Article GATE_4, MML version 4.99.1005
:: GATE_4:th 1
theorem
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
st b1 is not empty &
b13 is not empty &
(b26 is not empty implies XOR2(b38,AND2(b1,b25)) is not empty) &
(XOR2(b38,AND2(b1,b25)) is not empty implies b26 is not empty) &
(b27 is not empty implies XOR2(b14,AND2(b2,b25)) is not empty) &
(XOR2(b14,AND2(b2,b25)) is not empty implies b27 is not empty) &
(b28 is not empty implies XOR2(b15,AND2(b3,b25)) is not empty) &
(XOR2(b15,AND2(b3,b25)) is not empty implies b28 is not empty) &
(b29 is not empty implies XOR2(b16,AND2(b4,b25)) is not empty) &
(XOR2(b16,AND2(b4,b25)) is not empty implies b29 is not empty) &
(b30 is not empty implies XOR2(b17,AND2(b5,b25)) is not empty) &
(XOR2(b17,AND2(b5,b25)) is not empty implies b30 is not empty) &
(b31 is not empty implies XOR2(b18,AND2(b6,b25)) is not empty) &
(XOR2(b18,AND2(b6,b25)) is not empty implies b31 is not empty) &
(b32 is not empty implies XOR2(b19,AND2(b7,b25)) is not empty) &
(XOR2(b19,AND2(b7,b25)) is not empty implies b32 is not empty) &
(b33 is not empty implies XOR2(b20,AND2(b8,b25)) is not empty) &
(XOR2(b20,AND2(b8,b25)) is not empty implies b33 is not empty) &
(b34 is not empty implies XOR2(b21,AND2(b9,b25)) is not empty) &
(XOR2(b21,AND2(b9,b25)) is not empty implies b34 is not empty) &
(b35 is not empty implies XOR2(b22,AND2(b10,b25)) is not empty) &
(XOR2(b22,AND2(b10,b25)) is not empty implies b35 is not empty) &
(b36 is not empty implies XOR2(b23,AND2(b11,b25)) is not empty) &
(XOR2(b23,AND2(b11,b25)) is not empty implies b36 is not empty) &
(b37 is not empty implies XOR2(b24,AND2(b12,b25)) is not empty) &
(XOR2(b24,AND2(b12,b25)) is not empty implies b37 is not empty)
holds (b25 is not empty implies AND2(b13,b25) is not empty) &
(AND2(b13,b25) is not empty implies b25 is not empty) &
(b24 is not empty implies XOR2(b37,AND2(b12,b25)) is not empty) &
(XOR2(b37,AND2(b12,b25)) is not empty implies b24 is not empty) &
(b23 is not empty implies XOR2(b36,AND2(b11,b25)) is not empty) &
(XOR2(b36,AND2(b11,b25)) is not empty implies b23 is not empty) &
(b22 is not empty implies XOR2(b35,AND2(b10,b25)) is not empty) &
(XOR2(b35,AND2(b10,b25)) is not empty implies b22 is not empty) &
(b21 is not empty implies XOR2(b34,AND2(b9,b25)) is not empty) &
(XOR2(b34,AND2(b9,b25)) is not empty implies b21 is not empty) &
(b20 is not empty implies XOR2(b33,AND2(b8,b25)) is not empty) &
(XOR2(b33,AND2(b8,b25)) is not empty implies b20 is not empty) &
(b19 is not empty implies XOR2(b32,AND2(b7,b25)) is not empty) &
(XOR2(b32,AND2(b7,b25)) is not empty implies b19 is not empty) &
(b18 is not empty implies XOR2(b31,AND2(b6,b25)) is not empty) &
(XOR2(b31,AND2(b6,b25)) is not empty implies b18 is not empty) &
(b17 is not empty implies XOR2(b30,AND2(b5,b25)) is not empty) &
(XOR2(b30,AND2(b5,b25)) is not empty implies b17 is not empty) &
(b16 is not empty implies XOR2(b29,AND2(b4,b25)) is not empty) &
(XOR2(b29,AND2(b4,b25)) is not empty implies b16 is not empty) &
(b15 is not empty implies XOR2(b28,AND2(b3,b25)) is not empty) &
(XOR2(b28,AND2(b3,b25)) is not empty implies b15 is not empty) &
(b14 is not empty implies XOR2(b27,AND2(b2,b25)) is not empty) &
(XOR2(b27,AND2(b2,b25)) is not empty implies b14 is not empty) &
(b38 is not empty implies XOR2(b26,AND2(b1,b25)) is not empty) &
(XOR2(b26,AND2(b1,b25)) is not empty implies b38 is not empty);
:: GATE_4:th 2
theorem
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
st b1 is not empty &
b17 is not empty &
(b34 is not empty implies XOR2(b50,AND2(b1,b33)) is not empty) &
(XOR2(b50,AND2(b1,b33)) is not empty implies b34 is not empty) &
(b35 is not empty implies XOR2(b18,AND2(b2,b33)) is not empty) &
(XOR2(b18,AND2(b2,b33)) is not empty implies b35 is not empty) &
(b36 is not empty implies XOR2(b19,AND2(b3,b33)) is not empty) &
(XOR2(b19,AND2(b3,b33)) is not empty implies b36 is not empty) &
(b37 is not empty implies XOR2(b20,AND2(b4,b33)) is not empty) &
(XOR2(b20,AND2(b4,b33)) is not empty implies b37 is not empty) &
(b38 is not empty implies XOR2(b21,AND2(b5,b33)) is not empty) &
(XOR2(b21,AND2(b5,b33)) is not empty implies b38 is not empty) &
(b39 is not empty implies XOR2(b22,AND2(b6,b33)) is not empty) &
(XOR2(b22,AND2(b6,b33)) is not empty implies b39 is not empty) &
(b40 is not empty implies XOR2(b23,AND2(b7,b33)) is not empty) &
(XOR2(b23,AND2(b7,b33)) is not empty implies b40 is not empty) &
(b41 is not empty implies XOR2(b24,AND2(b8,b33)) is not empty) &
(XOR2(b24,AND2(b8,b33)) is not empty implies b41 is not empty) &
(b42 is not empty implies XOR2(b25,AND2(b9,b33)) is not empty) &
(XOR2(b25,AND2(b9,b33)) is not empty implies b42 is not empty) &
(b43 is not empty implies XOR2(b26,AND2(b10,b33)) is not empty) &
(XOR2(b26,AND2(b10,b33)) is not empty implies b43 is not empty) &
(b44 is not empty implies XOR2(b27,AND2(b11,b33)) is not empty) &
(XOR2(b27,AND2(b11,b33)) is not empty implies b44 is not empty) &
(b45 is not empty implies XOR2(b28,AND2(b12,b33)) is not empty) &
(XOR2(b28,AND2(b12,b33)) is not empty implies b45 is not empty) &
(b46 is not empty implies XOR2(b29,AND2(b13,b33)) is not empty) &
(XOR2(b29,AND2(b13,b33)) is not empty implies b46 is not empty) &
(b47 is not empty implies XOR2(b30,AND2(b14,b33)) is not empty) &
(XOR2(b30,AND2(b14,b33)) is not empty implies b47 is not empty) &
(b48 is not empty implies XOR2(b31,AND2(b15,b33)) is not empty) &
(XOR2(b31,AND2(b15,b33)) is not empty implies b48 is not empty) &
(b49 is not empty implies XOR2(b32,AND2(b16,b33)) is not empty) &
(XOR2(b32,AND2(b16,b33)) is not empty implies b49 is not empty)
holds (b33 is not empty implies AND2(b17,b33) is not empty) &
(AND2(b17,b33) is not empty implies b33 is not empty) &
(b32 is not empty implies XOR2(b49,AND2(b16,b33)) is not empty) &
(XOR2(b49,AND2(b16,b33)) is not empty implies b32 is not empty) &
(b31 is not empty implies XOR2(b48,AND2(b15,b33)) is not empty) &
(XOR2(b48,AND2(b15,b33)) is not empty implies b31 is not empty) &
(b30 is not empty implies XOR2(b47,AND2(b14,b33)) is not empty) &
(XOR2(b47,AND2(b14,b33)) is not empty implies b30 is not empty) &
(b29 is not empty implies XOR2(b46,AND2(b13,b33)) is not empty) &
(XOR2(b46,AND2(b13,b33)) is not empty implies b29 is not empty) &
(b28 is not empty implies XOR2(b45,AND2(b12,b33)) is not empty) &
(XOR2(b45,AND2(b12,b33)) is not empty implies b28 is not empty) &
(b27 is not empty implies XOR2(b44,AND2(b11,b33)) is not empty) &
(XOR2(b44,AND2(b11,b33)) is not empty implies b27 is not empty) &
(b26 is not empty implies XOR2(b43,AND2(b10,b33)) is not empty) &
(XOR2(b43,AND2(b10,b33)) is not empty implies b26 is not empty) &
(b25 is not empty implies XOR2(b42,AND2(b9,b33)) is not empty) &
(XOR2(b42,AND2(b9,b33)) is not empty implies b25 is not empty) &
(b24 is not empty implies XOR2(b41,AND2(b8,b33)) is not empty) &
(XOR2(b41,AND2(b8,b33)) is not empty implies b24 is not empty) &
(b23 is not empty implies XOR2(b40,AND2(b7,b33)) is not empty) &
(XOR2(b40,AND2(b7,b33)) is not empty implies b23 is not empty) &
(b22 is not empty implies XOR2(b39,AND2(b6,b33)) is not empty) &
(XOR2(b39,AND2(b6,b33)) is not empty implies b22 is not empty) &
(b21 is not empty implies XOR2(b38,AND2(b5,b33)) is not empty) &
(XOR2(b38,AND2(b5,b33)) is not empty implies b21 is not empty) &
(b20 is not empty implies XOR2(b37,AND2(b4,b33)) is not empty) &
(XOR2(b37,AND2(b4,b33)) is not empty implies b20 is not empty) &
(b19 is not empty implies XOR2(b36,AND2(b3,b33)) is not empty) &
(XOR2(b36,AND2(b3,b33)) is not empty implies b19 is not empty) &
(b18 is not empty implies XOR2(b35,AND2(b2,b33)) is not empty) &
(XOR2(b35,AND2(b2,b33)) is not empty implies b18 is not empty) &
(b50 is not empty implies XOR2(b34,AND2(b1,b33)) is not empty) &
(XOR2(b34,AND2(b1,b33)) is not empty implies b50 is not empty);
:: GATE_4:th 3
theorem
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
st b1 is not empty &
b13 is not empty &
b38 is empty &
(b26 is not empty implies XOR2(b39,b25) is not empty) &
(XOR2(b39,b25) is not empty implies b26 is not empty) &
(b27 is not empty implies XOR2(b14,AND2(b2,b26)) is not empty) &
(XOR2(b14,AND2(b2,b26)) is not empty implies b27 is not empty) &
(b28 is not empty implies XOR2(b15,AND2(b3,b26)) is not empty) &
(XOR2(b15,AND2(b3,b26)) is not empty implies b28 is not empty) &
(b29 is not empty implies XOR2(b16,AND2(b4,b26)) is not empty) &
(XOR2(b16,AND2(b4,b26)) is not empty implies b29 is not empty) &
(b30 is not empty implies XOR2(b17,AND2(b5,b26)) is not empty) &
(XOR2(b17,AND2(b5,b26)) is not empty implies b30 is not empty) &
(b31 is not empty implies XOR2(b18,AND2(b6,b26)) is not empty) &
(XOR2(b18,AND2(b6,b26)) is not empty implies b31 is not empty) &
(b32 is not empty implies XOR2(b19,AND2(b7,b26)) is not empty) &
(XOR2(b19,AND2(b7,b26)) is not empty implies b32 is not empty) &
(b33 is not empty implies XOR2(b20,AND2(b8,b26)) is not empty) &
(XOR2(b20,AND2(b8,b26)) is not empty implies b33 is not empty) &
(b34 is not empty implies XOR2(b21,AND2(b9,b26)) is not empty) &
(XOR2(b21,AND2(b9,b26)) is not empty implies b34 is not empty) &
(b35 is not empty implies XOR2(b22,AND2(b10,b26)) is not empty) &
(XOR2(b22,AND2(b10,b26)) is not empty implies b35 is not empty) &
(b36 is not empty implies XOR2(b23,AND2(b11,b26)) is not empty) &
(XOR2(b23,AND2(b11,b26)) is not empty implies b36 is not empty) &
(b37 is not empty implies XOR2(b24,AND2(b12,b26)) is not empty) &
(XOR2(b24,AND2(b12,b26)) is not empty implies b37 is not empty)
holds (b37 is not empty implies XOR2(XOR2(b24,AND2(b12,b25)),XOR2(b38,AND2(b12,b39))) is not empty) &
(XOR2(XOR2(b24,AND2(b12,b25)),XOR2(b38,AND2(b12,b39))) is not empty implies b37 is not empty) &
(b36 is not empty implies XOR2(XOR2(b23,AND2(b11,b25)),XOR2(b38,AND2(b11,b39))) is not empty) &
(XOR2(XOR2(b23,AND2(b11,b25)),XOR2(b38,AND2(b11,b39))) is not empty implies b36 is not empty) &
(b35 is not empty implies XOR2(XOR2(b22,AND2(b10,b25)),XOR2(b38,AND2(b10,b39))) is not empty) &
(XOR2(XOR2(b22,AND2(b10,b25)),XOR2(b38,AND2(b10,b39))) is not empty implies b35 is not empty) &
(b34 is not empty implies XOR2(XOR2(b21,AND2(b9,b25)),XOR2(b38,AND2(b9,b39))) is not empty) &
(XOR2(XOR2(b21,AND2(b9,b25)),XOR2(b38,AND2(b9,b39))) is not empty implies b34 is not empty) &
(b33 is not empty implies XOR2(XOR2(b20,AND2(b8,b25)),XOR2(b38,AND2(b8,b39))) is not empty) &
(XOR2(XOR2(b20,AND2(b8,b25)),XOR2(b38,AND2(b8,b39))) is not empty implies b33 is not empty) &
(b32 is not empty implies XOR2(XOR2(b19,AND2(b7,b25)),XOR2(b38,AND2(b7,b39))) is not empty) &
(XOR2(XOR2(b19,AND2(b7,b25)),XOR2(b38,AND2(b7,b39))) is not empty implies b32 is not empty) &
(b31 is not empty implies XOR2(XOR2(b18,AND2(b6,b25)),XOR2(b38,AND2(b6,b39))) is not empty) &
(XOR2(XOR2(b18,AND2(b6,b25)),XOR2(b38,AND2(b6,b39))) is not empty implies b31 is not empty) &
(b30 is not empty implies XOR2(XOR2(b17,AND2(b5,b25)),XOR2(b38,AND2(b5,b39))) is not empty) &
(XOR2(XOR2(b17,AND2(b5,b25)),XOR2(b38,AND2(b5,b39))) is not empty implies b30 is not empty) &
(b29 is not empty implies XOR2(XOR2(b16,AND2(b4,b25)),XOR2(b38,AND2(b4,b39))) is not empty) &
(XOR2(XOR2(b16,AND2(b4,b25)),XOR2(b38,AND2(b4,b39))) is not empty implies b29 is not empty) &
(b28 is not empty implies XOR2(XOR2(b15,AND2(b3,b25)),XOR2(b38,AND2(b3,b39))) is not empty) &
(XOR2(XOR2(b15,AND2(b3,b25)),XOR2(b38,AND2(b3,b39))) is not empty implies b28 is not empty) &
(b27 is not empty implies XOR2(XOR2(b14,AND2(b2,b25)),XOR2(b38,AND2(b2,b39))) is not empty) &
(XOR2(XOR2(b14,AND2(b2,b25)),XOR2(b38,AND2(b2,b39))) is not empty implies b27 is not empty) &
(b26 is not empty implies XOR2(XOR2(b38,AND2(b1,b25)),XOR2(b38,AND2(b1,b39))) is not empty) &
(XOR2(XOR2(b38,AND2(b1,b25)),XOR2(b38,AND2(b1,b39))) is not empty implies b26 is not empty);
:: GATE_4:th 4
theorem
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
st b1 is not empty &
b17 is not empty &
b50 is empty &
(b34 is not empty implies XOR2(b51,b33) is not empty) &
(XOR2(b51,b33) is not empty implies b34 is not empty) &
(b35 is not empty implies XOR2(b18,AND2(b2,b34)) is not empty) &
(XOR2(b18,AND2(b2,b34)) is not empty implies b35 is not empty) &
(b36 is not empty implies XOR2(b19,AND2(b3,b34)) is not empty) &
(XOR2(b19,AND2(b3,b34)) is not empty implies b36 is not empty) &
(b37 is not empty implies XOR2(b20,AND2(b4,b34)) is not empty) &
(XOR2(b20,AND2(b4,b34)) is not empty implies b37 is not empty) &
(b38 is not empty implies XOR2(b21,AND2(b5,b34)) is not empty) &
(XOR2(b21,AND2(b5,b34)) is not empty implies b38 is not empty) &
(b39 is not empty implies XOR2(b22,AND2(b6,b34)) is not empty) &
(XOR2(b22,AND2(b6,b34)) is not empty implies b39 is not empty) &
(b40 is not empty implies XOR2(b23,AND2(b7,b34)) is not empty) &
(XOR2(b23,AND2(b7,b34)) is not empty implies b40 is not empty) &
(b41 is not empty implies XOR2(b24,AND2(b8,b34)) is not empty) &
(XOR2(b24,AND2(b8,b34)) is not empty implies b41 is not empty) &
(b42 is not empty implies XOR2(b25,AND2(b9,b34)) is not empty) &
(XOR2(b25,AND2(b9,b34)) is not empty implies b42 is not empty) &
(b43 is not empty implies XOR2(b26,AND2(b10,b34)) is not empty) &
(XOR2(b26,AND2(b10,b34)) is not empty implies b43 is not empty) &
(b44 is not empty implies XOR2(b27,AND2(b11,b34)) is not empty) &
(XOR2(b27,AND2(b11,b34)) is not empty implies b44 is not empty) &
(b45 is not empty implies XOR2(b28,AND2(b12,b34)) is not empty) &
(XOR2(b28,AND2(b12,b34)) is not empty implies b45 is not empty) &
(b46 is not empty implies XOR2(b29,AND2(b13,b34)) is not empty) &
(XOR2(b29,AND2(b13,b34)) is not empty implies b46 is not empty) &
(b47 is not empty implies XOR2(b30,AND2(b14,b34)) is not empty) &
(XOR2(b30,AND2(b14,b34)) is not empty implies b47 is not empty) &
(b48 is not empty implies XOR2(b31,AND2(b15,b34)) is not empty) &
(XOR2(b31,AND2(b15,b34)) is not empty implies b48 is not empty) &
(b49 is not empty implies XOR2(b32,AND2(b16,b34)) is not empty) &
(XOR2(b32,AND2(b16,b34)) is not empty implies b49 is not empty)
holds (b49 is not empty implies XOR2(XOR2(b32,AND2(b16,b33)),XOR2(b50,AND2(b16,b51))) is not empty) &
(XOR2(XOR2(b32,AND2(b16,b33)),XOR2(b50,AND2(b16,b51))) is not empty implies b49 is not empty) &
(b48 is not empty implies XOR2(XOR2(b31,AND2(b15,b33)),XOR2(b50,AND2(b15,b51))) is not empty) &
(XOR2(XOR2(b31,AND2(b15,b33)),XOR2(b50,AND2(b15,b51))) is not empty implies b48 is not empty) &
(b47 is not empty implies XOR2(XOR2(b30,AND2(b14,b33)),XOR2(b50,AND2(b14,b51))) is not empty) &
(XOR2(XOR2(b30,AND2(b14,b33)),XOR2(b50,AND2(b14,b51))) is not empty implies b47 is not empty) &
(b46 is not empty implies XOR2(XOR2(b29,AND2(b13,b33)),XOR2(b50,AND2(b13,b51))) is not empty) &
(XOR2(XOR2(b29,AND2(b13,b33)),XOR2(b50,AND2(b13,b51))) is not empty implies b46 is not empty) &
(b45 is not empty implies XOR2(XOR2(b28,AND2(b12,b33)),XOR2(b50,AND2(b12,b51))) is not empty) &
(XOR2(XOR2(b28,AND2(b12,b33)),XOR2(b50,AND2(b12,b51))) is not empty implies b45 is not empty) &
(b44 is not empty implies XOR2(XOR2(b27,AND2(b11,b33)),XOR2(b50,AND2(b11,b51))) is not empty) &
(XOR2(XOR2(b27,AND2(b11,b33)),XOR2(b50,AND2(b11,b51))) is not empty implies b44 is not empty) &
(b43 is not empty implies XOR2(XOR2(b26,AND2(b10,b33)),XOR2(b50,AND2(b10,b51))) is not empty) &
(XOR2(XOR2(b26,AND2(b10,b33)),XOR2(b50,AND2(b10,b51))) is not empty implies b43 is not empty) &
(b42 is not empty implies XOR2(XOR2(b25,AND2(b9,b33)),XOR2(b50,AND2(b9,b51))) is not empty) &
(XOR2(XOR2(b25,AND2(b9,b33)),XOR2(b50,AND2(b9,b51))) is not empty implies b42 is not empty) &
(b41 is not empty implies XOR2(XOR2(b24,AND2(b8,b33)),XOR2(b50,AND2(b8,b51))) is not empty) &
(XOR2(XOR2(b24,AND2(b8,b33)),XOR2(b50,AND2(b8,b51))) is not empty implies b41 is not empty) &
(b40 is not empty implies XOR2(XOR2(b23,AND2(b7,b33)),XOR2(b50,AND2(b7,b51))) is not empty) &
(XOR2(XOR2(b23,AND2(b7,b33)),XOR2(b50,AND2(b7,b51))) is not empty implies b40 is not empty) &
(b39 is not empty implies XOR2(XOR2(b22,AND2(b6,b33)),XOR2(b50,AND2(b6,b51))) is not empty) &
(XOR2(XOR2(b22,AND2(b6,b33)),XOR2(b50,AND2(b6,b51))) is not empty implies b39 is not empty) &
(b38 is not empty implies XOR2(XOR2(b21,AND2(b5,b33)),XOR2(b50,AND2(b5,b51))) is not empty) &
(XOR2(XOR2(b21,AND2(b5,b33)),XOR2(b50,AND2(b5,b51))) is not empty implies b38 is not empty) &
(b37 is not empty implies XOR2(XOR2(b20,AND2(b4,b33)),XOR2(b50,AND2(b4,b51))) is not empty) &
(XOR2(XOR2(b20,AND2(b4,b33)),XOR2(b50,AND2(b4,b51))) is not empty implies b37 is not empty) &
(b36 is not empty implies XOR2(XOR2(b19,AND2(b3,b33)),XOR2(b50,AND2(b3,b51))) is not empty) &
(XOR2(XOR2(b19,AND2(b3,b33)),XOR2(b50,AND2(b3,b51))) is not empty implies b36 is not empty) &
(b35 is not empty implies XOR2(XOR2(b18,AND2(b2,b33)),XOR2(b50,AND2(b2,b51))) is not empty) &
(XOR2(XOR2(b18,AND2(b2,b33)),XOR2(b50,AND2(b2,b51))) is not empty implies b35 is not empty) &
(b34 is not empty implies XOR2(XOR2(b50,AND2(b1,b33)),XOR2(b50,AND2(b1,b51))) is not empty) &
(XOR2(XOR2(b50,AND2(b1,b33)),XOR2(b50,AND2(b1,b51))) is not empty implies b34 is not empty);