4,2) . (GFA0CarryOutput b1,b2,b3) = ((b5 '&' b6) 'or' (b6 '&' b7)) 'or' (b7 '&' b5) )
theorem Th48: :: GFACIRC1:48
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2 ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2 ] holds
for
b4 being
State of
(BitGFA0Circ b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA1CarryIStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 17
((1GateCircStr <*a1,a2*>,and2c ) +* (1GateCircStr <*a2,a3*>,and2a )) +* (1GateCircStr <*a3,a1*>,and2 );
coherence
((1GateCircStr <*c1,c2*>,and2c ) +* (1GateCircStr <*c2,c3*>,and2a )) +* (1GateCircStr <*c3,c1*>,and2 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def17 defines GFA1CarryIStr GFACIRC1:def 17 :
for
b1,
b2,
b3 being
set holds
GFA1CarryIStr b1,
b2,
b3 = ((1GateCircStr <*b1,b2*>,and2c ) +* (1GateCircStr <*b2,b3*>,and2a )) +* (1GateCircStr <*b3,b1*>,and2 );
definition
let c1,
c2,
c3 be
set ;
func GFA1CarryICirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA1CarryIStr a1,
a2,
a3 equals :: GFACIRC1:def 18
((1GateCircuit a1,a2,and2c ) +* (1GateCircuit a2,a3,and2a )) +* (1GateCircuit a3,a1,and2 );
coherence
((1GateCircuit c1,c2,and2c ) +* (1GateCircuit c2,c3,and2a )) +* (1GateCircuit c3,c1,and2 ) is strict gate`2=den Boolean Circuit of GFA1CarryIStr c1,c2,c3
;
end;
:: deftheorem Def18 defines GFA1CarryICirc GFACIRC1:def 18 :
for
b1,
b2,
b3 being
set holds
GFA1CarryICirc b1,
b2,
b3 = ((1GateCircuit b1,b2,and2c ) +* (1GateCircuit b2,b3,and2a )) +* (1GateCircuit b3,b1,and2 );
definition
let c1,
c2,
c3 be
set ;
func GFA1CarryStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 19
(GFA1CarryIStr a1,a2,a3) +* (1GateCircStr <*[<*a1,a2*>,and2c ],[<*a2,a3*>,and2a ],[<*a3,a1*>,and2 ]*>,or3 );
coherence
(GFA1CarryIStr c1,c2,c3) +* (1GateCircStr <*[<*c1,c2*>,and2c ],[<*c2,c3*>,and2a ],[<*c3,c1*>,and2 ]*>,or3 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def19 defines GFA1CarryStr GFACIRC1:def 19 :
for
b1,
b2,
b3 being
set holds
GFA1CarryStr b1,
b2,
b3 = (GFA1CarryIStr b1,b2,b3) +* (1GateCircStr <*[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]*>,or3 );
definition
let c1,
c2,
c3 be
set ;
func GFA1CarryCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA1CarryStr a1,
a2,
a3 equals :: GFACIRC1:def 20
(GFA1CarryICirc a1,a2,a3) +* (1GateCircuit [<*a1,a2*>,and2c ],[<*a2,a3*>,and2a ],[<*a3,a1*>,and2 ],or3 );
coherence
(GFA1CarryICirc c1,c2,c3) +* (1GateCircuit [<*c1,c2*>,and2c ],[<*c2,c3*>,and2a ],[<*c3,c1*>,and2 ],or3 ) is strict gate`2=den Boolean Circuit of GFA1CarryStr c1,c2,c3
;
end;
:: deftheorem Def20 defines GFA1CarryCirc GFACIRC1:def 20 :
for
b1,
b2,
b3 being
set holds
GFA1CarryCirc b1,
b2,
b3 = (GFA1CarryICirc b1,b2,b3) +* (1GateCircuit [<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ],or3 );
definition
let c1,
c2,
c3 be
set ;
func GFA1CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA1CarryStr a1,a2,a3) equals :: GFACIRC1:def 21
[<*[<*a1,a2*>,and2c ],[<*a2,a3*>,and2a ],[<*a3,a1*>,and2 ]*>,or3 ];
coherence
[<*[<*c1,c2*>,and2c ],[<*c2,c3*>,and2a ],[<*c3,c1*>,and2 ]*>,or3 ] is Element of InnerVertices (GFA1CarryStr c1,c2,c3)
end;
:: deftheorem Def21 defines GFA1CarryOutput GFACIRC1:def 21 :
for
b1,
b2,
b3 being
set holds
GFA1CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]*>,or3 ];
theorem Th49: :: GFACIRC1:49
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA1CarryIStr b1,b2,b3) = {[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]}
theorem Th50: :: GFACIRC1:50
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA1CarryStr b1,b2,b3) = {[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]} \/ {(GFA1CarryOutput b1,b2,b3)}
theorem Th51: :: GFACIRC1:51
theorem Th52: :: GFACIRC1:52
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
InputVertices (GFA1CarryIStr b1,b2,b3) = {b1,b2,b3}
theorem Th53: :: GFACIRC1:53
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
InputVertices (GFA1CarryStr b1,b2,b3) = {b1,b2,b3}
theorem Th54: :: GFACIRC1:54
theorem Th55: :: GFACIRC1:55
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA1CarryStr b1,b2,b3) &
b2 in the
carrier of
(GFA1CarryStr b1,b2,b3) &
b3 in the
carrier of
(GFA1CarryStr b1,b2,b3) &
[<*b1,b2*>,and2c ] in the
carrier of
(GFA1CarryStr b1,b2,b3) &
[<*b2,b3*>,and2a ] in the
carrier of
(GFA1CarryStr b1,b2,b3) &
[<*b3,b1*>,and2 ] in the
carrier of
(GFA1CarryStr b1,b2,b3) &
[<*[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]*>,or3 ] in the
carrier of
(GFA1CarryStr b1,b2,b3) )
theorem Th56: :: GFACIRC1:56
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,and2c ] in InnerVertices (GFA1CarryStr b1,b2,b3) &
[<*b2,b3*>,and2a ] in InnerVertices (GFA1CarryStr b1,b2,b3) &
[<*b3,b1*>,and2 ] in InnerVertices (GFA1CarryStr b1,b2,b3) &
GFA1CarryOutput b1,
b2,
b3 in InnerVertices (GFA1CarryStr b1,b2,b3) )
theorem Th57: :: GFACIRC1:57
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
(
b1 in InputVertices (GFA1CarryStr b1,b2,b3) &
b2 in InputVertices (GFA1CarryStr b1,b2,b3) &
b3 in InputVertices (GFA1CarryStr b1,b2,b3) )
theorem Th58: :: GFACIRC1:58
theorem Th59: :: GFACIRC1:59
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA1CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,and2c ] = b5 '&' ('not' b6) &
(Following b4) . [<*b2,b3*>,and2a ] = ('not' b6) '&' b7 &
(Following b4) . [<*b3,b1*>,and2 ] = b7 '&' b5 )
theorem Th60: :: GFACIRC1:60
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA1CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,and2c ] &
b6 = b4 . [<*b2,b3*>,and2a ] &
b7 = b4 . [<*b3,b1*>,and2 ] holds
(Following b4) . (GFA1CarryOutput b1,b2,b3) = (b5 'or' b6) 'or' b7
theorem Th61: :: GFACIRC1:61
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
for
b4 being
State of
(GFA1CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA1CarryOutput b1,b2,b3) = ((b5 '&' ('not' b6)) 'or' (('not' b6) '&' b7)) 'or' (b7 '&' b5) &
(Following b4,2) . [<*b1,b2*>,and2c ] = b5 '&' ('not' b6) &
(Following b4,2) . [<*b2,b3*>,and2a ] = ('not' b6) '&' b7 &
(Following b4,2) . [<*b3,b1*>,and2 ] = b7 '&' b5 )
theorem Th62: :: GFACIRC1:62
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
for
b4 being
State of
(GFA1CarryCirc b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA1AdderStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 22
2GatesCircStr a1,
a2,
a3,
xor2c ;
coherence
2GatesCircStr c1,c2,c3,xor2c is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def22 defines GFA1AdderStr GFACIRC1:def 22 :
definition
let c1,
c2,
c3 be
set ;
func GFA1AdderCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA1AdderStr a1,
a2,
a3 equals :: GFACIRC1:def 23
2GatesCircuit a1,
a2,
a3,
xor2c ;
coherence
2GatesCircuit c1,c2,c3,xor2c is strict gate`2=den Boolean Circuit of GFA1AdderStr c1,c2,c3
;
end;
:: deftheorem Def23 defines GFA1AdderCirc GFACIRC1:def 23 :
definition
let c1,
c2,
c3 be
set ;
func GFA1AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA1AdderStr a1,a2,a3) equals :: GFACIRC1:def 24
2GatesCircOutput a1,
a2,
a3,
xor2c ;
coherence
2GatesCircOutput c1,c2,c3,xor2c is Element of InnerVertices (GFA1AdderStr c1,c2,c3)
;
end;
:: deftheorem Def24 defines GFA1AdderOutput GFACIRC1:def 24 :
theorem Th63: :: GFACIRC1:63
theorem Th64: :: GFACIRC1:64
theorem Th65: :: GFACIRC1:65
theorem Th66: :: GFACIRC1:66
theorem Th67: :: GFACIRC1:67
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA1AdderStr b1,b2,b3) &
b2 in the
carrier of
(GFA1AdderStr b1,b2,b3) &
b3 in the
carrier of
(GFA1AdderStr b1,b2,b3) &
[<*b1,b2*>,xor2c ] in the
carrier of
(GFA1AdderStr b1,b2,b3) &
[<*[<*b1,b2*>,xor2c ],b3*>,xor2c ] in the
carrier of
(GFA1AdderStr b1,b2,b3) )
by FACIRC_1:60, FACIRC_1:61;
theorem Th68: :: GFACIRC1:68
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2c ] in InnerVertices (GFA1AdderStr b1,b2,b3) &
GFA1AdderOutput b1,
b2,
b3 in InnerVertices (GFA1AdderStr b1,b2,b3) )
theorem Th69: :: GFACIRC1:69
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
(
b1 in InputVertices (GFA1AdderStr b1,b2,b3) &
b2 in InputVertices (GFA1AdderStr b1,b2,b3) &
b3 in InputVertices (GFA1AdderStr b1,b2,b3) )
theorem Th70: :: GFACIRC1:70
theorem Th71: :: GFACIRC1:71
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA1AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,xor2c ] = b5 'xor' ('not' b6) &
(Following b4) . b1 = b5 &
(Following b4) . b2 = b6 &
(Following b4) . b3 = b7 )
theorem Th72: :: GFACIRC1:72
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA1AdderCirc b1,b2,b3)for
b5,
b6,
b7,
b8 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,xor2c ] &
b6 = b4 . b1 &
b7 = b4 . b2 &
b8 = b4 . b3 holds
(Following b4) . (GFA1AdderOutput b1,b2,b3) = b5 'xor' ('not' b8)
theorem Th73: :: GFACIRC1:73
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA1AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA1AdderOutput b1,b2,b3) = (b5 'xor' ('not' b6)) 'xor' ('not' b7) &
(Following b4,2) . [<*b1,b2*>,xor2c ] = b5 'xor' ('not' b6) &
(Following b4,2) . b1 = b5 &
(Following b4,2) . b2 = b6 &
(Following b4,2) . b3 = b7 )
theorem Th74: :: GFACIRC1:74
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA1AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(Following b4,2) . (GFA1AdderOutput b1,b2,b3) = 'not' ((b5 'xor' ('not' b6)) 'xor' b7)
theorem Th75: :: GFACIRC1:75
definition
let c1,
c2,
c3 be
set ;
func BitGFA1Str c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 25
(GFA1AdderStr a1,a2,a3) +* (GFA1CarryStr a1,a2,a3);
coherence
(GFA1AdderStr c1,c2,c3) +* (GFA1CarryStr c1,c2,c3) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def25 defines BitGFA1Str GFACIRC1:def 25 :
definition
let c1,
c2,
c3 be
set ;
func BitGFA1Circ c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
BitGFA1Str a1,
a2,
a3 equals :: GFACIRC1:def 26
(GFA1AdderCirc a1,a2,a3) +* (GFA1CarryCirc a1,a2,a3);
coherence
(GFA1AdderCirc c1,c2,c3) +* (GFA1CarryCirc c1,c2,c3) is strict gate`2=den Boolean Circuit of BitGFA1Str c1,c2,c3
;
end;
:: deftheorem Def26 defines BitGFA1Circ GFACIRC1:def 26 :
theorem Th76: :: GFACIRC1:76
for
b1,
b2,
b3 being
set holds
InnerVertices (BitGFA1Str b1,b2,b3) = (({[<*b1,b2*>,xor2c ]} \/ {(GFA1AdderOutput b1,b2,b3)}) \/ {[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]}) \/ {(GFA1CarryOutput b1,b2,b3)}
theorem Th77: :: GFACIRC1:77
theorem Th78: :: GFACIRC1:78
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
InputVertices (BitGFA1Str b1,b2,b3) = {b1,b2,b3}
theorem Th79: :: GFACIRC1:79
theorem Th80: :: GFACIRC1:80
theorem Th81: :: GFACIRC1:81
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(BitGFA1Str b1,b2,b3) &
b2 in the
carrier of
(BitGFA1Str b1,b2,b3) &
b3 in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*b1,b2*>,xor2c ] in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*[<*b1,b2*>,xor2c ],b3*>,xor2c ] in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*b1,b2*>,and2c ] in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*b2,b3*>,and2a ] in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*b3,b1*>,and2 ] in the
carrier of
(BitGFA1Str b1,b2,b3) &
[<*[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]*>,or3 ] in the
carrier of
(BitGFA1Str b1,b2,b3) )
theorem Th82: :: GFACIRC1:82
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2c ] in InnerVertices (BitGFA1Str b1,b2,b3) &
GFA1AdderOutput b1,
b2,
b3 in InnerVertices (BitGFA1Str b1,b2,b3) &
[<*b1,b2*>,and2c ] in InnerVertices (BitGFA1Str b1,b2,b3) &
[<*b2,b3*>,and2a ] in InnerVertices (BitGFA1Str b1,b2,b3) &
[<*b3,b1*>,and2 ] in InnerVertices (BitGFA1Str b1,b2,b3) &
GFA1CarryOutput b1,
b2,
b3 in InnerVertices (BitGFA1Str b1,b2,b3) )
theorem Th83: :: GFACIRC1:83
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
(
b1 in InputVertices (BitGFA1Str b1,b2,b3) &
b2 in InputVertices (BitGFA1Str b1,b2,b3) &
b3 in InputVertices (BitGFA1Str b1,b2,b3) )
definition
let c1,
c2,
c3 be
set ;
func BitGFA1CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA1Str a1,a2,a3) equals :: GFACIRC1:def 27
[<*[<*a1,a2*>,and2c ],[<*a2,a3*>,and2a ],[<*a3,a1*>,and2 ]*>,or3 ];
coherence
[<*[<*c1,c2*>,and2c ],[<*c2,c3*>,and2a ],[<*c3,c1*>,and2 ]*>,or3 ] is Element of InnerVertices (BitGFA1Str c1,c2,c3)
end;
:: deftheorem Def27 defines BitGFA1CarryOutput GFACIRC1:def 27 :
for
b1,
b2,
b3 being
set holds
BitGFA1CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2c ],[<*b2,b3*>,and2a ],[<*b3,b1*>,and2 ]*>,or3 ];
definition
let c1,
c2,
c3 be
set ;
func BitGFA1AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA1Str a1,a2,a3) equals :: GFACIRC1:def 28
2GatesCircOutput a1,
a2,
a3,
xor2c ;
coherence
2GatesCircOutput c1,c2,c3,xor2c is Element of InnerVertices (BitGFA1Str c1,c2,c3)
end;
:: deftheorem Def28 defines BitGFA1AdderOutput GFACIRC1:def 28 :
theorem Th84: :: GFACIRC1:84
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
for
b4 being
State of
(BitGFA1Circ b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA1AdderOutput b1,b2,b3) = 'not' ((b5 'xor' ('not' b6)) 'xor' b7) &
(Following b4,2) . (GFA1CarryOutput b1,b2,b3) = ((b5 '&' ('not' b6)) 'or' (('not' b6) '&' b7)) 'or' (b7 '&' b5) )
theorem Th85: :: GFACIRC1:85
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2a ] &
b2 <> [<*b3,b1*>,and2 ] &
b3 <> [<*b1,b2*>,and2c ] holds
for
b4 being
State of
(BitGFA1Circ b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA2CarryIStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 29
((1GateCircStr <*a1,a2*>,and2a ) +* (1GateCircStr <*a2,a3*>,and2c )) +* (1GateCircStr <*a3,a1*>,and2b );
coherence
((1GateCircStr <*c1,c2*>,and2a ) +* (1GateCircStr <*c2,c3*>,and2c )) +* (1GateCircStr <*c3,c1*>,and2b ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def29 defines GFA2CarryIStr GFACIRC1:def 29 :
for
b1,
b2,
b3 being
set holds
GFA2CarryIStr b1,
b2,
b3 = ((1GateCircStr <*b1,b2*>,and2a ) +* (1GateCircStr <*b2,b3*>,and2c )) +* (1GateCircStr <*b3,b1*>,and2b );
definition
let c1,
c2,
c3 be
set ;
func GFA2CarryICirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA2CarryIStr a1,
a2,
a3 equals :: GFACIRC1:def 30
((1GateCircuit a1,a2,and2a ) +* (1GateCircuit a2,a3,and2c )) +* (1GateCircuit a3,a1,and2b );
coherence
((1GateCircuit c1,c2,and2a ) +* (1GateCircuit c2,c3,and2c )) +* (1GateCircuit c3,c1,and2b ) is strict gate`2=den Boolean Circuit of GFA2CarryIStr c1,c2,c3
;
end;
:: deftheorem Def30 defines GFA2CarryICirc GFACIRC1:def 30 :
for
b1,
b2,
b3 being
set holds
GFA2CarryICirc b1,
b2,
b3 = ((1GateCircuit b1,b2,and2a ) +* (1GateCircuit b2,b3,and2c )) +* (1GateCircuit b3,b1,and2b );
definition
let c1,
c2,
c3 be
set ;
func GFA2CarryStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 31
(GFA2CarryIStr a1,a2,a3) +* (1GateCircStr <*[<*a1,a2*>,and2a ],[<*a2,a3*>,and2c ],[<*a3,a1*>,and2b ]*>,nor3 );
coherence
(GFA2CarryIStr c1,c2,c3) +* (1GateCircStr <*[<*c1,c2*>,and2a ],[<*c2,c3*>,and2c ],[<*c3,c1*>,and2b ]*>,nor3 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def31 defines GFA2CarryStr GFACIRC1:def 31 :
for
b1,
b2,
b3 being
set holds
GFA2CarryStr b1,
b2,
b3 = (GFA2CarryIStr b1,b2,b3) +* (1GateCircStr <*[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]*>,nor3 );
definition
let c1,
c2,
c3 be
set ;
func GFA2CarryCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA2CarryStr a1,
a2,
a3 equals :: GFACIRC1:def 32
(GFA2CarryICirc a1,a2,a3) +* (1GateCircuit [<*a1,a2*>,and2a ],[<*a2,a3*>,and2c ],[<*a3,a1*>,and2b ],nor3 );
coherence
(GFA2CarryICirc c1,c2,c3) +* (1GateCircuit [<*c1,c2*>,and2a ],[<*c2,c3*>,and2c ],[<*c3,c1*>,and2b ],nor3 ) is strict gate`2=den Boolean Circuit of GFA2CarryStr c1,c2,c3
;
end;
:: deftheorem Def32 defines GFA2CarryCirc GFACIRC1:def 32 :
for
b1,
b2,
b3 being
set holds
GFA2CarryCirc b1,
b2,
b3 = (GFA2CarryICirc b1,b2,b3) +* (1GateCircuit [<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ],nor3 );
definition
let c1,
c2,
c3 be
set ;
func GFA2CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA2CarryStr a1,a2,a3) equals :: GFACIRC1:def 33
[<*[<*a1,a2*>,and2a ],[<*a2,a3*>,and2c ],[<*a3,a1*>,and2b ]*>,nor3 ];
coherence
[<*[<*c1,c2*>,and2a ],[<*c2,c3*>,and2c ],[<*c3,c1*>,and2b ]*>,nor3 ] is Element of InnerVertices (GFA2CarryStr c1,c2,c3)
end;
:: deftheorem Def33 defines GFA2CarryOutput GFACIRC1:def 33 :
for
b1,
b2,
b3 being
set holds
GFA2CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]*>,nor3 ];
theorem Th86: :: GFACIRC1:86
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA2CarryIStr b1,b2,b3) = {[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]}
theorem Th87: :: GFACIRC1:87
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA2CarryStr b1,b2,b3) = {[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]} \/ {(GFA2CarryOutput b1,b2,b3)}
theorem Th88: :: GFACIRC1:88
theorem Th89: :: GFACIRC1:89
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
InputVertices (GFA2CarryIStr b1,b2,b3) = {b1,b2,b3}
theorem Th90: :: GFACIRC1:90
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
InputVertices (GFA2CarryStr b1,b2,b3) = {b1,b2,b3}
theorem Th91: :: GFACIRC1:91
theorem Th92: :: GFACIRC1:92
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA2CarryStr b1,b2,b3) &
b2 in the
carrier of
(GFA2CarryStr b1,b2,b3) &
b3 in the
carrier of
(GFA2CarryStr b1,b2,b3) &
[<*b1,b2*>,and2a ] in the
carrier of
(GFA2CarryStr b1,b2,b3) &
[<*b2,b3*>,and2c ] in the
carrier of
(GFA2CarryStr b1,b2,b3) &
[<*b3,b1*>,and2b ] in the
carrier of
(GFA2CarryStr b1,b2,b3) &
[<*[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]*>,nor3 ] in the
carrier of
(GFA2CarryStr b1,b2,b3) )
theorem Th93: :: GFACIRC1:93
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,and2a ] in InnerVertices (GFA2CarryStr b1,b2,b3) &
[<*b2,b3*>,and2c ] in InnerVertices (GFA2CarryStr b1,b2,b3) &
[<*b3,b1*>,and2b ] in InnerVertices (GFA2CarryStr b1,b2,b3) &
GFA2CarryOutput b1,
b2,
b3 in InnerVertices (GFA2CarryStr b1,b2,b3) )
theorem Th94: :: GFACIRC1:94
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
(
b1 in InputVertices (GFA2CarryStr b1,b2,b3) &
b2 in InputVertices (GFA2CarryStr b1,b2,b3) &
b3 in InputVertices (GFA2CarryStr b1,b2,b3) )
theorem Th95: :: GFACIRC1:95
theorem Th96: :: GFACIRC1:96
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA2CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,and2a ] = ('not' b5) '&' b6 &
(Following b4) . [<*b2,b3*>,and2c ] = b6 '&' ('not' b7) &
(Following b4) . [<*b3,b1*>,and2b ] = ('not' b7) '&' ('not' b5) )
theorem Th97: :: GFACIRC1:97
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA2CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,and2a ] &
b6 = b4 . [<*b2,b3*>,and2c ] &
b7 = b4 . [<*b3,b1*>,and2b ] holds
(Following b4) . (GFA2CarryOutput b1,b2,b3) = 'not' ((b5 'or' b6) 'or' b7)
theorem Th98: :: GFACIRC1:98
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
for
b4 being
State of
(GFA2CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA2CarryOutput b1,b2,b3) = 'not' (((('not' b5) '&' b6) 'or' (b6 '&' ('not' b7))) 'or' (('not' b7) '&' ('not' b5))) &
(Following b4,2) . [<*b1,b2*>,and2a ] = ('not' b5) '&' b6 &
(Following b4,2) . [<*b2,b3*>,and2c ] = b6 '&' ('not' b7) &
(Following b4,2) . [<*b3,b1*>,and2b ] = ('not' b7) '&' ('not' b5) )
theorem Th99: :: GFACIRC1:99
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
for
b4 being
State of
(GFA2CarryCirc b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA2AdderStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 34
2GatesCircStr a1,
a2,
a3,
xor2c ;
coherence
2GatesCircStr c1,c2,c3,xor2c is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def34 defines GFA2AdderStr GFACIRC1:def 34 :
definition
let c1,
c2,
c3 be
set ;
func GFA2AdderCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA2AdderStr a1,
a2,
a3 equals :: GFACIRC1:def 35
2GatesCircuit a1,
a2,
a3,
xor2c ;
coherence
2GatesCircuit c1,c2,c3,xor2c is strict gate`2=den Boolean Circuit of GFA2AdderStr c1,c2,c3
;
end;
:: deftheorem Def35 defines GFA2AdderCirc GFACIRC1:def 35 :
definition
let c1,
c2,
c3 be
set ;
func GFA2AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA2AdderStr a1,a2,a3) equals :: GFACIRC1:def 36
2GatesCircOutput a1,
a2,
a3,
xor2c ;
coherence
2GatesCircOutput c1,c2,c3,xor2c is Element of InnerVertices (GFA2AdderStr c1,c2,c3)
;
end;
:: deftheorem Def36 defines GFA2AdderOutput GFACIRC1:def 36 :
theorem Th100: :: GFACIRC1:100
theorem Th101: :: GFACIRC1:101
theorem Th102: :: GFACIRC1:102
theorem Th103: :: GFACIRC1:103
theorem Th104: :: GFACIRC1:104
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA2AdderStr b1,b2,b3) &
b2 in the
carrier of
(GFA2AdderStr b1,b2,b3) &
b3 in the
carrier of
(GFA2AdderStr b1,b2,b3) &
[<*b1,b2*>,xor2c ] in the
carrier of
(GFA2AdderStr b1,b2,b3) &
[<*[<*b1,b2*>,xor2c ],b3*>,xor2c ] in the
carrier of
(GFA2AdderStr b1,b2,b3) )
by FACIRC_1:60, FACIRC_1:61;
theorem Th105: :: GFACIRC1:105
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2c ] in InnerVertices (GFA2AdderStr b1,b2,b3) &
GFA2AdderOutput b1,
b2,
b3 in InnerVertices (GFA2AdderStr b1,b2,b3) )
theorem Th106: :: GFACIRC1:106
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
(
b1 in InputVertices (GFA2AdderStr b1,b2,b3) &
b2 in InputVertices (GFA2AdderStr b1,b2,b3) &
b3 in InputVertices (GFA2AdderStr b1,b2,b3) )
theorem Th107: :: GFACIRC1:107
theorem Th108: :: GFACIRC1:108
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA2AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,xor2c ] = b5 'xor' ('not' b6) &
(Following b4) . b1 = b5 &
(Following b4) . b2 = b6 &
(Following b4) . b3 = b7 )
theorem Th109: :: GFACIRC1:109
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA2AdderCirc b1,b2,b3)for
b5,
b6,
b7,
b8 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,xor2c ] &
b6 = b4 . b1 &
b7 = b4 . b2 &
b8 = b4 . b3 holds
(Following b4) . (GFA2AdderOutput b1,b2,b3) = b5 'xor' ('not' b8)
theorem Th110: :: GFACIRC1:110
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA2AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA2AdderOutput b1,b2,b3) = (b5 'xor' ('not' b6)) 'xor' ('not' b7) &
(Following b4,2) . [<*b1,b2*>,xor2c ] = b5 'xor' ('not' b6) &
(Following b4,2) . b1 = b5 &
(Following b4,2) . b2 = b6 &
(Following b4,2) . b3 = b7 )
theorem Th111: :: GFACIRC1:111
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] holds
for
b4 being
State of
(GFA2AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(Following b4,2) . (GFA2AdderOutput b1,b2,b3) = (('not' b5) 'xor' b6) 'xor' ('not' b7)
theorem Th112: :: GFACIRC1:112
definition
let c1,
c2,
c3 be
set ;
func BitGFA2Str c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 37
(GFA2AdderStr a1,a2,a3) +* (GFA2CarryStr a1,a2,a3);
coherence
(GFA2AdderStr c1,c2,c3) +* (GFA2CarryStr c1,c2,c3) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def37 defines BitGFA2Str GFACIRC1:def 37 :
definition
let c1,
c2,
c3 be
set ;
func BitGFA2Circ c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
BitGFA2Str a1,
a2,
a3 equals :: GFACIRC1:def 38
(GFA2AdderCirc a1,a2,a3) +* (GFA2CarryCirc a1,a2,a3);
coherence
(GFA2AdderCirc c1,c2,c3) +* (GFA2CarryCirc c1,c2,c3) is strict gate`2=den Boolean Circuit of BitGFA2Str c1,c2,c3
;
end;
:: deftheorem Def38 defines BitGFA2Circ GFACIRC1:def 38 :
theorem Th113: :: GFACIRC1:113
for
b1,
b2,
b3 being
set holds
InnerVertices (BitGFA2Str b1,b2,b3) = (({[<*b1,b2*>,xor2c ]} \/ {(GFA2AdderOutput b1,b2,b3)}) \/ {[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]}) \/ {(GFA2CarryOutput b1,b2,b3)}
theorem Th114: :: GFACIRC1:114
theorem Th115: :: GFACIRC1:115
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
InputVertices (BitGFA2Str b1,b2,b3) = {b1,b2,b3}
theorem Th116: :: GFACIRC1:116
theorem Th117: :: GFACIRC1:117
theorem Th118: :: GFACIRC1:118
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(BitGFA2Str b1,b2,b3) &
b2 in the
carrier of
(BitGFA2Str b1,b2,b3) &
b3 in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*b1,b2*>,xor2c ] in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*[<*b1,b2*>,xor2c ],b3*>,xor2c ] in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*b1,b2*>,and2a ] in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*b2,b3*>,and2c ] in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*b3,b1*>,and2b ] in the
carrier of
(BitGFA2Str b1,b2,b3) &
[<*[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]*>,nor3 ] in the
carrier of
(BitGFA2Str b1,b2,b3) )
theorem Th119: :: GFACIRC1:119
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2c ] in InnerVertices (BitGFA2Str b1,b2,b3) &
GFA2AdderOutput b1,
b2,
b3 in InnerVertices (BitGFA2Str b1,b2,b3) &
[<*b1,b2*>,and2a ] in InnerVertices (BitGFA2Str b1,b2,b3) &
[<*b2,b3*>,and2c ] in InnerVertices (BitGFA2Str b1,b2,b3) &
[<*b3,b1*>,and2b ] in InnerVertices (BitGFA2Str b1,b2,b3) &
GFA2CarryOutput b1,
b2,
b3 in InnerVertices (BitGFA2Str b1,b2,b3) )
theorem Th120: :: GFACIRC1:120
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
(
b1 in InputVertices (BitGFA2Str b1,b2,b3) &
b2 in InputVertices (BitGFA2Str b1,b2,b3) &
b3 in InputVertices (BitGFA2Str b1,b2,b3) )
definition
let c1,
c2,
c3 be
set ;
func BitGFA2CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA2Str a1,a2,a3) equals :: GFACIRC1:def 39
[<*[<*a1,a2*>,and2a ],[<*a2,a3*>,and2c ],[<*a3,a1*>,and2b ]*>,nor3 ];
coherence
[<*[<*c1,c2*>,and2a ],[<*c2,c3*>,and2c ],[<*c3,c1*>,and2b ]*>,nor3 ] is Element of InnerVertices (BitGFA2Str c1,c2,c3)
end;
:: deftheorem Def39 defines BitGFA2CarryOutput GFACIRC1:def 39 :
for
b1,
b2,
b3 being
set holds
BitGFA2CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2a ],[<*b2,b3*>,and2c ],[<*b3,b1*>,and2b ]*>,nor3 ];
definition
let c1,
c2,
c3 be
set ;
func BitGFA2AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA2Str a1,a2,a3) equals :: GFACIRC1:def 40
2GatesCircOutput a1,
a2,
a3,
xor2c ;
coherence
2GatesCircOutput c1,c2,c3,xor2c is Element of InnerVertices (BitGFA2Str c1,c2,c3)
end;
:: deftheorem Def40 defines BitGFA2AdderOutput GFACIRC1:def 40 :
theorem Th121: :: GFACIRC1:121
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
for
b4 being
State of
(BitGFA2Circ b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA2AdderOutput b1,b2,b3) = (('not' b5) 'xor' b6) 'xor' ('not' b7) &
(Following b4,2) . (GFA2CarryOutput b1,b2,b3) = 'not' (((('not' b5) '&' b6) 'or' (b6 '&' ('not' b7))) 'or' (('not' b7) '&' ('not' b5))) )
theorem Th122: :: GFACIRC1:122
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2c ] &
b1 <> [<*b2,b3*>,and2c ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2a ] holds
for
b4 being
State of
(BitGFA2Circ b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA3CarryIStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 41
((1GateCircStr <*a1,a2*>,and2b ) +* (1GateCircStr <*a2,a3*>,and2b )) +* (1GateCircStr <*a3,a1*>,and2b );
coherence
((1GateCircStr <*c1,c2*>,and2b ) +* (1GateCircStr <*c2,c3*>,and2b )) +* (1GateCircStr <*c3,c1*>,and2b ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def41 defines GFA3CarryIStr GFACIRC1:def 41 :
for
b1,
b2,
b3 being
set holds
GFA3CarryIStr b1,
b2,
b3 = ((1GateCircStr <*b1,b2*>,and2b ) +* (1GateCircStr <*b2,b3*>,and2b )) +* (1GateCircStr <*b3,b1*>,and2b );
definition
let c1,
c2,
c3 be
set ;
func GFA3CarryICirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA3CarryIStr a1,
a2,
a3 equals :: GFACIRC1:def 42
((1GateCircuit a1,a2,and2b ) +* (1GateCircuit a2,a3,and2b )) +* (1GateCircuit a3,a1,and2b );
coherence
((1GateCircuit c1,c2,and2b ) +* (1GateCircuit c2,c3,and2b )) +* (1GateCircuit c3,c1,and2b ) is strict gate`2=den Boolean Circuit of GFA3CarryIStr c1,c2,c3
;
end;
:: deftheorem Def42 defines GFA3CarryICirc GFACIRC1:def 42 :
for
b1,
b2,
b3 being
set holds
GFA3CarryICirc b1,
b2,
b3 = ((1GateCircuit b1,b2,and2b ) +* (1GateCircuit b2,b3,and2b )) +* (1GateCircuit b3,b1,and2b );
definition
let c1,
c2,
c3 be
set ;
func GFA3CarryStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 43
(GFA3CarryIStr a1,a2,a3) +* (1GateCircStr <*[<*a1,a2*>,and2b ],[<*a2,a3*>,and2b ],[<*a3,a1*>,and2b ]*>,nor3 );
coherence
(GFA3CarryIStr c1,c2,c3) +* (1GateCircStr <*[<*c1,c2*>,and2b ],[<*c2,c3*>,and2b ],[<*c3,c1*>,and2b ]*>,nor3 ) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def43 defines GFA3CarryStr GFACIRC1:def 43 :
for
b1,
b2,
b3 being
set holds
GFA3CarryStr b1,
b2,
b3 = (GFA3CarryIStr b1,b2,b3) +* (1GateCircStr <*[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]*>,nor3 );
definition
let c1,
c2,
c3 be
set ;
func GFA3CarryCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA3CarryStr a1,
a2,
a3 equals :: GFACIRC1:def 44
(GFA3CarryICirc a1,a2,a3) +* (1GateCircuit [<*a1,a2*>,and2b ],[<*a2,a3*>,and2b ],[<*a3,a1*>,and2b ],nor3 );
coherence
(GFA3CarryICirc c1,c2,c3) +* (1GateCircuit [<*c1,c2*>,and2b ],[<*c2,c3*>,and2b ],[<*c3,c1*>,and2b ],nor3 ) is strict gate`2=den Boolean Circuit of GFA3CarryStr c1,c2,c3
;
end;
:: deftheorem Def44 defines GFA3CarryCirc GFACIRC1:def 44 :
for
b1,
b2,
b3 being
set holds
GFA3CarryCirc b1,
b2,
b3 = (GFA3CarryICirc b1,b2,b3) +* (1GateCircuit [<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ],nor3 );
definition
let c1,
c2,
c3 be
set ;
func GFA3CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA3CarryStr a1,a2,a3) equals :: GFACIRC1:def 45
[<*[<*a1,a2*>,and2b ],[<*a2,a3*>,and2b ],[<*a3,a1*>,and2b ]*>,nor3 ];
coherence
[<*[<*c1,c2*>,and2b ],[<*c2,c3*>,and2b ],[<*c3,c1*>,and2b ]*>,nor3 ] is Element of InnerVertices (GFA3CarryStr c1,c2,c3)
end;
:: deftheorem Def45 defines GFA3CarryOutput GFACIRC1:def 45 :
for
b1,
b2,
b3 being
set holds
GFA3CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]*>,nor3 ];
theorem Th123: :: GFACIRC1:123
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA3CarryIStr b1,b2,b3) = {[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]}
theorem Th124: :: GFACIRC1:124
for
b1,
b2,
b3 being
set holds
InnerVertices (GFA3CarryStr b1,b2,b3) = {[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]} \/ {(GFA3CarryOutput b1,b2,b3)}
theorem Th125: :: GFACIRC1:125
theorem Th126: :: GFACIRC1:126
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
InputVertices (GFA3CarryIStr b1,b2,b3) = {b1,b2,b3}
theorem Th127: :: GFACIRC1:127
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
InputVertices (GFA3CarryStr b1,b2,b3) = {b1,b2,b3}
theorem Th128: :: GFACIRC1:128
theorem Th129: :: GFACIRC1:129
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA3CarryStr b1,b2,b3) &
b2 in the
carrier of
(GFA3CarryStr b1,b2,b3) &
b3 in the
carrier of
(GFA3CarryStr b1,b2,b3) &
[<*b1,b2*>,and2b ] in the
carrier of
(GFA3CarryStr b1,b2,b3) &
[<*b2,b3*>,and2b ] in the
carrier of
(GFA3CarryStr b1,b2,b3) &
[<*b3,b1*>,and2b ] in the
carrier of
(GFA3CarryStr b1,b2,b3) &
[<*[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]*>,nor3 ] in the
carrier of
(GFA3CarryStr b1,b2,b3) )
theorem Th130: :: GFACIRC1:130
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,and2b ] in InnerVertices (GFA3CarryStr b1,b2,b3) &
[<*b2,b3*>,and2b ] in InnerVertices (GFA3CarryStr b1,b2,b3) &
[<*b3,b1*>,and2b ] in InnerVertices (GFA3CarryStr b1,b2,b3) &
GFA3CarryOutput b1,
b2,
b3 in InnerVertices (GFA3CarryStr b1,b2,b3) )
theorem Th131: :: GFACIRC1:131
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
(
b1 in InputVertices (GFA3CarryStr b1,b2,b3) &
b2 in InputVertices (GFA3CarryStr b1,b2,b3) &
b3 in InputVertices (GFA3CarryStr b1,b2,b3) )
theorem Th132: :: GFACIRC1:132
theorem Th133: :: GFACIRC1:133
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA3CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,and2b ] = ('not' b5) '&' ('not' b6) &
(Following b4) . [<*b2,b3*>,and2b ] = ('not' b6) '&' ('not' b7) &
(Following b4) . [<*b3,b1*>,and2b ] = ('not' b7) '&' ('not' b5) )
theorem Th134: :: GFACIRC1:134
for
b1,
b2,
b3 being
set for
b4 being
State of
(GFA3CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,and2b ] &
b6 = b4 . [<*b2,b3*>,and2b ] &
b7 = b4 . [<*b3,b1*>,and2b ] holds
(Following b4) . (GFA3CarryOutput b1,b2,b3) = 'not' ((b5 'or' b6) 'or' b7)
theorem Th135: :: GFACIRC1:135
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
for
b4 being
State of
(GFA3CarryCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA3CarryOutput b1,b2,b3) = 'not' (((('not' b5) '&' ('not' b6)) 'or' (('not' b6) '&' ('not' b7))) 'or' (('not' b7) '&' ('not' b5))) &
(Following b4,2) . [<*b1,b2*>,and2b ] = ('not' b5) '&' ('not' b6) &
(Following b4,2) . [<*b2,b3*>,and2b ] = ('not' b6) '&' ('not' b7) &
(Following b4,2) . [<*b3,b1*>,and2b ] = ('not' b7) '&' ('not' b5) )
theorem Th136: :: GFACIRC1:136
for
b1,
b2,
b3 being
set st
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
for
b4 being
State of
(GFA3CarryCirc b1,b2,b3) holds
Following b4,2 is
stable
definition
let c1,
c2,
c3 be
set ;
func GFA3AdderStr c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 46
2GatesCircStr a1,
a2,
a3,
xor2 ;
coherence
2GatesCircStr c1,c2,c3,xor2 is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def46 defines GFA3AdderStr GFACIRC1:def 46 :
definition
let c1,
c2,
c3 be
set ;
func GFA3AdderCirc c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
GFA3AdderStr a1,
a2,
a3 equals :: GFACIRC1:def 47
2GatesCircuit a1,
a2,
a3,
xor2 ;
coherence
2GatesCircuit c1,c2,c3,xor2 is strict gate`2=den Boolean Circuit of GFA3AdderStr c1,c2,c3
;
end;
:: deftheorem Def47 defines GFA3AdderCirc GFACIRC1:def 47 :
definition
let c1,
c2,
c3 be
set ;
func GFA3AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (GFA3AdderStr a1,a2,a3) equals :: GFACIRC1:def 48
2GatesCircOutput a1,
a2,
a3,
xor2 ;
coherence
2GatesCircOutput c1,c2,c3,xor2 is Element of InnerVertices (GFA3AdderStr c1,c2,c3)
;
end;
:: deftheorem Def48 defines GFA3AdderOutput GFACIRC1:def 48 :
theorem Th137: :: GFACIRC1:137
theorem Th138: :: GFACIRC1:138
theorem Th139: :: GFACIRC1:139
theorem Th140: :: GFACIRC1:140
theorem Th141: :: GFACIRC1:141
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(GFA3AdderStr b1,b2,b3) &
b2 in the
carrier of
(GFA3AdderStr b1,b2,b3) &
b3 in the
carrier of
(GFA3AdderStr b1,b2,b3) &
[<*b1,b2*>,xor2 ] in the
carrier of
(GFA3AdderStr b1,b2,b3) &
[<*[<*b1,b2*>,xor2 ],b3*>,xor2 ] in the
carrier of
(GFA3AdderStr b1,b2,b3) )
by FACIRC_1:60, FACIRC_1:61;
theorem Th142: :: GFACIRC1:142
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2 ] in InnerVertices (GFA3AdderStr b1,b2,b3) &
GFA3AdderOutput b1,
b2,
b3 in InnerVertices (GFA3AdderStr b1,b2,b3) )
theorem Th143: :: GFACIRC1:143
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
(
b1 in InputVertices (GFA3AdderStr b1,b2,b3) &
b2 in InputVertices (GFA3AdderStr b1,b2,b3) &
b3 in InputVertices (GFA3AdderStr b1,b2,b3) )
theorem Th144: :: GFACIRC1:144
theorem Th145: :: GFACIRC1:145
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA3AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4) . [<*b1,b2*>,xor2 ] = b5 'xor' b6 &
(Following b4) . b1 = b5 &
(Following b4) . b2 = b6 &
(Following b4) . b3 = b7 )
theorem Th146: :: GFACIRC1:146
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA3AdderCirc b1,b2,b3)for
b5,
b6,
b7,
b8 being
Element of
BOOLEAN st
b5 = b4 . [<*b1,b2*>,xor2 ] &
b6 = b4 . b1 &
b7 = b4 . b2 &
b8 = b4 . b3 holds
(Following b4) . (GFA3AdderOutput b1,b2,b3) = b5 'xor' b8
theorem Th147: :: GFACIRC1:147
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA3AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA3AdderOutput b1,b2,b3) = (b5 'xor' b6) 'xor' b7 &
(Following b4,2) . [<*b1,b2*>,xor2 ] = b5 'xor' b6 &
(Following b4,2) . b1 = b5 &
(Following b4,2) . b2 = b6 &
(Following b4,2) . b3 = b7 )
theorem Th148: :: GFACIRC1:148
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] holds
for
b4 being
State of
(GFA3AdderCirc b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(Following b4,2) . (GFA3AdderOutput b1,b2,b3) = 'not' ((('not' b5) 'xor' ('not' b6)) 'xor' ('not' b7))
theorem Th149: :: GFACIRC1:149
definition
let c1,
c2,
c3 be
set ;
func BitGFA3Str c1,
c2,
c3 -> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign equals :: GFACIRC1:def 49
(GFA3AdderStr a1,a2,a3) +* (GFA3CarryStr a1,a2,a3);
coherence
(GFA3AdderStr c1,c2,c3) +* (GFA3CarryStr c1,c2,c3) is non empty strict non void unsplit gate`1=arity gate`2isBoolean ManySortedSign
;
end;
:: deftheorem Def49 defines BitGFA3Str GFACIRC1:def 49 :
definition
let c1,
c2,
c3 be
set ;
func BitGFA3Circ c1,
c2,
c3 -> strict gate`2=den Boolean Circuit of
BitGFA3Str a1,
a2,
a3 equals :: GFACIRC1:def 50
(GFA3AdderCirc a1,a2,a3) +* (GFA3CarryCirc a1,a2,a3);
coherence
(GFA3AdderCirc c1,c2,c3) +* (GFA3CarryCirc c1,c2,c3) is strict gate`2=den Boolean Circuit of BitGFA3Str c1,c2,c3
;
end;
:: deftheorem Def50 defines BitGFA3Circ GFACIRC1:def 50 :
theorem Th150: :: GFACIRC1:150
for
b1,
b2,
b3 being
set holds
InnerVertices (BitGFA3Str b1,b2,b3) = (({[<*b1,b2*>,xor2 ]} \/ {(GFA3AdderOutput b1,b2,b3)}) \/ {[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]}) \/ {(GFA3CarryOutput b1,b2,b3)}
theorem Th151: :: GFACIRC1:151
theorem Th152: :: GFACIRC1:152
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
InputVertices (BitGFA3Str b1,b2,b3) = {b1,b2,b3}
theorem Th153: :: GFACIRC1:153
theorem Th154: :: GFACIRC1:154
theorem Th155: :: GFACIRC1:155
for
b1,
b2,
b3 being
set holds
(
b1 in the
carrier of
(BitGFA3Str b1,b2,b3) &
b2 in the
carrier of
(BitGFA3Str b1,b2,b3) &
b3 in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*b1,b2*>,xor2 ] in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*[<*b1,b2*>,xor2 ],b3*>,xor2 ] in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*b1,b2*>,and2b ] in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*b2,b3*>,and2b ] in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*b3,b1*>,and2b ] in the
carrier of
(BitGFA3Str b1,b2,b3) &
[<*[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]*>,nor3 ] in the
carrier of
(BitGFA3Str b1,b2,b3) )
theorem Th156: :: GFACIRC1:156
for
b1,
b2,
b3 being
set holds
(
[<*b1,b2*>,xor2 ] in InnerVertices (BitGFA3Str b1,b2,b3) &
GFA3AdderOutput b1,
b2,
b3 in InnerVertices (BitGFA3Str b1,b2,b3) &
[<*b1,b2*>,and2b ] in InnerVertices (BitGFA3Str b1,b2,b3) &
[<*b2,b3*>,and2b ] in InnerVertices (BitGFA3Str b1,b2,b3) &
[<*b3,b1*>,and2b ] in InnerVertices (BitGFA3Str b1,b2,b3) &
GFA3CarryOutput b1,
b2,
b3 in InnerVertices (BitGFA3Str b1,b2,b3) )
theorem Th157: :: GFACIRC1:157
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
(
b1 in InputVertices (BitGFA3Str b1,b2,b3) &
b2 in InputVertices (BitGFA3Str b1,b2,b3) &
b3 in InputVertices (BitGFA3Str b1,b2,b3) )
definition
let c1,
c2,
c3 be
set ;
func BitGFA3CarryOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA3Str a1,a2,a3) equals :: GFACIRC1:def 51
[<*[<*a1,a2*>,and2b ],[<*a2,a3*>,and2b ],[<*a3,a1*>,and2b ]*>,nor3 ];
coherence
[<*[<*c1,c2*>,and2b ],[<*c2,c3*>,and2b ],[<*c3,c1*>,and2b ]*>,nor3 ] is Element of InnerVertices (BitGFA3Str c1,c2,c3)
end;
:: deftheorem Def51 defines BitGFA3CarryOutput GFACIRC1:def 51 :
for
b1,
b2,
b3 being
set holds
BitGFA3CarryOutput b1,
b2,
b3 = [<*[<*b1,b2*>,and2b ],[<*b2,b3*>,and2b ],[<*b3,b1*>,and2b ]*>,nor3 ];
definition
let c1,
c2,
c3 be
set ;
func BitGFA3AdderOutput c1,
c2,
c3 -> Element of
InnerVertices (BitGFA3Str a1,a2,a3) equals :: GFACIRC1:def 52
2GatesCircOutput a1,
a2,
a3,
xor2 ;
coherence
2GatesCircOutput c1,c2,c3,xor2 is Element of InnerVertices (BitGFA3Str c1,c2,c3)
end;
:: deftheorem Def52 defines BitGFA3AdderOutput GFACIRC1:def 52 :
theorem Th158: :: GFACIRC1:158
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
for
b4 being
State of
(BitGFA3Circ b1,b2,b3)for
b5,
b6,
b7 being
Element of
BOOLEAN st
b5 = b4 . b1 &
b6 = b4 . b2 &
b7 = b4 . b3 holds
(
(Following b4,2) . (GFA3AdderOutput b1,b2,b3) = 'not' ((('not' b5) 'xor' ('not' b6)) 'xor' ('not' b7)) &
(Following b4,2) . (GFA3CarryOutput b1,b2,b3) = 'not' (((('not' b5) '&' ('not' b6)) 'or' (('not' b6) '&' ('not' b7))) 'or' (('not' b7) '&' ('not' b5))) )
theorem Th159: :: GFACIRC1:159
for
b1,
b2,
b3 being
set st
b3 <> [<*b1,b2*>,xor2 ] &
b1 <> [<*b2,b3*>,and2b ] &
b2 <> [<*b3,b1*>,and2b ] &
b3 <> [<*b1,b2*>,and2b ] holds
for
b4 being
State of
(BitGFA3Circ b1,b2,b3) holds
Following b4,2 is
stable