:: Correctness of a Cyclic Redundancy Check Code Generator :: by Yuguang Yang , Katsumi Wasaki , Yasushi Fuwa and Yatsuka Nakamura :: :: Received April 16, 1999 :: Copyright (c) 1999-2018 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies XBOOLE_0, GATE_1; notations XBOOLE_0, GATE_1; constructors XBOOLE_0, GATE_1; registrations GATE_1; begin :: Correctness of Division Circuits with 12-bit Register and 16-bit Register :: :: Assumptions: :: g0,g1,g2,...,g12: the coefficients of divident polynomial; :: a0,a1,a2,...,a11: the present state of register; :: p: input; :: b0,b1,b2,...,b11: the state of register with input p; :: A=(a11, a10, ..., a1, a0), B=(b11, b10, ..., b1,b0); :: G=(g12,g11,g10,...,g1,g0); :: A + B = (a11+b11, a10+b10, ..., a1+b1, a0+b0); :: note: the operator + here means exclusive or. :: :: In a division circuit, a shift is expected to complete one step division by :: divident polynomial. So we can derive the relation of A, B, G and p as following: :: A*2+p=G*a11+B :: here, A*2=(a11, a10, ..., a1, a0)*2=(a11, a10, ..., a1, a0,0), :: G*a11=(g12 & a11, g11 & a11, g10 & a11,...,g1 & a11, g0 & a11). theorem for g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12, a0,a1,a2,a3,a4,a5,a6,a7 ,a8,a9,a10,a11, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11, p being set holds g12 is not empty & (b0 is not empty iff XOR2(p,AND2(g0,a11)) is not empty)& (b1 is not empty iff XOR2(a0,AND2(g1,a11)) is not empty)& (b2 is not empty iff XOR2(a1, AND2(g2,a11)) is not empty)& (b3 is not empty iff XOR2(a2,AND2(g3,a11)) is not empty)& (b4 is not empty iff XOR2(a3,AND2(g4,a11)) is not empty)& (b5 is not empty iff XOR2(a4,AND2(g5,a11)) is not empty)& (b6 is not empty iff XOR2(a5, AND2(g6,a11)) is not empty)& (b7 is not empty iff XOR2(a6,AND2(g7,a11)) is not empty)& (b8 is not empty iff XOR2(a7,AND2(g8,a11)) is not empty)& (b9 is not empty iff XOR2(a8,AND2(g9,a11)) is not empty)& (b10 is not empty iff XOR2(a9, AND2(g10,a11)) is not empty)& (b11 is not empty iff XOR2(a10,AND2(g11,a11)) is not empty) implies (a11 is not empty iff AND2(g12,a11) is not empty) & (a10 is not empty iff XOR2(b11,AND2(g11,a11)) is not empty)& (a9 is not empty iff XOR2( b10,AND2(g10,a11)) is not empty)& (a8 is not empty iff XOR2(b9,AND2(g9,a11)) is not empty)& (a7 is not empty iff XOR2(b8,AND2(g8,a11)) is not empty)& (a6 is not empty iff XOR2(b7,AND2(g7,a11)) is not empty)& (a5 is not empty iff XOR2(b6 ,AND2(g6,a11)) is not empty)& (a4 is not empty iff XOR2(b5,AND2(g5,a11)) is not empty)& (a3 is not empty iff XOR2(b4,AND2(g4,a11)) is not empty)& (a2 is not empty iff XOR2(b3,AND2(g3,a11)) is not empty)& (a1 is not empty iff XOR2(b2, AND2(g2,a11)) is not empty)& (a0 is not empty iff XOR2(b1,AND2(g1,a11)) is not empty)& (p is not empty iff XOR2(b0,AND2(g0,a11)) is not empty) proof let g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12, a0,a1,a2,a3,a4,a5,a6,a7,a8,a9 ,a10,a11, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11, p be set; assume that A1: g12 is not empty and A2: b0 is not empty iff XOR2(p,AND2(g0,a11)) is not empty and A3: b1 is not empty iff XOR2(a0,AND2(g1,a11)) is not empty and A4: b2 is not empty iff XOR2(a1,AND2(g2,a11)) is not empty and A5: b3 is not empty iff XOR2(a2,AND2(g3,a11)) is not empty and A6: b4 is not empty iff XOR2(a3,AND2(g4,a11)) is not empty and A7: b5 is not empty iff XOR2(a4,AND2(g5,a11)) is not empty and A8: b6 is not empty iff XOR2(a5,AND2(g6,a11)) is not empty and A9: b7 is not empty iff XOR2(a6,AND2(g7,a11)) is not empty and A10: b8 is not empty iff XOR2(a7,AND2(g8,a11)) is not empty and A11: b9 is not empty iff XOR2(a8,AND2(g9,a11)) is not empty and A12: b10 is not empty iff XOR2(a9,AND2(g10,a11)) is not empty and A13: b11 is not empty iff XOR2(a10,AND2(g11,a11)) is not empty; thus a11 is not empty iff AND2(g12,a11) is not empty by A1; XOR2(b11,AND2(g11,a11)) is not empty iff b11 is not empty & not AND2( g11,a11) is not empty or not b11 is not empty & AND2(g11,a11) is not empty; hence a10 is not empty iff XOR2(b11,AND2(g11,a11)) is not empty by A13; XOR2(b10,AND2(g10,a11)) is not empty iff b10 is not empty & not AND2( g10,a11) is not empty or not b10 is not empty & AND2(g10,a11) is not empty; hence a9 is not empty iff XOR2(b10,AND2(g10,a11)) is not empty by A12; XOR2(b9,AND2(g9,a11)) is not empty iff b9 is not empty & not AND2(g9, a11) is not empty or not b9 is not empty & AND2(g9,a11) is not empty; hence a8 is not empty iff XOR2(b9,AND2(g9,a11)) is not empty by A11; XOR2(b8,AND2(g8,a11)) is not empty iff b8 is not empty & not AND2(g8, a11) is not empty or not b8 is not empty & AND2(g8,a11) is not empty; hence a7 is not empty iff XOR2(b8,AND2(g8,a11)) is not empty by A10; XOR2(b7,AND2(g7,a11)) is not empty iff b7 is not empty & not AND2(g7, a11) is not empty or not b7 is not empty & AND2(g7,a11) is not empty; hence a6 is not empty iff XOR2(b7,AND2(g7,a11)) is not empty by A9; XOR2(b6,AND2(g6,a11)) is not empty iff b6 is not empty & not AND2(g6, a11) is not empty or not b6 is not empty & AND2(g6,a11) is not empty; hence a5 is not empty iff XOR2(b6,AND2(g6,a11)) is not empty by A8; XOR2(b5,AND2(g5,a11)) is not empty iff b5 is not empty & not AND2(g5, a11) is not empty or not b5 is not empty & AND2(g5,a11) is not empty; hence a4 is not empty iff XOR2(b5,AND2(g5,a11)) is not empty by A7; XOR2(b4,AND2(g4,a11)) is not empty iff b4 is not empty & not AND2(g4, a11) is not empty or not b4 is not empty & AND2(g4,a11) is not empty; hence a3 is not empty iff XOR2(b4,AND2(g4,a11)) is not empty by A6; XOR2(b3,AND2(g3,a11)) is not empty iff b3 is not empty & not AND2(g3, a11) is not empty or not b3 is not empty & AND2(g3,a11) is not empty; hence a2 is not empty iff XOR2(b3,AND2(g3,a11)) is not empty by A5; XOR2(b2,AND2(g2,a11)) is not empty iff b2 is not empty & not AND2(g2, a11) is not empty or not b2 is not empty & AND2(g2,a11) is not empty; hence a1 is not empty iff XOR2(b2,AND2(g2,a11)) is not empty by A4; XOR2(b1,AND2(g1,a11)) is not empty iff b1 is not empty & not AND2(g1, a11) is not empty or not b1 is not empty & AND2(g1,a11) is not empty; hence a0 is not empty iff XOR2(b1,AND2(g1,a11)) is not empty by A3; XOR2(b0,AND2(g0,a11)) is not empty iff b0 is not empty & not AND2(g0, a11) is not empty or not b0 is not empty & AND2(g0,a11) is not empty; hence thesis by A2; end; :: Assumptions: :: g0,g1,g2,...,g16: the coefficients of divident polynomial; :: a0,a1,a2,...,a15: the present state of register; :: p: input; :: b0,b1,b2,...,b15: the state of register with input p; :: A=(a15, a14, ..., a1, a0), B=(b15, b14, ..., b1,b0); :: G=(g16,g15,g14,...,g1,g0); :: A + B = (a15+b15, a14+b14, ..., a1+b1, a0+b0); :: note: the operator + here means exclusive or. :: :: In a division circuit, a shift is expected to complete one step division by :: divident polynomial. So we can derive the relation of A, B, G and p as following: :: A*2+p=G*a15+B :: here, A*2=(a15, a14, ..., a1, a0)*2=(a15, a14, ..., a1, a0,0), :: G*a15=(g16 & a15, g15 & a15, g14 & a15,...,g1 & a15, g0 is not empty & a15). theorem for g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16, a0,a1, a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9, b10,b11,b12,b13,b14,b15, p being set holds g16 is not empty & (b0 is not empty iff XOR2(p,AND2(g0,a15)) is not empty)& (b1 is not empty iff XOR2(a0,AND2(g1, a15)) is not empty)& (b2 is not empty iff XOR2(a1,AND2(g2,a15)) is not empty)& (b3 is not empty iff XOR2(a2,AND2(g3,a15)) is not empty)& (b4 is not empty iff XOR2(a3,AND2(g4,a15)) is not empty)& (b5 is not empty iff XOR2(a4,AND2(g5,a15)) is not empty)& (b6 is not empty iff XOR2(a5,AND2(g6,a15)) is not empty)& (b7 is not empty iff XOR2(a6,AND2(g7,a15)) is not empty)& (b8 is not empty iff XOR2(a7 ,AND2(g8,a15)) is not empty)& (b9 is not empty iff XOR2(a8,AND2(g9,a15)) is not empty)& (b10 is not empty iff XOR2(a9,AND2(g10,a15)) is not empty)& (b11 is not empty iff XOR2(a10,AND2(g11,a15)) is not empty)& (b12 is not empty iff XOR2(a11 ,AND2(g12,a15)) is not empty)& (b13 is not empty iff XOR2(a12,AND2(g13,a15)) is not empty)& (b14 is not empty iff XOR2(a13,AND2(g14,a15)) is not empty)& (b15 is not empty iff XOR2(a14,AND2(g15,a15)) is not empty) implies (a15 is not empty iff AND2(g16,a15) is not empty) & (a14 is not empty iff XOR2(b15,AND2(g15 ,a15)) is not empty)& (a13 is not empty iff XOR2(b14,AND2(g14,a15)) is not empty)& (a12 is not empty iff XOR2(b13,AND2(g13,a15)) is not empty)& (a11 is not empty iff XOR2(b12,AND2(g12,a15)) is not empty)& (a10 is not empty iff XOR2 (b11,AND2(g11,a15)) is not empty)& (a9 is not empty iff XOR2(b10,AND2(g10,a15)) is not empty)& (a8 is not empty iff XOR2(b9,AND2(g9,a15)) is not empty)& (a7 is not empty iff XOR2(b8,AND2(g8,a15)) is not empty)& (a6 is not empty iff XOR2(b7 ,AND2(g7,a15)) is not empty)& (a5 is not empty iff XOR2(b6,AND2(g6,a15)) is not empty)& (a4 is not empty iff XOR2(b5,AND2(g5,a15)) is not empty)& (a3 is not empty iff XOR2(b4,AND2(g4,a15)) is not empty)& (a2 is not empty iff XOR2(b3, AND2(g3,a15)) is not empty)& (a1 is not empty iff XOR2(b2,AND2(g2,a15)) is not empty)& (a0 is not empty iff XOR2(b1,AND2(g1,a15)) is not empty)& (p is not empty iff XOR2(b0,AND2(g0,a15)) is not empty) proof let g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16, a0,a1,a2,a3, a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10, b11,b12,b13,b14,b15, p be set; assume that A1: g16 is not empty and A2: b0 is not empty iff XOR2(p,AND2(g0,a15)) is not empty and A3: b1 is not empty iff XOR2(a0,AND2(g1,a15)) is not empty and A4: b2 is not empty iff XOR2(a1,AND2(g2,a15)) is not empty and A5: b3 is not empty iff XOR2(a2,AND2(g3,a15)) is not empty and A6: b4 is not empty iff XOR2(a3,AND2(g4,a15)) is not empty and A7: b5 is not empty iff XOR2(a4,AND2(g5,a15)) is not empty and A8: b6 is not empty iff XOR2(a5,AND2(g6,a15)) is not empty and A9: b7 is not empty iff XOR2(a6,AND2(g7,a15)) is not empty and A10: b8 is not empty iff XOR2(a7,AND2(g8,a15)) is not empty and A11: b9 is not empty iff XOR2(a8,AND2(g9,a15)) is not empty and A12: b10 is not empty iff XOR2(a9,AND2(g10,a15)) is not empty and A13: b11 is not empty iff XOR2(a10,AND2(g11,a15)) is not empty and A14: b12 is not empty iff XOR2(a11,AND2(g12,a15)) is not empty and A15: b13 is not empty iff XOR2(a12,AND2(g13,a15)) is not empty and A16: b14 is not empty iff XOR2(a13,AND2(g14,a15)) is not empty and A17: b15 is not empty iff XOR2(a14,AND2(g15,a15)) is not empty; thus a15 is not empty iff AND2(g16,a15) is not empty by A1; XOR2(b15,AND2(g15,a15)) is not empty iff b15 is not empty & not AND2( g15,a15) is not empty or not b15 is not empty & AND2(g15,a15) is not empty; hence a14 is not empty iff XOR2(b15,AND2(g15,a15)) is not empty by A17; XOR2(b14,AND2(g14,a15)) is not empty iff b14 is not empty & not AND2( g14,a15) is not empty or not b14 is not empty & AND2(g14,a15) is not empty; hence a13 is not empty iff XOR2(b14,AND2(g14,a15)) is not empty by A16; XOR2(b13,AND2(g13,a15)) is not empty iff b13 is not empty & not AND2( g13,a15) is not empty or not b13 is not empty & AND2(g13,a15) is not empty; hence a12 is not empty iff XOR2(b13,AND2(g13,a15)) is not empty by A15; XOR2(b12,AND2(g12,a15)) is not empty iff b12 is not empty & not AND2( g12,a15) is not empty or not b12 is not empty & AND2(g12,a15) is not empty; hence a11 is not empty iff XOR2(b12,AND2(g12,a15)) is not empty by A14; XOR2(b11,AND2(g11,a15)) is not empty iff b11 is not empty & not AND2( g11,a15) is not empty or not b11 is not empty & AND2(g11,a15) is not empty; hence a10 is not empty iff XOR2(b11,AND2(g11,a15)) is not empty by A13; XOR2(b10,AND2(g10,a15)) is not empty iff b10 is not empty & not AND2( g10,a15) is not empty or not b10 is not empty & AND2(g10,a15) is not empty; hence a9 is not empty iff XOR2(b10,AND2(g10,a15)) is not empty by A12; XOR2(b9,AND2(g9,a15)) is not empty iff b9 is not empty & not AND2(g9, a15) is not empty or not b9 is not empty & AND2(g9,a15) is not empty; hence a8 is not empty iff XOR2(b9,AND2(g9,a15)) is not empty by A11; XOR2(b8,AND2(g8,a15)) is not empty iff b8 is not empty & not AND2(g8, a15) is not empty or not b8 is not empty & AND2(g8,a15) is not empty; hence a7 is not empty iff XOR2(b8,AND2(g8,a15)) is not empty by A10; XOR2(b7,AND2(g7,a15)) is not empty iff b7 is not empty & not AND2(g7, a15) is not empty or not b7 is not empty & AND2(g7,a15) is not empty; hence a6 is not empty iff XOR2(b7,AND2(g7,a15)) is not empty by A9; XOR2(b6,AND2(g6,a15)) is not empty iff b6 is not empty & not AND2(g6, a15) is not empty or not b6 is not empty & AND2(g6,a15) is not empty; hence a5 is not empty iff XOR2(b6,AND2(g6,a15)) is not empty by A8; XOR2(b5,AND2(g5,a15)) is not empty iff b5 is not empty & not AND2(g5, a15) is not empty or not b5 is not empty & AND2(g5,a15) is not empty; hence a4 is not empty iff XOR2(b5,AND2(g5,a15)) is not empty by A7; XOR2(b4,AND2(g4,a15)) is not empty iff b4 is not empty & not AND2(g4, a15) is not empty or not b4 is not empty & AND2(g4,a15) is not empty; hence a3 is not empty iff XOR2(b4,AND2(g4,a15)) is not empty by A6; XOR2(b3,AND2(g3,a15)) is not empty iff b3 is not empty & not AND2(g3, a15) is not empty or not b3 is not empty & AND2(g3,a15) is not empty; hence a2 is not empty iff XOR2(b3,AND2(g3,a15)) is not empty by A5; XOR2(b2,AND2(g2,a15)) is not empty iff b2 is not empty & not AND2(g2, a15) is not empty or not b2 is not empty & AND2(g2,a15) is not empty; hence a1 is not empty iff XOR2(b2,AND2(g2,a15)) is not empty by A4; XOR2(b1,AND2(g1,a15)) is not empty iff b1 is not empty & not AND2(g1, a15) is not empty or not b1 is not empty & AND2(g1,a15) is not empty; hence a0 is not empty iff XOR2(b1,AND2(g1,a15)) is not empty by A3; XOR2(b0,AND2(g0,a15)) is not empty iff b0 is not empty & not AND2(g0, a15) is not empty or not b0 is not empty & AND2(g0,a15) is not empty; hence thesis by A2; end; begin:: Correctness of CRC Circuits with Generator Polynomial of :: Degree 12 and 16 :: Assumptions: :: g0,g1,g2,...,g12: the coefficients of generator polynomial; :: a0,a1,a2,...,a11: the present state of register; :: p: input; :: b0,b1,b2,...,b11: the state of register after a shift with input p; :: A=(a11, a10, ..., a1, a0), B=(b11, b10, ..., b1,b0); :: G=(g12,g11,g10,...,g1,g0) :: A + B=(a11+b11, a10+b10, ..., a1+b1, a0+b0); :: note: the operator + here means exclusive or; :: A*2=(a11, a10, ..., a1, a0)*2=(a11, a10, ..., a1, a0,0); :: G*a11=(g12 is not empty & a11, g11 & a11, g10 & a11,...,g1 & a11, g0 is not empty & a11); :: 2^11 means 2 to the 11th power. :: :: :: In a CRC circuit, with assumption that A is the remainder of message M*2^11 :: divided by G, B which is the contents of register after one shift is expected :: to be the remainder of (M*2 + p)*2^11 divided by G. That means :: B= mod((M*2 + p)*2^11,G) :: = mod(M*2^12+ p*2^11,G) :: = mod(M*2^12,G) + mod(p*2^11,G) :: = mod(A*2,G) + mod(p*2^11,G) :: = (A*2+G*a11)+(p*2^11 + G*p); :: :: Note: The initial state of register can be regarded as M=0. theorem for g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12, a0,a1,a2,a3,a4,a5,a6,a7 ,a8,a9,a10,a11, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11, z,p being set holds g0 is not empty & not z is not empty & (b0 is not empty iff XOR2(p,a11) is not empty)& (b1 is not empty iff XOR2(a0,AND2(g1,b0)) is not empty)& (b2 is not empty iff XOR2(a1,AND2(g2,b0)) is not empty)& (b3 is not empty iff XOR2(a2,AND2 (g3,b0)) is not empty)& (b4 is not empty iff XOR2(a3,AND2(g4,b0)) is not empty) & (b5 is not empty iff XOR2(a4,AND2(g5,b0)) is not empty)& (b6 is not empty iff XOR2(a5,AND2(g6,b0)) is not empty)& (b7 is not empty iff XOR2(a6,AND2(g7,b0)) is not empty)& (b8 is not empty iff XOR2(a7,AND2(g8,b0)) is not empty)& (b9 is not empty iff XOR2(a8,AND2(g9,b0)) is not empty)& (b10 is not empty iff XOR2(a9 ,AND2(g10,b0)) is not empty)& (b11 is not empty iff XOR2(a10,AND2(g11,b0)) is not empty) implies (b11 is not empty iff XOR2(XOR2(a10,AND2(g11,a11)),XOR2(z, AND2(g11,p))) is not empty) & (b10 is not empty iff XOR2(XOR2(a9,AND2(g10,a11)) ,XOR2(z,AND2(g10,p))) is not empty) & (b9 is not empty iff XOR2(XOR2(a8,AND2(g9 ,a11)),XOR2(z,AND2(g9,p))) is not empty) & (b8 is not empty iff XOR2(XOR2(a7, AND2(g8,a11)),XOR2(z,AND2(g8,p))) is not empty) & (b7 is not empty iff XOR2( XOR2(a6,AND2(g7,a11)),XOR2(z,AND2(g7,p))) is not empty) & (b6 is not empty iff XOR2(XOR2(a5,AND2(g6,a11)),XOR2(z,AND2(g6,p))) is not empty) & (b5 is not empty iff XOR2(XOR2(a4,AND2(g5,a11)),XOR2(z,AND2(g5,p))) is not empty) & (b4 is not empty iff XOR2(XOR2(a3,AND2(g4,a11)),XOR2(z,AND2(g4,p))) is not empty) & (b3 is not empty iff XOR2(XOR2(a2,AND2(g3,a11)),XOR2(z,AND2(g3,p))) is not empty) & ( b2 is not empty iff XOR2(XOR2(a1,AND2(g2,a11)),XOR2(z,AND2(g2,p))) is not empty ) & (b1 is not empty iff XOR2(XOR2(a0,AND2(g1,a11)),XOR2(z,AND2(g1,p))) is not empty) & (b0 is not empty iff XOR2(XOR2(z, AND2(g0,a11)),XOR2(z,AND2(g0,p))) is not empty) proof let g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12, a0,a1,a2,a3,a4,a5,a6,a7,a8,a9 ,a10,a11, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11, z,p be set; assume that A1: g0 is not empty and A2: not z is not empty and A3: b0 is not empty iff XOR2(p,a11) is not empty and A4: b1 is not empty iff XOR2(a0,AND2(g1,b0)) is not empty and A5: b2 is not empty iff XOR2(a1,AND2(g2,b0)) is not empty and A6: b3 is not empty iff XOR2(a2,AND2(g3,b0)) is not empty and A7: b4 is not empty iff XOR2(a3,AND2(g4,b0)) is not empty and A8: b5 is not empty iff XOR2(a4,AND2(g5,b0)) is not empty and A9: b6 is not empty iff XOR2(a5,AND2(g6,b0)) is not empty and A10: b7 is not empty iff XOR2(a6,AND2(g7,b0)) is not empty and A11: b8 is not empty iff XOR2(a7,AND2(g8,b0)) is not empty and A12: b9 is not empty iff XOR2(a8,AND2(g9,b0)) is not empty and A13: b10 is not empty iff XOR2(a9,AND2(g10,b0)) is not empty and A14: b11 is not empty iff XOR2(a10,AND2(g11,b0)) is not empty; XOR2(XOR2(a10,AND2(g11,a11)),XOR2(z,AND2(g11,p))) is not empty iff ( a10 is not empty & (not g11 is not empty or not a11 is not empty) & (not g11 is not empty or not p is not empty) or not a10 is not empty & g11 is not empty & a11 is not empty & (not g11 is not empty or not p is not empty) ) or (not a10 is not empty or g11 is not empty & a11 is not empty ) & (a10 is not empty or not (g11 is not empty & a11 is not empty)) & g11 is not empty & p is not empty by A2; hence b11 is not empty iff XOR2(XOR2(a10,AND2(g11,a11)),XOR2(z,AND2(g11,p))) is not empty by A3,A14; XOR2(XOR2(a9,AND2(g10,a11)),XOR2(z,AND2(g10,p))) is not empty iff ( a9 is not empty & (not g10 is not empty or not a11 is not empty) & (not g10 is not empty or not p is not empty) or not a9 is not empty & g10 is not empty & a11 is not empty & (not g10 is not empty or not p is not empty) ) or (not a9 is not empty or g10 is not empty & a11 is not empty ) & (a9 is not empty or not (g10 is not empty & a11 is not empty)) & g10 is not empty & p is not empty by A2; hence b10 is not empty iff XOR2(XOR2(a9,AND2(g10,a11)),XOR2(z,AND2(g10,p))) is not empty by A3,A13; XOR2(XOR2(a8,AND2(g9,a11)),XOR2(z,AND2(g9,p))) is not empty iff ( a8 is not empty & (not g9 is not empty or not a11 is not empty) & (not g9 is not empty or not p is not empty) or not a8 is not empty & g9 is not empty & a11 is not empty & (not g9 is not empty or not p is not empty) ) or (not a8 is not empty or g9 is not empty & a11 is not empty ) & (a8 is not empty or not (g9 is not empty & a11 is not empty)) & g9 is not empty & p is not empty by A2; hence b9 is not empty iff XOR2(XOR2(a8,AND2(g9,a11)),XOR2(z,AND2(g9,p))) is not empty by A3,A12; XOR2(XOR2(a7,AND2(g8,a11)),XOR2(z,AND2(g8,p))) is not empty iff ( a7 is not empty & (not g8 is not empty or not a11 is not empty) & (not g8 is not empty or not p is not empty) or not a7 is not empty & g8 is not empty & a11 is not empty & (not g8 is not empty or not p is not empty) ) or (not a7 is not empty or g8 is not empty & a11 is not empty ) & (a7 is not empty or not (g8 is not empty & a11 is not empty)) & g8 is not empty & p is not empty by A2; hence b8 is not empty iff XOR2(XOR2(a7,AND2(g8,a11)),XOR2(z,AND2(g8,p))) is not empty by A3,A11; XOR2(XOR2(a6,AND2(g7,a11)),XOR2(z,AND2(g7,p))) is not empty iff ( a6 is not empty & (not g7 is not empty or not a11 is not empty) & (not g7 is not empty or not p is not empty) or not a6 is not empty & g7 is not empty & a11 is not empty & (not g7 is not empty or not p is not empty) ) or (not a6 is not empty or g7 is not empty & a11 is not empty ) & (a6 is not empty or not (g7 is not empty & a11 is not empty)) & g7 is not empty & p is not empty by A2; hence b7 is not empty iff XOR2(XOR2(a6,AND2(g7,a11)),XOR2(z,AND2(g7,p))) is not empty by A3,A10; XOR2(XOR2(a5,AND2(g6,a11)),XOR2(z,AND2(g6,p))) is not empty iff ( a5 is not empty & (not g6 is not empty or not a11 is not empty) & (not g6 is not empty or not p is not empty) or not a5 is not empty & g6 is not empty & a11 is not empty & (not g6 is not empty or not p is not empty) ) or (not a5 is not empty or g6 is not empty & a11 is not empty ) & (a5 is not empty or not (g6 is not empty & a11 is not empty)) & g6 is not empty & p is not empty by A2; hence b6 is not empty iff XOR2(XOR2(a5,AND2(g6,a11)),XOR2(z,AND2(g6,p))) is not empty by A3,A9; XOR2(XOR2(a4,AND2(g5,a11)),XOR2(z,AND2(g5,p))) is not empty iff ( a4 is not empty & (not g5 is not empty or not a11 is not empty) & (not g5 is not empty or not p is not empty) or not a4 is not empty & g5 is not empty & a11 is not empty & (not g5 is not empty or not p is not empty) ) or (not a4 is not empty or g5 is not empty & a11 is not empty ) & (a4 is not empty or not (g5 is not empty & a11 is not empty)) & g5 is not empty & p is not empty by A2; hence b5 is not empty iff XOR2(XOR2(a4,AND2(g5,a11)),XOR2(z,AND2(g5,p))) is not empty by A3,A8; XOR2(XOR2(a3,AND2(g4,a11)),XOR2(z,AND2(g4,p))) is not empty iff ( a3 is not empty & (not g4 is not empty or not a11 is not empty) & (not g4 is not empty or not p is not empty) or not a3 is not empty & g4 is not empty & a11 is not empty & (not g4 is not empty or not p is not empty) ) or (not a3 is not empty or g4 is not empty & a11 is not empty ) & (a3 is not empty or not (g4 is not empty & a11 is not empty)) & g4 is not empty & p is not empty by A2; hence b4 is not empty iff XOR2(XOR2(a3,AND2(g4,a11)),XOR2(z,AND2(g4,p))) is not empty by A3,A7; XOR2(XOR2(a2,AND2(g3,a11)),XOR2(z,AND2(g3,p))) is not empty iff ( a2 is not empty & (not g3 is not empty or not a11 is not empty) & (not g3 is not empty or not p is not empty) or not a2 is not empty & g3 is not empty & a11 is not empty & (not g3 is not empty or not p is not empty) ) or (not a2 is not empty or g3 is not empty & a11 is not empty ) & (a2 is not empty or not (g3 is not empty & a11 is not empty)) & g3 is not empty & p is not empty by A2; hence b3 is not empty iff XOR2(XOR2(a2,AND2(g3,a11)),XOR2(z,AND2(g3,p))) is not empty by A3,A6; XOR2(XOR2(a1,AND2(g2,a11)),XOR2(z,AND2(g2,p))) is not empty iff ( a1 is not empty & (not g2 is not empty or not a11 is not empty) & (not g2 is not empty or not p is not empty) or not a1 is not empty & g2 is not empty & a11 is not empty & (not g2 is not empty or not p is not empty) ) or (not a1 is not empty or g2 is not empty & a11 is not empty ) & (a1 is not empty or not (g2 is not empty & a11 is not empty)) & g2 is not empty & p is not empty by A2; hence b2 is not empty iff XOR2(XOR2(a1,AND2(g2,a11)),XOR2(z,AND2(g2,p))) is not empty by A3,A5; XOR2(XOR2(a0,AND2(g1,a11)),XOR2(z,AND2(g1,p))) is not empty iff ( a0 is not empty & (not g1 is not empty or not a11 is not empty) & (not g1 is not empty or not p is not empty) or not a0 is not empty & g1 is not empty & a11 is not empty & (not g1 is not empty or not p is not empty) ) or (not a0 is not empty or g1 is not empty & a11 is not empty ) & (a0 is not empty or not (g1 is not empty & a11 is not empty)) & g1 is not empty & p is not empty by A2; hence b1 is not empty iff XOR2(XOR2(a0,AND2(g1,a11)),XOR2(z,AND2(g1,p))) is not empty by A3,A4; XOR2(XOR2(z,AND2(g0,a11)),XOR2(z,AND2(g0,p))) is not empty iff ( z is not empty & (not g0 is not empty or not a11 is not empty) & (not g0 is not empty or not p is not empty) or not z is not empty & g0 is not empty & a11 is not empty & (not g0 is not empty or not p is not empty) ) or (not z is not empty or g0 is not empty & a11 is not empty ) & (z is not empty or not (g0 is not empty & a11 is not empty)) & g0 is not empty & p is not empty by A2; hence thesis by A1,A3; end; :: Assumptions: :: g0,g1,g2,...,g16: the coefficients of generator polynomial; :: a0,a1,a2,...,a15: the present state of register; :: p: input; :: b0,b1,b2,...,b15: the state of register with input p; :: A=(a15, a10, ..., a1, a0), B=(b11, b10, ..., b1,b0); :: G=(g16,g15,g14,...,g1,g0) :: A + B=(a15+b15, a14+b14, ..., a1+b1, a0+b0); :: note: the operator + here means exclusive or; :: A*2= (a15, a14, ..., a1, a0)*2=(a15, a14, ..., a1, a0,0); :: G*a15=(g16 is not empty & a15, g15 & a15, g14 & a15,...,g1 is not empty & a15, g0 is not empty & a15); :: 2^15 means 2 to the 15th power. :: :: In a CRC circuit, with assumption that A is the remainder of message M*2^15 :: divided by G, B which is the contents of register after one shift is expected :: to be the remainder of (M*2 + p)*2^15 divided by G. That means :: B= mod((M*2 + p)*2^15,G) :: = mod(M*2^16+ p*2^15,G) :: = mod(M*2^16,G) + mod(p*2^15,G) :: = mod(A*2,G) + mod(p*2^15,G) :: = (A*2+G*a15)+(p*2^15 + G*p); :: :: Note: The initial state of register can be regarded as M=0. theorem for g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16, a0,a1, a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9, b10,b11,b12,b13,b14,b15, z,p being set holds g0 is not empty & not z is not empty & (b0 is not empty iff XOR2(p,a15) is not empty)& (b1 is not empty iff XOR2(a0,AND2(g1,b0)) is not empty)& (b2 is not empty iff XOR2(a1,AND2(g2,b0)) is not empty)& (b3 is not empty iff XOR2(a2,AND2(g3,b0)) is not empty)& (b4 is not empty iff XOR2(a3,AND2(g4,b0)) is not empty)& (b5 is not empty iff XOR2(a4, AND2(g5,b0)) is not empty)& (b6 is not empty iff XOR2(a5,AND2(g6,b0)) is not empty)& (b7 is not empty iff XOR2(a6,AND2(g7,b0)) is not empty)& (b8 is not empty iff XOR2(a7,AND2(g8,b0)) is not empty)& (b9 is not empty iff XOR2(a8,AND2 (g9,b0)) is not empty)& (b10 is not empty iff XOR2(a9,AND2(g10,b0)) is not empty)& (b11 is not empty iff XOR2(a10,AND2(g11,b0)) is not empty)& (b12 is not empty iff XOR2(a11,AND2(g12,b0)) is not empty)& (b13 is not empty iff XOR2(a12, AND2(g13,b0)) is not empty)& (b14 is not empty iff XOR2(a13,AND2(g14,b0)) is not empty)& (b15 is not empty iff XOR2(a14,AND2(g15,b0)) is not empty) implies (b15 is not empty iff XOR2(XOR2(a14,AND2(g15,a15)),XOR2(z,AND2(g15,p))) is not empty) & (b14 is not empty iff XOR2(XOR2(a13,AND2(g14,a15)),XOR2(z,AND2(g14,p)) ) is not empty) & (b13 is not empty iff XOR2(XOR2(a12,AND2(g13,a15)),XOR2(z, AND2(g13,p))) is not empty) & (b12 is not empty iff XOR2(XOR2(a11,AND2(g12,a15) ),XOR2(z,AND2(g12,p))) is not empty) & (b11 is not empty iff XOR2(XOR2(a10,AND2 (g11,a15)),XOR2(z,AND2(g11,p))) is not empty) & (b10 is not empty iff XOR2(XOR2 (a9,AND2(g10,a15)),XOR2(z,AND2(g10,p))) is not empty) & (b9 is not empty iff XOR2(XOR2(a8,AND2(g9,a15)),XOR2(z,AND2(g9,p))) is not empty) & (b8 is not empty iff XOR2(XOR2(a7,AND2(g8,a15)),XOR2(z,AND2(g8,p))) is not empty) & (b7 is not empty iff XOR2(XOR2(a6,AND2(g7,a15)),XOR2(z,AND2(g7,p))) is not empty) & (b6 is not empty iff XOR2(XOR2(a5,AND2(g6,a15)),XOR2(z,AND2(g6,p))) is not empty) & ( b5 is not empty iff XOR2(XOR2(a4,AND2(g5,a15)),XOR2(z,AND2(g5,p))) is not empty ) & (b4 is not empty iff XOR2(XOR2(a3,AND2(g4,a15)),XOR2(z,AND2(g4,p))) is not empty) & (b3 is not empty iff XOR2(XOR2(a2,AND2(g3,a15)),XOR2(z,AND2(g3,p))) is not empty) & (b2 is not empty iff XOR2(XOR2(a1,AND2(g2,a15)),XOR2(z,AND2(g2,p)) ) is not empty) & (b1 is not empty iff XOR2(XOR2(a0,AND2(g1,a15)),XOR2(z,AND2( g1,p))) is not empty) & (b0 is not empty iff XOR2(XOR2(z, AND2(g0,a15)),XOR2(z, AND2(g0,p))) is not empty) proof let g0,g1,g2,g3,g4,g5,g6,g7,g8,g9,g10,g11,g12,g13,g14,g15,g16, a0,a1,a2,a3, a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15, b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10, b11,b12,b13,b14,b15, z,p be set; assume that A1: g0 is not empty and A2: not z is not empty and A3: b0 is not empty iff XOR2(p,a15) is not empty and A4: b1 is not empty iff XOR2(a0,AND2(g1,b0)) is not empty and A5: b2 is not empty iff XOR2(a1,AND2(g2,b0)) is not empty and A6: b3 is not empty iff XOR2(a2,AND2(g3,b0)) is not empty and A7: b4 is not empty iff XOR2(a3,AND2(g4,b0)) is not empty and A8: b5 is not empty iff XOR2(a4,AND2(g5,b0)) is not empty and A9: b6 is not empty iff XOR2(a5,AND2(g6,b0)) is not empty and A10: b7 is not empty iff XOR2(a6,AND2(g7,b0)) is not empty and A11: b8 is not empty iff XOR2(a7,AND2(g8,b0)) is not empty and A12: b9 is not empty iff XOR2(a8,AND2(g9,b0)) is not empty and A13: b10 is not empty iff XOR2(a9,AND2(g10,b0)) is not empty and A14: b11 is not empty iff XOR2(a10,AND2(g11,b0)) is not empty and A15: b12 is not empty iff XOR2(a11,AND2(g12,b0)) is not empty and A16: b13 is not empty iff XOR2(a12,AND2(g13,b0)) is not empty and A17: b14 is not empty iff XOR2(a13,AND2(g14,b0)) is not empty and A18: b15 is not empty iff XOR2(a14,AND2(g15,b0)) is not empty; XOR2(XOR2(a14,AND2(g15,a15)),XOR2(z,AND2(g15,p))) is not empty iff ( a14 is not empty & (not g15 is not empty or not a15 is not empty) & (not g15 is not empty or not p is not empty) or not a14 is not empty & g15 is not empty & a15 is not empty & (not g15 is not empty or not p is not empty) ) or (not a14 is not empty or g15 is not empty & a15 is not empty ) & (a14 is not empty or not (g15 is not empty & a15 is not empty)) & g15 is not empty & p is not empty by A2; hence b15 is not empty iff XOR2(XOR2(a14,AND2(g15,a15)),XOR2(z,AND2(g15,p))) is not empty by A3,A18; XOR2(XOR2(a13,AND2(g14,a15)),XOR2(z,AND2(g14,p))) is not empty iff ( a13 is not empty & (not g14 is not empty or not a15 is not empty) & (not g14 is not empty or not p is not empty) or not a13 is not empty & g14 is not empty & a15 is not empty & (not g14 is not empty or not p is not empty) ) or (not a13 is not empty or g14 is not empty & a15 is not empty ) & (a13 is not empty or not (g14 is not empty & a15 is not empty)) & g14 is not empty & p is not empty by A2; hence b14 is not empty iff XOR2(XOR2(a13,AND2(g14,a15)),XOR2(z,AND2(g14,p))) is not empty by A3,A17; XOR2(XOR2(a12,AND2(g13,a15)),XOR2(z,AND2(g13,p))) is not empty iff ( a12 is not empty & (not g13 is not empty or not a15 is not empty) & (not g13 is not empty or not p is not empty) or not a12 is not empty & g13 is not empty & a15 is not empty & (not g13 is not empty or not p is not empty) ) or (not a12 is not empty or g13 is not empty & a15 is not empty ) & (a12 is not empty or not (g13 is not empty & a15 is not empty)) & g13 is not empty & p is not empty by A2; hence b13 is not empty iff XOR2(XOR2(a12,AND2(g13,a15)),XOR2(z,AND2(g13,p))) is not empty by A3,A16; XOR2(XOR2(a11,AND2(g12,a15)),XOR2(z,AND2(g12,p))) is not empty iff ( a11 is not empty & (not g12 is not empty or not a15 is not empty) & (not g12 is not empty or not p is not empty) or not a11 is not empty & g12 is not empty & a15 is not empty & (not g12 is not empty or not p is not empty) ) or (not a11 is not empty or g12 is not empty & a15 is not empty ) & (a11 is not empty or not (g12 is not empty & a15 is not empty)) & g12 is not empty & p is not empty by A2; hence b12 is not empty iff XOR2(XOR2(a11,AND2(g12,a15)),XOR2(z,AND2(g12,p))) is not empty by A3,A15; XOR2(XOR2(a10,AND2(g11,a15)),XOR2(z,AND2(g11,p))) is not empty iff ( a10 is not empty & (not g11 is not empty or not a15 is not empty) & (not g11 is not empty or not p is not empty) or not a10 is not empty & g11 is not empty & a15 is not empty & (not g11 is not empty or not p is not empty) ) or (not a10 is not empty or g11 is not empty & a15 is not empty ) & (a10 is not empty or not (g11 is not empty & a15 is not empty)) & g11 is not empty & p is not empty by A2; hence b11 is not empty iff XOR2(XOR2(a10,AND2(g11,a15)),XOR2(z,AND2(g11,p))) is not empty by A3,A14; XOR2(XOR2(a9,AND2(g10,a15)),XOR2(z,AND2(g10,p))) is not empty iff ( a9 is not empty & (not g10 is not empty or not a15 is not empty) & (not g10 is not empty or not p is not empty) or not a9 is not empty & g10 is not empty & a15 is not empty & (not g10 is not empty or not p is not empty) ) or (not a9 is not empty or g10 is not empty & a15 is not empty ) & (a9 is not empty or not (g10 is not empty & a15 is not empty)) & g10 is not empty & p is not empty by A2; hence b10 is not empty iff XOR2(XOR2(a9,AND2(g10,a15)),XOR2(z,AND2(g10,p))) is not empty by A3,A13; XOR2(XOR2(a8,AND2(g9,a15)),XOR2(z,AND2(g9,p))) is not empty iff ( a8 is not empty & (not g9 is not empty or not a15 is not empty) & (not g9 is not empty or not p is not empty) or not a8 is not empty & g9 is not empty & a15 is not empty & (not g9 is not empty or not p is not empty) ) or (not a8 is not empty or g9 is not empty & a15 is not empty ) & (a8 is not empty or not (g9 is not empty & a15 is not empty)) & g9 is not empty & p is not empty by A2; hence b9 is not empty iff XOR2(XOR2(a8,AND2(g9,a15)),XOR2(z,AND2(g9,p))) is not empty by A3,A12; XOR2(XOR2(a7,AND2(g8,a15)),XOR2(z,AND2(g8,p))) is not empty iff ( a7 is not empty & (not g8 is not empty or not a15 is not empty) & (not g8 is not empty or not p is not empty) or not a7 is not empty & g8 is not empty & a15 is not empty & (not g8 is not empty or not p is not empty) ) or (not a7 is not empty or g8 is not empty & a15 is not empty ) & (a7 is not empty or not (g8 is not empty & a15 is not empty)) & g8 is not empty & p is not empty by A2; hence b8 is not empty iff XOR2(XOR2(a7,AND2(g8,a15)),XOR2(z,AND2(g8,p))) is not empty by A3,A11; XOR2(XOR2(a6,AND2(g7,a15)),XOR2(z,AND2(g7,p))) is not empty iff ( a6 is not empty & (not g7 is not empty or not a15 is not empty) & (not g7 is not empty or not p is not empty) or not a6 is not empty & g7 is not empty & a15 is not empty & (not g7 is not empty or not p is not empty) ) or (not a6 is not empty or g7 is not empty & a15 is not empty ) & (a6 is not empty or not (g7 is not empty & a15 is not empty)) & g7 is not empty & p is not empty by A2; hence b7 is not empty iff XOR2(XOR2(a6,AND2(g7,a15)),XOR2(z,AND2(g7,p))) is not empty by A3,A10; XOR2(XOR2(a5,AND2(g6,a15)),XOR2(z,AND2(g6,p))) is not empty iff ( a5 is not empty & (not g6 is not empty or not a15 is not empty) & (not g6 is not empty or not p is not empty) or not a5 is not empty & g6 is not empty & a15 is not empty & (not g6 is not empty or not p is not empty) ) or (not a5 is not empty or g6 is not empty & a15 is not empty ) & (a5 is not empty or not (g6 is not empty & a15 is not empty)) & g6 is not empty & p is not empty by A2; hence b6 is not empty iff XOR2(XOR2(a5,AND2(g6,a15)),XOR2(z,AND2(g6,p))) is not empty by A3,A9; XOR2(XOR2(a4,AND2(g5,a15)),XOR2(z,AND2(g5,p))) is not empty iff ( a4 is not empty & (not g5 is not empty or not a15 is not empty) & (not g5 is not empty or not p is not empty) or not a4 is not empty & g5 is not empty & a15 is not empty & (not g5 is not empty or not p is not empty) ) or (not a4 is not empty or g5 is not empty & a15 is not empty ) & (a4 is not empty or not (g5 is not empty & a15 is not empty)) & g5 is not empty & p is not empty by A2; hence b5 is not empty iff XOR2(XOR2(a4,AND2(g5,a15)),XOR2(z,AND2(g5,p))) is not empty by A3,A8; XOR2(XOR2(a3,AND2(g4,a15)),XOR2(z,AND2(g4,p))) is not empty iff ( a3 is not empty & (not g4 is not empty or not a15 is not empty) & (not g4 is not empty or not p is not empty) or not a3 is not empty & g4 is not empty & a15 is not empty & (not g4 is not empty or not p is not empty) ) or (not a3 is not empty or g4 is not empty & a15 is not empty ) & (a3 is not empty or not (g4 is not empty & a15 is not empty)) & g4 is not empty & p is not empty by A2; hence b4 is not empty iff XOR2(XOR2(a3,AND2(g4,a15)),XOR2(z,AND2(g4,p))) is not empty by A3,A7; XOR2(XOR2(a2,AND2(g3,a15)),XOR2(z,AND2(g3,p))) is not empty iff ( a2 is not empty & (not g3 is not empty or not a15 is not empty) & (not g3 is not empty or not p is not empty) or not a2 is not empty & g3 is not empty & a15 is not empty & (not g3 is not empty or not p is not empty) ) or (not a2 is not empty or g3 is not empty & a15 is not empty ) & (a2 is not empty or not (g3 is not empty & a15 is not empty)) & g3 is not empty & p is not empty by A2; hence b3 is not empty iff XOR2(XOR2(a2,AND2(g3,a15)),XOR2(z,AND2(g3,p))) is not empty by A3,A6; XOR2(XOR2(a1,AND2(g2,a15)),XOR2(z,AND2(g2,p))) is not empty iff ( a1 is not empty & (not g2 is not empty or not a15 is not empty) & (not g2 is not empty or not p is not empty) or not a1 is not empty & g2 is not empty & a15 is not empty & (not g2 is not empty or not p is not empty) ) or (not a1 is not empty or g2 is not empty & a15 is not empty ) & (a1 is not empty or not (g2 is not empty & a15 is not empty)) & g2 is not empty & p is not empty by A2; hence b2 is not empty iff XOR2(XOR2(a1,AND2(g2,a15)),XOR2(z,AND2(g2,p))) is not empty by A3,A5; XOR2(XOR2(a0,AND2(g1,a15)),XOR2(z,AND2(g1,p))) is not empty iff ( a0 is not empty & (not g1 is not empty or not a15 is not empty) & (not g1 is not empty or not p is not empty) or not a0 is not empty & g1 is not empty & a15 is not empty & (not g1 is not empty or not p is not empty) ) or (not a0 is not empty or g1 is not empty & a15 is not empty ) & (a0 is not empty or not (g1 is not empty & a15 is not empty)) & g1 is not empty & p is not empty by A2; hence b1 is not empty iff XOR2(XOR2(a0,AND2(g1,a15)),XOR2(z,AND2(g1,p))) is not empty by A3,A4; XOR2(XOR2(z,AND2(g0,a15)),XOR2(z,AND2(g0,p))) is not empty iff ( z is not empty & (not g0 is not empty or not a15 is not empty) & (not g0 is not empty or not p is not empty) or not z is not empty & g0 is not empty & a15 is not empty & (not g0 is not empty or not p is not empty) ) or (not z is not empty or g0 is not empty & a15 is not empty ) & (z is not empty or not (g0 is not empty & a15 is not empty)) & g0 is not empty & p is not empty by A2; hence thesis by A1,A3; end;