:: GATE_2 semantic presentation
theorem Th1: :: GATE_2:1
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 being
set holds
not ( not (
$ b
1 & not
$ AND3 (NOT1 b19),
(NOT1 b18),
(NOT1 b17) ) & not (
$ AND3 (NOT1 b19),
(NOT1 b18),
(NOT1 b17) & not
$ b
1 ) & not (
$ b
2 & not
$ AND3 (NOT1 b19),
(NOT1 b18),b
17 ) & not (
$ AND3 (NOT1 b19),
(NOT1 b18),b
17 & not
$ b
2 ) & not (
$ b
3 & not
$ AND3 (NOT1 b19),b
18,
(NOT1 b17) ) & not (
$ AND3 (NOT1 b19),b
18,
(NOT1 b17) & not
$ b
3 ) & not (
$ b
4 & not
$ AND3 (NOT1 b19),b
18,b
17 ) & not (
$ AND3 (NOT1 b19),b
18,b
17 & not
$ b
4 ) & not (
$ b
5 & not
$ AND3 b
19,
(NOT1 b18),
(NOT1 b17) ) & not (
$ AND3 b
19,
(NOT1 b18),
(NOT1 b17) & not
$ b
5 ) & not (
$ b
6 & not
$ AND3 b
19,
(NOT1 b18),b
17 ) & not (
$ AND3 b
19,
(NOT1 b18),b
17 & not
$ b
6 ) & not (
$ b
7 & not
$ AND3 b
19,b
18,
(NOT1 b17) ) & not (
$ AND3 b
19,b
18,
(NOT1 b17) & not
$ b
7 ) & not (
$ b
8 & not
$ AND3 b
19,b
18,b
17 ) & not (
$ AND3 b
19,b
18,b
17 & not
$ b
8 ) & not (
$ b
9 & not
$ AND3 (NOT1 b22),
(NOT1 b21),
(NOT1 b20) ) & not (
$ AND3 (NOT1 b22),
(NOT1 b21),
(NOT1 b20) & not
$ b
9 ) & not (
$ b
10 & not
$ AND3 (NOT1 b22),
(NOT1 b21),b
20 ) & not (
$ AND3 (NOT1 b22),
(NOT1 b21),b
20 & not
$ b
10 ) & not (
$ b
11 & not
$ AND3 (NOT1 b22),b
21,
(NOT1 b20) ) & not (
$ AND3 (NOT1 b22),b
21,
(NOT1 b20) & not
$ b
11 ) & not (
$ b
12 & not
$ AND3 (NOT1 b22),b
21,b
20 ) & not (
$ AND3 (NOT1 b22),b
21,b
20 & not
$ b
12 ) & not (
$ b
13 & not
$ AND3 b
22,
(NOT1 b21),
(NOT1 b20) ) & not (
$ AND3 b
22,
(NOT1 b21),
(NOT1 b20) & not
$ b
13 ) & not (
$ b
14 & not
$ AND3 b
22,
(NOT1 b21),b
20 ) & not (
$ AND3 b
22,
(NOT1 b21),b
20 & not
$ b
14 ) & not (
$ b
15 & not
$ AND3 b
22,b
21,
(NOT1 b20) ) & not (
$ AND3 b
22,b
21,
(NOT1 b20) & not
$ b
15 ) & not (
$ b
16 & not
$ AND3 b
22,b
21,b
20 ) & not (
$ AND3 b
22,b
21,b
20 & not
$ b
16 ) & not (
$ b
20 & not
$ NOT1 b
17 ) & not (
$ NOT1 b
17 & not
$ b
20 ) & not (
$ b
21 & not
$ XOR2 b
17,b
18 ) & not (
$ XOR2 b
17,b
18 & not
$ b
21 ) & not (
$ b
22 & not
$ OR2 (AND2 b19,(NOT1 b17)),
(AND2 b17,(XOR2 b18,b19)) ) & not (
$ OR2 (AND2 b19,(NOT1 b17)),
(AND2 b17,(XOR2 b18,b19)) & not
$ b
22 ) & not ( not (
$ b
10 & not
$ b
1 ) & not (
$ b
1 & not
$ b
10 ) & not (
$ b
11 & not
$ b
2 ) & not (
$ b
2 & not
$ b
11 ) & not (
$ b
12 & not
$ b
3 ) & not (
$ b
3 & not
$ b
12 ) & not (
$ b
13 & not
$ b
4 ) & not (
$ b
4 & not
$ b
13 ) & not (
$ b
14 & not
$ b
5 ) & not (
$ b
5 & not
$ b
14 ) & not (
$ b
15 & not
$ b
6 ) & not (
$ b
6 & not
$ b
15 ) & not (
$ b
16 & not
$ b
7 ) & not (
$ b
7 & not
$ b
16 ) & not (
$ b
9 & not
$ b
8 ) & not (
$ b
8 & not
$ b
9 ) ) )
theorem Th2: :: GATE_2:2
for b
1, b
2, b
3, b
4 being
set holds
( not (
$ AND3 (AND2 b1,b2),
(AND2 b3,b2),
(AND2 b4,b2) & not
$ AND2 (AND3 b1,b3,b4),b
2 ) & not (
$ AND2 (AND3 b1,b3,b4),b
2 & not
$ AND3 (AND2 b1,b2),
(AND2 b3,b2),
(AND2 b4,b2) ) )
theorem Th3: :: GATE_2:3
for b
1, b
2, b
3, b
4 being
set holds
( not ( not
$ AND2 b
1,b
2 & not
$ OR2 (NOT1 b1),
(NOT1 b2) ) & (
$ OR2 (NOT1 b1),
(NOT1 b2) implies not
$ AND2 b
1,b
2 ) & not (
$ OR2 b
1,b
2 &
$ OR2 b
3,b
2 & not
$ OR2 (AND2 b1,b3),b
2 ) & (
$ OR2 (AND2 b1,b3),b
2 implies (
$ OR2 b
1,b
2 &
$ OR2 b
3,b
2 ) ) & not (
$ OR2 b
1,b
2 &
$ OR2 b
3,b
2 &
$ OR2 b
4,b
2 & not
$ OR2 (AND3 b1,b3,b4),b
2 ) & (
$ OR2 (AND3 b1,b3,b4),b
2 implies (
$ OR2 b
1,b
2 &
$ OR2 b
3,b
2 &
$ OR2 b
4,b
2 ) ) & not (
$ OR2 b
1,b
2 & not (
$ b
1 & not
$ b
3 ) & not (
$ b
3 & not
$ b
1 ) & not
$ OR2 b
3,b
2 ) )
theorem Th4: :: GATE_2:4
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 being
set holds
not ( not (
$ b
1 & not
$ AND3 (NOT1 b19),
(NOT1 b18),
(NOT1 b17) ) & not (
$ AND3 (NOT1 b19),
(NOT1 b18),
(NOT1 b17) & not
$ b
1 ) & not (
$ b
2 & not
$ AND3 (NOT1 b19),
(NOT1 b18),b
17 ) & not (
$ AND3 (NOT1 b19),
(NOT1 b18),b
17 & not
$ b
2 ) & not (
$ b
3 & not
$ AND3 (NOT1 b19),b
18,
(NOT1 b17) ) & not (
$ AND3 (NOT1 b19),b
18,
(NOT1 b17) & not
$ b
3 ) & not (
$ b
4 & not
$ AND3 (NOT1 b19),b
18,b
17 ) & not (
$ AND3 (NOT1 b19),b
18,b
17 & not
$ b
4 ) & not (
$ b
5 & not
$ AND3 b
19,
(NOT1 b18),
(NOT1 b17) ) & not (
$ AND3 b
19,
(NOT1 b18),
(NOT1 b17) & not
$ b
5 ) & not (
$ b
6 & not
$ AND3 b
19,
(NOT1 b18),b
17 ) & not (
$ AND3 b
19,
(NOT1 b18),b
17 & not
$ b
6 ) & not (
$ b
7 & not
$ AND3 b
19,b
18,
(NOT1 b17) ) & not (
$ AND3 b
19,b
18,
(NOT1 b17) & not
$ b
7 ) & not (
$ b
8 & not
$ AND3 b
19,b
18,b
17 ) & not (
$ AND3 b
19,b
18,b
17 & not
$ b
8 ) & not (
$ b
9 & not
$ AND3 (NOT1 b22),
(NOT1 b21),
(NOT1 b20) ) & not (
$ AND3 (NOT1 b22),
(NOT1 b21),
(NOT1 b20) & not
$ b
9 ) & not (
$ b
10 & not
$ AND3 (NOT1 b22),
(NOT1 b21),b
20 ) & not (
$ AND3 (NOT1 b22),
(NOT1 b21),b
20 & not
$ b
10 ) & not (
$ b
11 & not
$ AND3 (NOT1 b22),b
21,
(NOT1 b20) ) & not (
$ AND3 (NOT1 b22),b
21,
(NOT1 b20) & not
$ b
11 ) & not (
$ b
12 & not
$ AND3 (NOT1 b22),b
21,b
20 ) & not (
$ AND3 (NOT1 b22),b
21,b
20 & not
$ b
12 ) & not (
$ b
13 & not
$ AND3 b
22,
(NOT1 b21),
(NOT1 b20) ) & not (
$ AND3 b
22,
(NOT1 b21),
(NOT1 b20) & not
$ b
13 ) & not (
$ b
14 & not
$ AND3 b
22,
(NOT1 b21),b
20 ) & not (
$ AND3 b
22,
(NOT1 b21),b
20 & not
$ b
14 ) & not (
$ b
15 & not
$ AND3 b
22,b
21,
(NOT1 b20) ) & not (
$ AND3 b
22,b
21,
(NOT1 b20) & not
$ b
15 ) & not (
$ b
16 & not
$ AND3 b
22,b
21,b
20 ) & not (
$ AND3 b
22,b
21,b
20 & not
$ b
16 ) & not (
$ b
20 & not
$ AND2 (NOT1 b17),b
23 ) & not (
$ AND2 (NOT1 b17),b
23 & not
$ b
20 ) & not (
$ b
21 & not
$ AND2 (XOR2 b17,b18),b
23 ) & not (
$ AND2 (XOR2 b17,b18),b
23 & not
$ b
21 ) & not (
$ b
22 & not
$ AND2 (OR2 (AND2 b19,(NOT1 b17)),(AND2 b17,(XOR2 b18,b19))),b
23 ) & not (
$ AND2 (OR2 (AND2 b19,(NOT1 b17)),(AND2 b17,(XOR2 b18,b19))),b
23 & not
$ b
22 ) & not ( not (
$ b
10 & not
$ AND2 b
1,b
23 ) & not (
$ AND2 b
1,b
23 & not
$ b
10 ) & not (
$ b
11 & not
$ AND2 b
2,b
23 ) & not (
$ AND2 b
2,b
23 & not
$ b
11 ) & not (
$ b
12 & not
$ AND2 b
3,b
23 ) & not (
$ AND2 b
3,b
23 & not
$ b
12 ) & not (
$ b
13 & not
$ AND2 b
4,b
23 ) & not (
$ AND2 b
4,b
23 & not
$ b
13 ) & not (
$ b
14 & not
$ AND2 b
5,b
23 ) & not (
$ AND2 b
5,b
23 & not
$ b
14 ) & not (
$ b
15 & not
$ AND2 b
6,b
23 ) & not (
$ AND2 b
6,b
23 & not
$ b
15 ) & not (
$ b
16 & not
$ AND2 b
7,b
23 ) & not (
$ AND2 b
7,b
23 & not
$ b
16 ) & not (
$ b
9 & not
$ OR2 b
8,
(NOT1 b23) ) & not (
$ OR2 b
8,
(NOT1 b23) & not
$ b
9 ) ) )