:: GATE_4 semantic presentation

theorem Th1: :: GATE_4:1
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38 being set holds
( $ b1 & $ b13 & not ( $ b26 & not $ XOR2 b38,(AND2 b1,b25) ) & not ( $ XOR2 b38,(AND2 b1,b25) & not $ b26 ) & not ( $ b27 & not $ XOR2 b14,(AND2 b2,b25) ) & not ( $ XOR2 b14,(AND2 b2,b25) & not $ b27 ) & not ( $ b28 & not $ XOR2 b15,(AND2 b3,b25) ) & not ( $ XOR2 b15,(AND2 b3,b25) & not $ b28 ) & not ( $ b29 & not $ XOR2 b16,(AND2 b4,b25) ) & not ( $ XOR2 b16,(AND2 b4,b25) & not $ b29 ) & not ( $ b30 & not $ XOR2 b17,(AND2 b5,b25) ) & not ( $ XOR2 b17,(AND2 b5,b25) & not $ b30 ) & not ( $ b31 & not $ XOR2 b18,(AND2 b6,b25) ) & not ( $ XOR2 b18,(AND2 b6,b25) & not $ b31 ) & not ( $ b32 & not $ XOR2 b19,(AND2 b7,b25) ) & not ( $ XOR2 b19,(AND2 b7,b25) & not $ b32 ) & not ( $ b33 & not $ XOR2 b20,(AND2 b8,b25) ) & not ( $ XOR2 b20,(AND2 b8,b25) & not $ b33 ) & not ( $ b34 & not $ XOR2 b21,(AND2 b9,b25) ) & not ( $ XOR2 b21,(AND2 b9,b25) & not $ b34 ) & not ( $ b35 & not $ XOR2 b22,(AND2 b10,b25) ) & not ( $ XOR2 b22,(AND2 b10,b25) & not $ b35 ) & not ( $ b36 & not $ XOR2 b23,(AND2 b11,b25) ) & not ( $ XOR2 b23,(AND2 b11,b25) & not $ b36 ) & not ( $ b37 & not $ XOR2 b24,(AND2 b12,b25) ) & not ( $ XOR2 b24,(AND2 b12,b25) & not $ b37 ) implies ( not ( $ b25 & not $ AND2 b13,b25 ) & not ( $ AND2 b13,b25 & not $ b25 ) & not ( $ b24 & not $ XOR2 b37,(AND2 b12,b25) ) & not ( $ XOR2 b37,(AND2 b12,b25) & not $ b24 ) & not ( $ b23 & not $ XOR2 b36,(AND2 b11,b25) ) & not ( $ XOR2 b36,(AND2 b11,b25) & not $ b23 ) & not ( $ b22 & not $ XOR2 b35,(AND2 b10,b25) ) & not ( $ XOR2 b35,(AND2 b10,b25) & not $ b22 ) & not ( $ b21 & not $ XOR2 b34,(AND2 b9,b25) ) & not ( $ XOR2 b34,(AND2 b9,b25) & not $ b21 ) & not ( $ b20 & not $ XOR2 b33,(AND2 b8,b25) ) & not ( $ XOR2 b33,(AND2 b8,b25) & not $ b20 ) & not ( $ b19 & not $ XOR2 b32,(AND2 b7,b25) ) & not ( $ XOR2 b32,(AND2 b7,b25) & not $ b19 ) & not ( $ b18 & not $ XOR2 b31,(AND2 b6,b25) ) & not ( $ XOR2 b31,(AND2 b6,b25) & not $ b18 ) & not ( $ b17 & not $ XOR2 b30,(AND2 b5,b25) ) & not ( $ XOR2 b30,(AND2 b5,b25) & not $ b17 ) & not ( $ b16 & not $ XOR2 b29,(AND2 b4,b25) ) & not ( $ XOR2 b29,(AND2 b4,b25) & not $ b16 ) & not ( $ b15 & not $ XOR2 b28,(AND2 b3,b25) ) & not ( $ XOR2 b28,(AND2 b3,b25) & not $ b15 ) & not ( $ b14 & not $ XOR2 b27,(AND2 b2,b25) ) & not ( $ XOR2 b27,(AND2 b2,b25) & not $ b14 ) & not ( $ b38 & not $ XOR2 b26,(AND2 b1,b25) ) & not ( $ XOR2 b26,(AND2 b1,b25) & not $ b38 ) ) )
proof end;

theorem Th2: :: GATE_4:2
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38, b39, b40, b41, b42, b43, b44, b45, b46, b47, b48, b49, b50 being set holds
( $ b1 & $ b17 & not ( $ b34 & not $ XOR2 b50,(AND2 b1,b33) ) & not ( $ XOR2 b50,(AND2 b1,b33) & not $ b34 ) & not ( $ b35 & not $ XOR2 b18,(AND2 b2,b33) ) & not ( $ XOR2 b18,(AND2 b2,b33) & not $ b35 ) & not ( $ b36 & not $ XOR2 b19,(AND2 b3,b33) ) & not ( $ XOR2 b19,(AND2 b3,b33) & not $ b36 ) & not ( $ b37 & not $ XOR2 b20,(AND2 b4,b33) ) & not ( $ XOR2 b20,(AND2 b4,b33) & not $ b37 ) & not ( $ b38 & not $ XOR2 b21,(AND2 b5,b33) ) & not ( $ XOR2 b21,(AND2 b5,b33) & not $ b38 ) & not ( $ b39 & not $ XOR2 b22,(AND2 b6,b33) ) & not ( $ XOR2 b22,(AND2 b6,b33) & not $ b39 ) & not ( $ b40 & not $ XOR2 b23,(AND2 b7,b33) ) & not ( $ XOR2 b23,(AND2 b7,b33) & not $ b40 ) & not ( $ b41 & not $ XOR2 b24,(AND2 b8,b33) ) & not ( $ XOR2 b24,(AND2 b8,b33) & not $ b41 ) & not ( $ b42 & not $ XOR2 b25,(AND2 b9,b33) ) & not ( $ XOR2 b25,(AND2 b9,b33) & not $ b42 ) & not ( $ b43 & not $ XOR2 b26,(AND2 b10,b33) ) & not ( $ XOR2 b26,(AND2 b10,b33) & not $ b43 ) & not ( $ b44 & not $ XOR2 b27,(AND2 b11,b33) ) & not ( $ XOR2 b27,(AND2 b11,b33) & not $ b44 ) & not ( $ b45 & not $ XOR2 b28,(AND2 b12,b33) ) & not ( $ XOR2 b28,(AND2 b12,b33) & not $ b45 ) & not ( $ b46 & not $ XOR2 b29,(AND2 b13,b33) ) & not ( $ XOR2 b29,(AND2 b13,b33) & not $ b46 ) & not ( $ b47 & not $ XOR2 b30,(AND2 b14,b33) ) & not ( $ XOR2 b30,(AND2 b14,b33) & not $ b47 ) & not ( $ b48 & not $ XOR2 b31,(AND2 b15,b33) ) & not ( $ XOR2 b31,(AND2 b15,b33) & not $ b48 ) & not ( $ b49 & not $ XOR2 b32,(AND2 b16,b33) ) & not ( $ XOR2 b32,(AND2 b16,b33) & not $ b49 ) implies ( not ( $ b33 & not $ AND2 b17,b33 ) & not ( $ AND2 b17,b33 & not $ b33 ) & not ( $ b32 & not $ XOR2 b49,(AND2 b16,b33) ) & not ( $ XOR2 b49,(AND2 b16,b33) & not $ b32 ) & not ( $ b31 & not $ XOR2 b48,(AND2 b15,b33) ) & not ( $ XOR2 b48,(AND2 b15,b33) & not $ b31 ) & not ( $ b30 & not $ XOR2 b47,(AND2 b14,b33) ) & not ( $ XOR2 b47,(AND2 b14,b33) & not $ b30 ) & not ( $ b29 & not $ XOR2 b46,(AND2 b13,b33) ) & not ( $ XOR2 b46,(AND2 b13,b33) & not $ b29 ) & not ( $ b28 & not $ XOR2 b45,(AND2 b12,b33) ) & not ( $ XOR2 b45,(AND2 b12,b33) & not $ b28 ) & not ( $ b27 & not $ XOR2 b44,(AND2 b11,b33) ) & not ( $ XOR2 b44,(AND2 b11,b33) & not $ b27 ) & not ( $ b26 & not $ XOR2 b43,(AND2 b10,b33) ) & not ( $ XOR2 b43,(AND2 b10,b33) & not $ b26 ) & not ( $ b25 & not $ XOR2 b42,(AND2 b9,b33) ) & not ( $ XOR2 b42,(AND2 b9,b33) & not $ b25 ) & not ( $ b24 & not $ XOR2 b41,(AND2 b8,b33) ) & not ( $ XOR2 b41,(AND2 b8,b33) & not $ b24 ) & not ( $ b23 & not $ XOR2 b40,(AND2 b7,b33) ) & not ( $ XOR2 b40,(AND2 b7,b33) & not $ b23 ) & not ( $ b22 & not $ XOR2 b39,(AND2 b6,b33) ) & not ( $ XOR2 b39,(AND2 b6,b33) & not $ b22 ) & not ( $ b21 & not $ XOR2 b38,(AND2 b5,b33) ) & not ( $ XOR2 b38,(AND2 b5,b33) & not $ b21 ) & not ( $ b20 & not $ XOR2 b37,(AND2 b4,b33) ) & not ( $ XOR2 b37,(AND2 b4,b33) & not $ b20 ) & not ( $ b19 & not $ XOR2 b36,(AND2 b3,b33) ) & not ( $ XOR2 b36,(AND2 b3,b33) & not $ b19 ) & not ( $ b18 & not $ XOR2 b35,(AND2 b2,b33) ) & not ( $ XOR2 b35,(AND2 b2,b33) & not $ b18 ) & not ( $ b50 & not $ XOR2 b34,(AND2 b1,b33) ) & not ( $ XOR2 b34,(AND2 b1,b33) & not $ b50 ) ) )
proof end;

theorem Th3: :: GATE_4:3
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38, b39 being set holds
( $ b1 & $ b13 & not $ b38 & not ( $ b26 & not $ XOR2 b39,b25 ) & not ( $ XOR2 b39,b25 & not $ b26 ) & not ( $ b27 & not $ XOR2 b14,(AND2 b2,b26) ) & not ( $ XOR2 b14,(AND2 b2,b26) & not $ b27 ) & not ( $ b28 & not $ XOR2 b15,(AND2 b3,b26) ) & not ( $ XOR2 b15,(AND2 b3,b26) & not $ b28 ) & not ( $ b29 & not $ XOR2 b16,(AND2 b4,b26) ) & not ( $ XOR2 b16,(AND2 b4,b26) & not $ b29 ) & not ( $ b30 & not $ XOR2 b17,(AND2 b5,b26) ) & not ( $ XOR2 b17,(AND2 b5,b26) & not $ b30 ) & not ( $ b31 & not $ XOR2 b18,(AND2 b6,b26) ) & not ( $ XOR2 b18,(AND2 b6,b26) & not $ b31 ) & not ( $ b32 & not $ XOR2 b19,(AND2 b7,b26) ) & not ( $ XOR2 b19,(AND2 b7,b26) & not $ b32 ) & not ( $ b33 & not $ XOR2 b20,(AND2 b8,b26) ) & not ( $ XOR2 b20,(AND2 b8,b26) & not $ b33 ) & not ( $ b34 & not $ XOR2 b21,(AND2 b9,b26) ) & not ( $ XOR2 b21,(AND2 b9,b26) & not $ b34 ) & not ( $ b35 & not $ XOR2 b22,(AND2 b10,b26) ) & not ( $ XOR2 b22,(AND2 b10,b26) & not $ b35 ) & not ( $ b36 & not $ XOR2 b23,(AND2 b11,b26) ) & not ( $ XOR2 b23,(AND2 b11,b26) & not $ b36 ) & not ( $ b37 & not $ XOR2 b24,(AND2 b12,b26) ) & not ( $ XOR2 b24,(AND2 b12,b26) & not $ b37 ) implies ( not ( $ b37 & not $ XOR2 (XOR2 b24,(AND2 b12,b25)),(XOR2 b38,(AND2 b12,b39)) ) & not ( $ XOR2 (XOR2 b24,(AND2 b12,b25)),(XOR2 b38,(AND2 b12,b39)) & not $ b37 ) & not ( $ b36 & not $ XOR2 (XOR2 b23,(AND2 b11,b25)),(XOR2 b38,(AND2 b11,b39)) ) & not ( $ XOR2 (XOR2 b23,(AND2 b11,b25)),(XOR2 b38,(AND2 b11,b39)) & not $ b36 ) & not ( $ b35 & not $ XOR2 (XOR2 b22,(AND2 b10,b25)),(XOR2 b38,(AND2 b10,b39)) ) & not ( $ XOR2 (XOR2 b22,(AND2 b10,b25)),(XOR2 b38,(AND2 b10,b39)) & not $ b35 ) & not ( $ b34 & not $ XOR2 (XOR2 b21,(AND2 b9,b25)),(XOR2 b38,(AND2 b9,b39)) ) & not ( $ XOR2 (XOR2 b21,(AND2 b9,b25)),(XOR2 b38,(AND2 b9,b39)) & not $ b34 ) & not ( $ b33 & not $ XOR2 (XOR2 b20,(AND2 b8,b25)),(XOR2 b38,(AND2 b8,b39)) ) & not ( $ XOR2 (XOR2 b20,(AND2 b8,b25)),(XOR2 b38,(AND2 b8,b39)) & not $ b33 ) & not ( $ b32 & not $ XOR2 (XOR2 b19,(AND2 b7,b25)),(XOR2 b38,(AND2 b7,b39)) ) & not ( $ XOR2 (XOR2 b19,(AND2 b7,b25)),(XOR2 b38,(AND2 b7,b39)) & not $ b32 ) & not ( $ b31 & not $ XOR2 (XOR2 b18,(AND2 b6,b25)),(XOR2 b38,(AND2 b6,b39)) ) & not ( $ XOR2 (XOR2 b18,(AND2 b6,b25)),(XOR2 b38,(AND2 b6,b39)) & not $ b31 ) & not ( $ b30 & not $ XOR2 (XOR2 b17,(AND2 b5,b25)),(XOR2 b38,(AND2 b5,b39)) ) & not ( $ XOR2 (XOR2 b17,(AND2 b5,b25)),(XOR2 b38,(AND2 b5,b39)) & not $ b30 ) & not ( $ b29 & not $ XOR2 (XOR2 b16,(AND2 b4,b25)),(XOR2 b38,(AND2 b4,b39)) ) & not ( $ XOR2 (XOR2 b16,(AND2 b4,b25)),(XOR2 b38,(AND2 b4,b39)) & not $ b29 ) & not ( $ b28 & not $ XOR2 (XOR2 b15,(AND2 b3,b25)),(XOR2 b38,(AND2 b3,b39)) ) & not ( $ XOR2 (XOR2 b15,(AND2 b3,b25)),(XOR2 b38,(AND2 b3,b39)) & not $ b28 ) & not ( $ b27 & not $ XOR2 (XOR2 b14,(AND2 b2,b25)),(XOR2 b38,(AND2 b2,b39)) ) & not ( $ XOR2 (XOR2 b14,(AND2 b2,b25)),(XOR2 b38,(AND2 b2,b39)) & not $ b27 ) & not ( $ b26 & not $ XOR2 (XOR2 b38,(AND2 b1,b25)),(XOR2 b38,(AND2 b1,b39)) ) & not ( $ XOR2 (XOR2 b38,(AND2 b1,b25)),(XOR2 b38,(AND2 b1,b39)) & not $ b26 ) ) )
proof end;

theorem Th4: :: GATE_4:4
for b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38, b39, b40, b41, b42, b43, b44, b45, b46, b47, b48, b49, b50, b51 being set holds
( $ b1 & $ b17 & not $ b50 & not ( $ b34 & not $ XOR2 b51,b33 ) & not ( $ XOR2 b51,b33 & not $ b34 ) & not ( $ b35 & not $ XOR2 b18,(AND2 b2,b34) ) & not ( $ XOR2 b18,(AND2 b2,b34) & not $ b35 ) & not ( $ b36 & not $ XOR2 b19,(AND2 b3,b34) ) & not ( $ XOR2 b19,(AND2 b3,b34) & not $ b36 ) & not ( $ b37 & not $ XOR2 b20,(AND2 b4,b34) ) & not ( $ XOR2 b20,(AND2 b4,b34) & not $ b37 ) & not ( $ b38 & not $ XOR2 b21,(AND2 b5,b34) ) & not ( $ XOR2 b21,(AND2 b5,b34) & not $ b38 ) & not ( $ b39 & not $ XOR2 b22,(AND2 b6,b34) ) & not ( $ XOR2 b22,(AND2 b6,b34) & not $ b39 ) & not ( $ b40 & not $ XOR2 b23,(AND2 b7,b34) ) & not ( $ XOR2 b23,(AND2 b7,b34) & not $ b40 ) & not ( $ b41 & not $ XOR2 b24,(AND2 b8,b34) ) & not ( $ XOR2 b24,(AND2 b8,b34) & not $ b41 ) & not ( $ b42 & not $ XOR2 b25,(AND2 b9,b34) ) & not ( $ XOR2 b25,(AND2 b9,b34) & not $ b42 ) & not ( $ b43 & not $ XOR2 b26,(AND2 b10,b34) ) & not ( $ XOR2 b26,(AND2 b10,b34) & not $ b43 ) & not ( $ b44 & not $ XOR2 b27,(AND2 b11,b34) ) & not ( $ XOR2 b27,(AND2 b11,b34) & not $ b44 ) & not ( $ b45 & not $ XOR2 b28,(AND2 b12,b34) ) & not ( $ XOR2 b28,(AND2 b12,b34) & not $ b45 ) & not ( $ b46 & not $ XOR2 b29,(AND2 b13,b34) ) & not ( $ XOR2 b29,(AND2 b13,b34) & not $ b46 ) & not ( $ b47 & not $ XOR2 b30,(AND2 b14,b34) ) & not ( $ XOR2 b30,(AND2 b14,b34) & not $ b47 ) & not ( $ b48 & not $ XOR2 b31,(AND2 b15,b34) ) & not ( $ XOR2 b31,(AND2 b15,b34) & not $ b48 ) & not ( $ b49 & not $ XOR2 b32,(AND2 b16,b34) ) & not ( $ XOR2 b32,(AND2 b16,b34) & not $ b49 ) implies ( not ( $ b49 & not $ XOR2 (XOR2 b32,(AND2 b16,b33)),(XOR2 b50,(AND2 b16,b51)) ) & not ( $ XOR2 (XOR2 b32,(AND2 b16,b33)),(XOR2 b50,(AND2 b16,b51)) & not $ b49 ) & not ( $ b48 & not $ XOR2 (XOR2 b31,(AND2 b15,b33)),(XOR2 b50,(AND2 b15,b51)) ) & not ( $ XOR2 (XOR2 b31,(AND2 b15,b33)),(XOR2 b50,(AND2 b15,b51)) & not $ b48 ) & not ( $ b47 & not $ XOR2 (XOR2 b30,(AND2 b14,b33)),(XOR2 b50,(AND2 b14,b51)) ) & not ( $ XOR2 (XOR2 b30,(AND2 b14,b33)),(XOR2 b50,(AND2 b14,b51)) & not $ b47 ) & not ( $ b46 & not $ XOR2 (XOR2 b29,(AND2 b13,b33)),(XOR2 b50,(AND2 b13,b51)) ) & not ( $ XOR2 (XOR2 b29,(AND2 b13,b33)),(XOR2 b50,(AND2 b13,b51)) & not $ b46 ) & not ( $ b45 & not $ XOR2 (XOR2 b28,(AND2 b12,b33)),(XOR2 b50,(AND2 b12,b51)) ) & not ( $ XOR2 (XOR2 b28,(AND2 b12,b33)),(XOR2 b50,(AND2 b12,b51)) & not $ b45 ) & not ( $ b44 & not $ XOR2 (XOR2 b27,(AND2 b11,b33)),(XOR2 b50,(AND2 b11,b51)) ) & not ( $ XOR2 (XOR2 b27,(AND2 b11,b33)),(XOR2 b50,(AND2 b11,b51)) & not $ b44 ) & not ( $ b43 & not $ XOR2 (XOR2 b26,(AND2 b10,b33)),(XOR2 b50,(AND2 b10,b51)) ) & not ( $ XOR2 (XOR2 b26,(AND2 b10,b33)),(XOR2 b50,(AND2 b10,b51)) & not $ b43 ) & not ( $ b42 & not $ XOR2 (XOR2 b25,(AND2 b9,b33)),(XOR2 b50,(AND2 b9,b51)) ) & not ( $ XOR2 (XOR2 b25,(AND2 b9,b33)),(XOR2 b50,(AND2 b9,b51)) & not $ b42 ) & not ( $ b41 & not $ XOR2 (XOR2 b24,(AND2 b8,b33)),(XOR2 b50,(AND2 b8,b51)) ) & not ( $ XOR2 (XOR2 b24,(AND2 b8,b33)),(XOR2 b50,(AND2 b8,b51)) & not $ b41 ) & not ( $ b40 & not $ XOR2 (XOR2 b23,(AND2 b7,b33)),(XOR2 b50,(AND2 b7,b51)) ) & not ( $ XOR2 (XOR2 b23,(AND2 b7,b33)),(XOR2 b50,(AND2 b7,b51)) & not $ b40 ) & not ( $ b39 & not $ XOR2 (XOR2 b22,(AND2 b6,b33)),(XOR2 b50,(AND2 b6,b51)) ) & not ( $ XOR2 (XOR2 b22,(AND2 b6,b33)),(XOR2 b50,(AND2 b6,b51)) & not $ b39 ) & not ( $ b38 & not $ XOR2 (XOR2 b21,(AND2 b5,b33)),(XOR2 b50,(AND2 b5,b51)) ) & not ( $ XOR2 (XOR2 b21,(AND2 b5,b33)),(XOR2 b50,(AND2 b5,b51)) & not $ b38 ) & not ( $ b37 & not $ XOR2 (XOR2 b20,(AND2 b4,b33)),(XOR2 b50,(AND2 b4,b51)) ) & not ( $ XOR2 (XOR2 b20,(AND2 b4,b33)),(XOR2 b50,(AND2 b4,b51)) & not $ b37 ) & not ( $ b36 & not $ XOR2 (XOR2 b19,(AND2 b3,b33)),(XOR2 b50,(AND2 b3,b51)) ) & not ( $ XOR2 (XOR2 b19,(AND2 b3,b33)),(XOR2 b50,(AND2 b3,b51)) & not $ b36 ) & not ( $ b35 & not $ XOR2 (XOR2 b18,(AND2 b2,b33)),(XOR2 b50,(AND2 b2,b51)) ) & not ( $ XOR2 (XOR2 b18,(AND2 b2,b33)),(XOR2 b50,(AND2 b2,b51)) & not $ b35 ) & not ( $ b34 & not $ XOR2 (XOR2 b50,(AND2 b1,b33)),(XOR2 b50,(AND2 b1,b51)) ) & not ( $ XOR2 (XOR2 b50,(AND2 b1,b33)),(XOR2 b50,(AND2 b1,b51)) & not $ b34 ) ) )
proof end;