:: GATE_1 semantic presentation
theorem Th1: :: GATE_1:1
for b
1 being
set holds
not ( b
1 = {{} } & not
$ b
1 ) ;
theorem Th2: :: GATE_1:2
not for b
1 being
set holds not
$ b
1
:: deftheorem Def1 defines NOT1 GATE_1:def 1 :
theorem Th3: :: GATE_1:3
theorem Th4: :: GATE_1:4
theorem Th5: :: GATE_1:5
:: deftheorem Def2 defines AND2 GATE_1:def 2 :
for b
1, b
2 being
set holds
( (
$ b
1 &
$ b
2 implies
AND2 b
1,b
2 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 ) implies
AND2 b
1,b
2 = {} ) );
theorem Th6: :: GATE_1:6
:: deftheorem Def3 defines OR2 GATE_1:def 3 :
for b
1, b
2 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 ) implies
OR2 b
1,b
2 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
OR2 b
1,b
2 = {} ) );
theorem Th7: :: GATE_1:7
for b
1, b
2 being
set holds
( not (
$ OR2 b
1,b
2 & not
$ b
1 & not
$ b
2 ) & not ( not ( not
$ b
1 & not
$ b
2 ) & not
$ OR2 b
1,b
2 ) )
by Def3, Th5;
:: deftheorem Def4 defines XOR2 GATE_1:def 4 :
for b
1, b
2 being
set holds
( ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) implies
XOR2 b
1,b
2 = NOT1 {} ) & not ( not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) & not
XOR2 b
1,b
2 = {} ) );
theorem Th8: :: GATE_1:8
for b
1, b
2 being
set holds
( not (
$ XOR2 b
1,b
2 & not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) ) & not ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) & not
$ XOR2 b
1,b
2 ) )
by Def4, Th5;
theorem Th9: :: GATE_1:9
theorem Th10: :: GATE_1:10
theorem Th11: :: GATE_1:11
definition
let c
1, c
2 be
set ;
func EQV2 c
1,c
2 -> set equals :
Def5:
:: GATE_1:def 5
NOT1 {} if ( not (
$ a
1 & not
$ a
2 ) & not (
$ a
2 & not
$ a
1 ) )
otherwise {} ;
correctness
coherence
( not ( not ( $ c1 & not $ c2 ) & not ( $ c2 & not $ c1 ) & not NOT1 {} is set ) & ( not ( not ( $ c1 & not $ c2 ) & not ( $ c2 & not $ c1 ) ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
commutativity
for b1 being set
for b2, b3 being set holds
( not ( not ( $ b2 & not $ b3 ) & not ( $ b3 & not $ b2 ) & not b1 = NOT1 {} ) & ( not ( not ( $ b2 & not $ b3 ) & not ( $ b3 & not $ b2 ) ) implies b1 = {} ) implies ( not ( not ( $ b3 & not $ b2 ) & not ( $ b2 & not $ b3 ) & not b1 = NOT1 {} ) & ( not ( not ( $ b3 & not $ b2 ) & not ( $ b2 & not $ b3 ) ) implies b1 = {} ) ) )
;
end;
:: deftheorem Def5 defines EQV2 GATE_1:def 5 :
for b
1, b
2 being
set holds
( not ( not (
$ b
1 & not
$ b
2 ) & not (
$ b
2 & not
$ b
1 ) & not
EQV2 b
1,b
2 = NOT1 {} ) & ( not ( not (
$ b
1 & not
$ b
2 ) & not (
$ b
2 & not
$ b
1 ) ) implies
EQV2 b
1,b
2 = {} ) );
theorem Th12: :: GATE_1:12
for b
1, b
2 being
set holds
(
$ EQV2 b
1,b
2 iff ( not (
$ b
1 & not
$ b
2 ) & not (
$ b
2 & not
$ b
1 ) ) )
by Def5, Th5;
theorem Th13: :: GATE_1:13
:: deftheorem Def6 defines NAND2 GATE_1:def 6 :
theorem Th14: :: GATE_1:14
for b
1, b
2 being
set holds
( not (
$ NAND2 b
1,b
2 &
$ b
1 &
$ b
2 ) & not ( not (
$ b
1 &
$ b
2 ) & not
$ NAND2 b
1,b
2 ) )
by Def6, Th5;
:: deftheorem Def7 defines NOR2 GATE_1:def 7 :
for b
1, b
2 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
NOR2 b
1,b
2 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 ) implies
NOR2 b
1,b
2 = {} ) );
theorem Th15: :: GATE_1:15
for b
1, b
2 being
set holds
(
$ NOR2 b
1,b
2 iff ( not
$ b
1 & not
$ b
2 ) )
by Def7, Th5;
:: deftheorem Def8 defines AND3 GATE_1:def 8 :
for b
1, b
2, b
3 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 implies
AND3 b
1,b
2,b
3 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 ) implies
AND3 b
1,b
2,b
3 = {} ) );
theorem Th16: :: GATE_1:16
for b
1, b
2, b
3 being
set holds
(
$ AND3 b
1,b
2,b
3 iff (
$ b
1 &
$ b
2 &
$ b
3 ) )
by Def8, Th5;
:: deftheorem Def9 defines OR3 GATE_1:def 9 :
for b
1, b
2, b
3 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 ) implies
OR3 b
1,b
2,b
3 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
OR3 b
1,b
2,b
3 = {} ) );
theorem Th17: :: GATE_1:17
for b
1, b
2, b
3 being
set holds
( not (
$ OR3 b
1,b
2,b
3 & not
$ b
1 & not
$ b
2 & not
$ b
3 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 ) & not
$ OR3 b
1,b
2,b
3 ) )
by Def9, Th5;
definition
let c
1, c
2, c
3 be
set ;
func XOR3 c
1,c
2,c
3 -> set equals :
Def10:
:: GATE_1:def 10
NOT1 {} if ( ( ( (
$ a
1 & not
$ a
2 ) or ( not
$ a
1 &
$ a
2 ) ) & not
$ a
3 ) or ( not (
$ a
1 & not
$ a
2 ) & not ( not
$ a
1 &
$ a
2 ) &
$ a
3 ) )
otherwise {} ;
correctness
coherence
( ( ( ( ( ( $ c1 & not $ c2 ) or ( not $ c1 & $ c2 ) ) & not $ c3 ) or ( not ( $ c1 & not $ c2 ) & not ( not $ c1 & $ c2 ) & $ c3 ) ) implies NOT1 {} is set ) & not ( not ( ( ( $ c1 & not $ c2 ) or ( not $ c1 & $ c2 ) ) & not $ c3 ) & not ( not ( $ c1 & not $ c2 ) & not ( not $ c1 & $ c2 ) & $ c3 ) & not {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def10 defines XOR3 GATE_1:def 10 :
for b
1, b
2, b
3 being
set holds
( ( ( ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) & not
$ b
3 ) or ( not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) &
$ b
3 ) ) implies
XOR3 b
1,b
2,b
3 = NOT1 {} ) & not ( not ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) & not
$ b
3 ) & not ( not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) &
$ b
3 ) & not
XOR3 b
1,b
2,b
3 = {} ) );
theorem Th18: :: GATE_1:18
for b
1, b
2, b
3 being
set holds
( not (
$ XOR3 b
1,b
2,b
3 & not ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) & not
$ b
3 ) & not ( not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) &
$ b
3 ) ) & not ( ( ( ( (
$ b
1 & not
$ b
2 ) or ( not
$ b
1 &
$ b
2 ) ) & not
$ b
3 ) or ( not (
$ b
1 & not
$ b
2 ) & not ( not
$ b
1 &
$ b
2 ) &
$ b
3 ) ) & not
$ XOR3 b
1,b
2,b
3 ) )
by Def10, Th5;
:: deftheorem Def11 defines MAJ3 GATE_1:def 11 :
for b
1, b
2, b
3 being
set holds
( ( not ( not (
$ b
1 &
$ b
2 ) & not (
$ b
2 &
$ b
3 ) & not (
$ b
3 &
$ b
1 ) ) implies
MAJ3 b
1,b
2,b
3 = NOT1 {} ) & not ( not (
$ b
1 &
$ b
2 ) & not (
$ b
2 &
$ b
3 ) & not (
$ b
3 &
$ b
1 ) & not
MAJ3 b
1,b
2,b
3 = {} ) );
theorem Th19: :: GATE_1:19
for b
1, b
2, b
3 being
set holds
( not (
$ MAJ3 b
1,b
2,b
3 & not (
$ b
1 &
$ b
2 ) & not (
$ b
2 &
$ b
3 ) & not (
$ b
3 &
$ b
1 ) ) & not ( not ( not (
$ b
1 &
$ b
2 ) & not (
$ b
2 &
$ b
3 ) & not (
$ b
3 &
$ b
1 ) ) & not
$ MAJ3 b
1,b
2,b
3 ) )
by Def11, Th5;
:: deftheorem Def12 defines NAND3 GATE_1:def 12 :
for b
1, b
2, b
3 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 ) implies
NAND3 b
1,b
2,b
3 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 implies
NAND3 b
1,b
2,b
3 = {} ) );
theorem Th20: :: GATE_1:20
for b
1, b
2, b
3 being
set holds
( not (
$ NAND3 b
1,b
2,b
3 &
$ b
1 &
$ b
2 &
$ b
3 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 ) & not
$ NAND3 b
1,b
2,b
3 ) )
by Def12, Th5;
:: deftheorem Def13 defines NOR3 GATE_1:def 13 :
for b
1, b
2, b
3 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
NOR3 b
1,b
2,b
3 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 ) implies
NOR3 b
1,b
2,b
3 = {} ) );
theorem Th21: :: GATE_1:21
for b
1, b
2, b
3 being
set holds
(
$ NOR3 b
1,b
2,b
3 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 ) )
by Def13, Th5;
:: deftheorem Def14 defines AND4 GATE_1:def 14 :
for b
1, b
2, b
3, b
4 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 implies
AND4 b
1,b
2,b
3,b
4 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 ) implies
AND4 b
1,b
2,b
3,b
4 = {} ) );
theorem Th22: :: GATE_1:22
for b
1, b
2, b
3, b
4 being
set holds
(
$ AND4 b
1,b
2,b
3,b
4 iff (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 ) )
by Def14, Th5;
:: deftheorem Def15 defines OR4 GATE_1:def 15 :
for b
1, b
2, b
3, b
4 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 ) implies
OR4 b
1,b
2,b
3,b
4 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
OR4 b
1,b
2,b
3,b
4 = {} ) );
theorem Th23: :: GATE_1:23
for b
1, b
2, b
3, b
4 being
set holds
( not (
$ OR4 b
1,b
2,b
3,b
4 & not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 ) & not
$ OR4 b
1,b
2,b
3,b
4 ) )
by Def15, Th5;
:: deftheorem Def16 defines NAND4 GATE_1:def 16 :
for b
1, b
2, b
3, b
4 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 ) implies
NAND4 b
1,b
2,b
3,b
4 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 implies
NAND4 b
1,b
2,b
3,b
4 = {} ) );
theorem Th24: :: GATE_1:24
for b
1, b
2, b
3, b
4 being
set holds
( not (
$ NAND4 b
1,b
2,b
3,b
4 &
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 ) & not
$ NAND4 b
1,b
2,b
3,b
4 ) )
by Def16, Th5;
:: deftheorem Def17 defines NOR4 GATE_1:def 17 :
for b
1, b
2, b
3, b
4 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
NOR4 b
1,b
2,b
3,b
4 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 ) implies
NOR4 b
1,b
2,b
3,b
4 = {} ) );
theorem Th25: :: GATE_1:25
for b
1, b
2, b
3, b
4 being
set holds
(
$ NOR4 b
1,b
2,b
3,b
4 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 ) )
by Def17, Th5;
:: deftheorem Def18 defines AND5 GATE_1:def 18 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 implies
AND5 b
1,b
2,b
3,b
4,b
5 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 ) implies
AND5 b
1,b
2,b
3,b
4,b
5 = {} ) );
theorem Th26: :: GATE_1:26
for b
1, b
2, b
3, b
4, b
5 being
set holds
(
$ AND5 b
1,b
2,b
3,b
4,b
5 iff (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 ) )
by Def18, Th5;
:: deftheorem Def19 defines OR5 GATE_1:def 19 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 ) implies
OR5 b
1,b
2,b
3,b
4,b
5 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
OR5 b
1,b
2,b
3,b
4,b
5 = {} ) );
theorem Th27: :: GATE_1:27
for b
1, b
2, b
3, b
4, b
5 being
set holds
( not (
$ OR5 b
1,b
2,b
3,b
4,b
5 & not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 ) & not
$ OR5 b
1,b
2,b
3,b
4,b
5 ) )
by Def19, Th5;
:: deftheorem Def20 defines NAND5 GATE_1:def 20 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 ) implies
NAND5 b
1,b
2,b
3,b
4,b
5 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 implies
NAND5 b
1,b
2,b
3,b
4,b
5 = {} ) );
theorem Th28: :: GATE_1:28
for b
1, b
2, b
3, b
4, b
5 being
set holds
( not (
$ NAND5 b
1,b
2,b
3,b
4,b
5 &
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 ) & not
$ NAND5 b
1,b
2,b
3,b
4,b
5 ) )
by Def20, Th5;
:: deftheorem Def21 defines NOR5 GATE_1:def 21 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
NOR5 b
1,b
2,b
3,b
4,b
5 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 ) implies
NOR5 b
1,b
2,b
3,b
4,b
5 = {} ) );
theorem Th29: :: GATE_1:29
for b
1, b
2, b
3, b
4, b
5 being
set holds
(
$ NOR5 b
1,b
2,b
3,b
4,b
5 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 ) )
by Def21, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
func AND6 c
1,c
2,c
3,c
4,c
5,c
6 -> set equals :
Def22:
:: GATE_1:def 22
NOT1 {} if (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 )
otherwise {} ;
correctness
coherence
( ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 implies NOT1 {} is set ) & ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def22 defines AND6 GATE_1:def 22 :
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 implies
AND6 b
1,b
2,b
3,b
4,b
5,b
6 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 ) implies
AND6 b
1,b
2,b
3,b
4,b
5,b
6 = {} ) );
theorem Th30: :: GATE_1:30
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
$ AND6 b
1,b
2,b
3,b
4,b
5,b
6 iff (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 ) )
by Def22, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
func OR6 c
1,c
2,c
3,c
4,c
5,c
6 -> set equals :
Def23:
:: GATE_1:def 23
NOT1 {} if not ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 )
otherwise {} ;
correctness
coherence
( ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 ) implies NOT1 {} is set ) & not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def23 defines OR6 GATE_1:def 23 :
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 ) implies
OR6 b
1,b
2,b
3,b
4,b
5,b
6 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
OR6 b
1,b
2,b
3,b
4,b
5,b
6 = {} ) );
theorem Th31: :: GATE_1:31
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( not (
$ OR6 b
1,b
2,b
3,b
4,b
5,b
6 & not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 ) & not
$ OR6 b
1,b
2,b
3,b
4,b
5,b
6 ) )
by Def23, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
func NAND6 c
1,c
2,c
3,c
4,c
5,c
6 -> set equals :
Def24:
:: GATE_1:def 24
NOT1 {} if not (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 )
otherwise {} ;
correctness
coherence
( ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 ) implies NOT1 {} is set ) & ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def24 defines NAND6 GATE_1:def 24 :
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 ) implies
NAND6 b
1,b
2,b
3,b
4,b
5,b
6 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 implies
NAND6 b
1,b
2,b
3,b
4,b
5,b
6 = {} ) );
theorem Th32: :: GATE_1:32
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( not (
$ NAND6 b
1,b
2,b
3,b
4,b
5,b
6 &
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 ) & not
$ NAND6 b
1,b
2,b
3,b
4,b
5,b
6 ) )
by Def24, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6 be
set ;
func NOR6 c
1,c
2,c
3,c
4,c
5,c
6 -> set equals :
Def25:
:: GATE_1:def 25
NOT1 {} if ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 )
otherwise {} ;
correctness
coherence
( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not NOT1 {} is set ) & ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def25 defines NOR6 GATE_1:def 25 :
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
NOR6 b
1,b
2,b
3,b
4,b
5,b
6 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 ) implies
NOR6 b
1,b
2,b
3,b
4,b
5,b
6 = {} ) );
theorem Th33: :: GATE_1:33
for b
1, b
2, b
3, b
4, b
5, b
6 being
set holds
(
$ NOR6 b
1,b
2,b
3,b
4,b
5,b
6 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 ) )
by Def25, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func AND7 c
1,c
2,c
3,c
4,c
5,c
6,c
7 -> set equals :
Def26:
:: GATE_1:def 26
NOT1 {} if (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 &
$ a
7 )
otherwise {} ;
correctness
coherence
( ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 implies NOT1 {} is set ) & ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def26 defines AND7 GATE_1:def 26 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 implies
AND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 ) implies
AND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = {} ) );
theorem Th34: :: GATE_1:34
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
(
$ AND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 iff (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 ) )
by Def26, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func OR7 c
1,c
2,c
3,c
4,c
5,c
6,c
7 -> set equals :
Def27:
:: GATE_1:def 27
NOT1 {} if not ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 & not
$ a
7 )
otherwise {} ;
correctness
coherence
( ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 ) implies NOT1 {} is set ) & not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def27 defines OR7 GATE_1:def 27 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 ) implies
OR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
OR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = {} ) );
theorem Th35: :: GATE_1:35
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( not (
$ OR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 & not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 ) & not
$ OR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 ) )
by Def27, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func NAND7 c
1,c
2,c
3,c
4,c
5,c
6,c
7 -> set equals :
Def28:
:: GATE_1:def 28
NOT1 {} if not (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 &
$ a
7 )
otherwise {} ;
correctness
coherence
( ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 ) implies NOT1 {} is set ) & ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def28 defines NAND7 GATE_1:def 28 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 ) implies
NAND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 implies
NAND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = {} ) );
theorem Th36: :: GATE_1:36
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( not (
$ NAND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 &
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 ) & not
$ NAND7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 ) )
by Def28, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func NOR7 c
1,c
2,c
3,c
4,c
5,c
6,c
7 -> set equals :
Def29:
:: GATE_1:def 29
NOT1 {} if ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 & not
$ a
7 )
otherwise {} ;
correctness
coherence
( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not NOT1 {} is set ) & ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def29 defines NOR7 GATE_1:def 29 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
NOR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 ) implies
NOR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 = {} ) );
theorem Th37: :: GATE_1:37
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
(
$ NOR7 b
1,b
2,b
3,b
4,b
5,b
6,b
7 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 ) )
by Def29, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
func AND8 c
1,c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> set equals :
Def30:
:: GATE_1:def 30
NOT1 {} if (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 &
$ a
7 &
$ a
8 )
otherwise {} ;
correctness
coherence
( ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 & $ c8 implies NOT1 {} is set ) & ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 & $ c8 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def30 defines AND8 GATE_1:def 30 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 implies
AND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = NOT1 {} ) & ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 ) implies
AND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = {} ) );
theorem Th38: :: GATE_1:38
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
$ AND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 iff (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 ) )
by Def30, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
func OR8 c
1,c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> set equals :
Def31:
:: GATE_1:def 31
NOT1 {} if not ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 & not
$ a
7 & not
$ a
8 )
otherwise {} ;
correctness
coherence
( ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not $ c8 ) implies NOT1 {} is set ) & not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not $ c8 & not {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def31 defines OR8 GATE_1:def 31 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 ) implies
OR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = NOT1 {} ) & not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 & not
OR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = {} ) );
theorem Th39: :: GATE_1:39
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( not (
$ OR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 & not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 ) & not ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 ) & not
$ OR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 ) )
by Def31, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
func NAND8 c
1,c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> set equals :
Def32:
:: GATE_1:def 32
NOT1 {} if not (
$ a
1 &
$ a
2 &
$ a
3 &
$ a
4 &
$ a
5 &
$ a
6 &
$ a
7 &
$ a
8 )
otherwise {} ;
correctness
coherence
( ( not ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 & $ c8 ) implies NOT1 {} is set ) & ( $ c1 & $ c2 & $ c3 & $ c4 & $ c5 & $ c6 & $ c7 & $ c8 implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def32 defines NAND8 GATE_1:def 32 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 ) implies
NAND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = NOT1 {} ) & (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 implies
NAND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = {} ) );
theorem Th40: :: GATE_1:40
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( not (
$ NAND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 &
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 ) & not ( not (
$ b
1 &
$ b
2 &
$ b
3 &
$ b
4 &
$ b
5 &
$ b
6 &
$ b
7 &
$ b
8 ) & not
$ NAND8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 ) )
by Def32, Th5;
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8 be
set ;
func NOR8 c
1,c
2,c
3,c
4,c
5,c
6,c
7,c
8 -> set equals :
Def33:
:: GATE_1:def 33
NOT1 {} if ( not
$ a
1 & not
$ a
2 & not
$ a
3 & not
$ a
4 & not
$ a
5 & not
$ a
6 & not
$ a
7 & not
$ a
8 )
otherwise {} ;
correctness
coherence
( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not $ c8 & not NOT1 {} is set ) & ( not ( not $ c1 & not $ c2 & not $ c3 & not $ c4 & not $ c5 & not $ c6 & not $ c7 & not $ c8 ) implies {} is set ) );
consistency
for b1 being set holds
verum;
;
end;
:: deftheorem Def33 defines NOR8 GATE_1:def 33 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 & not
NOR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = NOT1 {} ) & ( not ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 ) implies
NOR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 = {} ) );
theorem Th41: :: GATE_1:41
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
set holds
(
$ NOR8 b
1,b
2,b
3,b
4,b
5,b
6,b
7,b
8 iff ( not
$ b
1 & not
$ b
2 & not
$ b
3 & not
$ b
4 & not
$ b
5 & not
$ b
6 & not
$ b
7 & not
$ b
8 ) )
by Def33, Th5;
theorem Th42: :: GATE_1:42
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19 being
set holds
( not (
$ MAJ3 b
2,b
6,b
1 & not
$ b
10 ) & not (
$ MAJ3 b
3,b
7,b
10 & not
$ b
11 ) & not (
$ MAJ3 b
4,b
8,b
11 & not
$ b
12 ) & not (
$ MAJ3 b
5,b
9,b
12 & not
$ b
13 ) & not (
$ b
14 & not
$ OR2 b
2,b
6 ) & not (
$ b
15 & not
$ OR2 b
3,b
7 ) & not (
$ b
16 & not
$ OR2 b
4,b
8 ) & not (
$ b
17 & not
$ OR2 b
5,b
9 ) & not (
$ b
18 & not
$ AND5 b
1,b
14,b
15,b
16,b
17 ) & not (
$ b
19 & not
$ OR2 b
13,b
18 ) & not (
$ OR2 b
13,b
18 & not
$ b
19 ) implies ( not (
$ b
13 & not
$ b
19 ) & not (
$ b
19 & not
$ b
13 ) ) )
:: deftheorem Def34 defines MODADD2 GATE_1:def 34 :
for b
1, b
2 being
set holds
( ( not ( not
$ b
1 & not
$ b
2 ) & not (
$ b
1 &
$ b
2 ) implies
MODADD2 b
1,b
2 = NOT1 {} ) & ( not ( not ( not
$ b
1 & not
$ b
2 ) & not (
$ b
1 &
$ b
2 ) ) implies
MODADD2 b
1,b
2 = {} ) );
theorem Th43: :: GATE_1:43
for b
1, b
2 being
set holds
(
$ MODADD2 b
1,b
2 iff ( not ( not
$ b
1 & not
$ b
2 ) & not (
$ b
1 &
$ b
2 ) ) )
by Def34, Th5;
notation
let c
1, c
2, c
3 be
set ;
synonym ADD1 c
1,c
2,c
3 for XOR3 c
1,c
2,c
3;
synonym CARR1 c
1,c
2,c
3 for MAJ3 c
1,c
2,c
3;
end;
definition
let c
1, c
2, c
3, c
4, c
5 be
set ;
canceled;canceled;func ADD2 c
3,c
4,c
1,c
2,c
5 -> set equals :: GATE_1:def 37
XOR3 a
3,a
4,
(CARR1 a1,a2,a5);
coherence
XOR3 c3,c4,(CARR1 c1,c2,c5) is set
;
end;
:: deftheorem Def35 GATE_1:def 35 :
canceled;
:: deftheorem Def36 GATE_1:def 36 :
canceled;
:: deftheorem Def37 defines ADD2 GATE_1:def 37 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
ADD2 b
3,b
4,b
1,b
2,b
5 = XOR3 b
3,b
4,
(CARR1 b1,b2,b5);
definition
let c
1, c
2, c
3, c
4, c
5 be
set ;
func CARR2 c
3,c
4,c
1,c
2,c
5 -> set equals :: GATE_1:def 38
MAJ3 a
3,a
4,
(CARR1 a1,a2,a5);
coherence
MAJ3 c3,c4,(CARR1 c1,c2,c5) is set
;
end;
:: deftheorem Def38 defines CARR2 GATE_1:def 38 :
for b
1, b
2, b
3, b
4, b
5 being
set holds
CARR2 b
3,b
4,b
1,b
2,b
5 = MAJ3 b
3,b
4,
(CARR1 b1,b2,b5);
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func ADD3 c
5,c
6,c
3,c
4,c
1,c
2,c
7 -> set equals :: GATE_1:def 39
XOR3 a
5,a
6,
(CARR2 a3,a4,a1,a2,a7);
coherence
XOR3 c5,c6,(CARR2 c3,c4,c1,c2,c7) is set
;
end;
:: deftheorem Def39 defines ADD3 GATE_1:def 39 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
ADD3 b
5,b
6,b
3,b
4,b
1,b
2,b
7 = XOR3 b
5,b
6,
(CARR2 b3,b4,b1,b2,b7);
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7 be
set ;
func CARR3 c
5,c
6,c
3,c
4,c
1,c
2,c
7 -> set equals :: GATE_1:def 40
MAJ3 a
5,a
6,
(CARR2 a3,a4,a1,a2,a7);
coherence
MAJ3 c5,c6,(CARR2 c3,c4,c1,c2,c7) is set
;
end;
:: deftheorem Def40 defines CARR3 GATE_1:def 40 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7 being
set holds
CARR3 b
5,b
6,b
3,b
4,b
1,b
2,b
7 = MAJ3 b
5,b
6,
(CARR2 b3,b4,b1,b2,b7);
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9 be
set ;
func ADD4 c
7,c
8,c
5,c
6,c
3,c
4,c
1,c
2,c
9 -> set equals :: GATE_1:def 41
XOR3 a
7,a
8,
(CARR3 a5,a6,a3,a4,a1,a2,a9);
coherence
XOR3 c7,c8,(CARR3 c5,c6,c3,c4,c1,c2,c9) is set
;
end;
:: deftheorem Def41 defines ADD4 GATE_1:def 41 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
ADD4 b
7,b
8,b
5,b
6,b
3,b
4,b
1,b
2,b
9 = XOR3 b
7,b
8,
(CARR3 b5,b6,b3,b4,b1,b2,b9);
definition
let c
1, c
2, c
3, c
4, c
5, c
6, c
7, c
8, c
9 be
set ;
func CARR4 c
7,c
8,c
5,c
6,c
3,c
4,c
1,c
2,c
9 -> set equals :: GATE_1:def 42
MAJ3 a
7,a
8,
(CARR3 a5,a6,a3,a4,a1,a2,a9);
coherence
MAJ3 c7,c8,(CARR3 c5,c6,c3,c4,c1,c2,c9) is set
;
end;
:: deftheorem Def42 defines CARR4 GATE_1:def 42 :
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
set holds
CARR4 b
7,b
8,b
5,b
6,b
3,b
4,b
1,b
2,b
9 = MAJ3 b
7,b
8,
(CARR3 b5,b6,b3,b4,b1,b2,b9);
theorem Th44: :: GATE_1:44
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13, b
14, b
15, b
16, b
17, b
18, b
19, b
20, b
21, b
22, b
23, b
24, b
25, b
26, b
27, b
28, b
29, b
30, b
31, b
32, b
33, b
34, b
35, b
36, b
37, b
38, b
39, b
40, b
41 being
set holds
not ( not (
$ b
11 & not
$ NOR2 b
2,b
3 ) & not (
$ NOR2 b
2,b
3 & not
$ b
11 ) & not (
$ b
12 & not
$ NAND2 b
2,b
3 ) & not (
$ NAND2 b
2,b
3 & not
$ b
12 ) & not (
$ b
13 & not
$ MODADD2 b
2,b
3 ) & not (
$ MODADD2 b
2,b
3 & not
$ b
13 ) & not (
$ b
14 & not
$ NOR2 b
4,b
5 ) & not (
$ NOR2 b
4,b
5 & not
$ b
14 ) & not (
$ b
15 & not
$ NAND2 b
4,b
5 ) & not (
$ NAND2 b
4,b
5 & not
$ b
15 ) & not (
$ b
16 & not
$ MODADD2 b
4,b
5 ) & not (
$ MODADD2 b
4,b
5 & not
$ b
16 ) & not (
$ b
17 & not
$ NOR2 b
6,b
7 ) & not (
$ NOR2 b
6,b
7 & not
$ b
17 ) & not (
$ b
18 & not
$ NAND2 b
6,b
7 ) & not (
$ NAND2 b
6,b
7 & not
$ b
18 ) & not (
$ b
19 & not
$ MODADD2 b
6,b
7 ) & not (
$ MODADD2 b
6,b
7 & not
$ b
19 ) & not (
$ b
20 & not
$ NOR2 b
8,b
9 ) & not (
$ NOR2 b
8,b
9 & not
$ b
20 ) & not (
$ b
21 & not
$ NAND2 b
8,b
9 ) & not (
$ NAND2 b
8,b
9 & not
$ b
21 ) & not (
$ b
22 & not
$ MODADD2 b
8,b
9 ) & not (
$ MODADD2 b
8,b
9 & not
$ b
22 ) & not (
$ b
23 & not
$ NOT1 b
1 ) & not (
$ NOT1 b
1 & not
$ b
23 ) & not (
$ b
24 & not
$ NOT1 b
23 ) & not (
$ NOT1 b
23 & not
$ b
24 ) & not (
$ b
38 & not
$ XOR2 b
24,b
13 ) & not (
$ XOR2 b
24,b
13 & not
$ b
38 ) & not (
$ b
25 & not
$ AND2 b
23,b
12 ) & not (
$ AND2 b
23,b
12 & not
$ b
25 ) & not (
$ b
26 & not
$ NOR2 b
25,b
11 ) & not (
$ NOR2 b
25,b
11 & not
$ b
26 ) & not (
$ b
39 & not
$ XOR2 b
26,b
16 ) & not (
$ XOR2 b
26,b
16 & not
$ b
39 ) & not (
$ b
27 & not
$ AND2 b
11,b
15 ) & not (
$ AND2 b
11,b
15 & not
$ b
27 ) & not (
$ b
28 & not
$ AND3 b
15,b
12,b
23 ) & not (
$ AND3 b
15,b
12,b
23 & not
$ b
28 ) & not (
$ b
29 & not
$ NOR3 b
27,b
28,b
14 ) & not (
$ NOR3 b
27,b
28,b
14 & not
$ b
29 ) & not (
$ b
40 & not
$ XOR2 b
29,b
19 ) & not (
$ XOR2 b
29,b
19 & not
$ b
40 ) & not (
$ b
30 & not
$ AND2 b
14,b
18 ) & not (
$ AND2 b
14,b
18 & not
$ b
30 ) & not (
$ b
31 & not
$ AND3 b
11,b
18,b
15 ) & not (
$ AND3 b
11,b
18,b
15 & not
$ b
31 ) & not (
$ b
32 & not
$ AND4 b
18,b
15,b
12,b
23 ) & not (
$ AND4 b
18,b
15,b
12,b
23 & not
$ b
32 ) & not (
$ b
33 & not
$ NOR4 b
30,b
31,b
32,b
17 ) & not (
$ NOR4 b
30,b
31,b
32,b
17 & not
$ b
33 ) & not (
$ b
41 & not
$ XOR2 b
33,b
22 ) & not (
$ XOR2 b
33,b
22 & not
$ b
41 ) & not (
$ b
34 & not
$ AND2 b
17,b
21 ) & not (
$ AND2 b
17,b
21 & not
$ b
34 ) & not (
$ b
35 & not
$ AND3 b
14,b
21,b
18 ) & not (
$ AND3 b
14,b
21,b
18 & not
$ b
35 ) & not (
$ b
36 & not
$ AND4 b
11,b
21,b
18,b
15 ) & not (
$ AND4 b
11,b
21,b
18,b
15 & not
$ b
36 ) & not (
$ b
37 & not
$ AND5 b
21,b
18,b
15,b
12,b
23 ) & not (
$ AND5 b
21,b
18,b
15,b
12,b
23 & not
$ b
37 ) & not (
$ b
10 & not
$ NOR5 b
20,b
34,b
35,b
36,b
37 ) & not (
$ NOR5 b
20,b
34,b
35,b
36,b
37 & not
$ b
10 ) & not ( not (
$ b
38 & not
$ ADD1 b
2,b
3,b
1 ) & not (
$ ADD1 b
2,b
3,b
1 & not
$ b
38 ) & not (
$ b
39 & not
$ ADD2 b
4,b
5,b
2,b
3,b
1 ) & not (
$ ADD2 b
4,b
5,b
2,b
3,b
1 & not
$ b
39 ) & not (
$ b
40 & not
$ ADD3 b
6,b
7,b
4,b
5,b
2,b
3,b
1 ) & not (
$ ADD3 b
6,b
7,b
4,b
5,b
2,b
3,b
1 & not
$ b
40 ) & not (
$ b
41 & not
$ ADD4 b
8,b
9,b
6,b
7,b
4,b
5,b
2,b
3,b
1 ) & not (
$ ADD4 b
8,b
9,b
6,b
7,b
4,b
5,b
2,b
3,b
1 & not
$ b
41 ) & not (
$ b
10 & not
$ CARR4 b
8,b
9,b
6,b
7,b
4,b
5,b
2,b
3,b
1 ) & not (
$ CARR4 b
8,b
9,b
6,b
7,b
4,b
5,b
2,b
3,b
1 & not
$ b
10 ) ) )