3.68/6.43 YES 3.68/6.43 3.68/6.43 Problem 1: 3.68/6.43 3.68/6.43 (VAR x y z) 3.68/6.43 (THEORY 3.68/6.43 (AC and or xor)) 3.68/6.43 (RULES 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 ) 3.68/6.43 3.68/6.43 Problem 1: 3.68/6.43 3.68/6.43 Dependency Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 OR(or(x3,x4),x5) = OR(x3,or(x4,x5)) 3.68/6.43 OR(x3,x4) = OR(x4,x3) 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(x,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> XOR(and(x,z),and(y,z)) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 AND(xor(x,y),z) -> XOR(and(x,z),and(y,z)) 3.68/6.43 EQUIV(x,y) -> XOR(xor(T,y),x) 3.68/6.43 EQUIV(x,y) -> XOR(T,y) 3.68/6.43 IMPL(x,y) -> AND(x,y) 3.68/6.43 IMPL(x,y) -> XOR(and(x,y),xor(T,x)) 3.68/6.43 IMPL(x,y) -> XOR(T,x) 3.68/6.43 NEG(x) -> XOR(T,x) 3.68/6.43 OR(or(x,y),x3) -> AND(x,y) 3.68/6.43 OR(or(x,y),x3) -> OR(xor(and(x,y),xor(x,y)),x3) 3.68/6.43 OR(or(x,y),x3) -> XOR(and(x,y),xor(x,y)) 3.68/6.43 OR(or(x,y),x3) -> XOR(x,y) 3.68/6.43 OR(x,y) -> AND(x,y) 3.68/6.43 OR(x,y) -> XOR(and(x,y),xor(x,y)) 3.68/6.43 OR(x,y) -> XOR(x,y) 3.68/6.43 XOR(xor(neg(x),x),x3) -> XOR(F,x3) 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,x4) 3.68/6.43 OR(x3,or(x4,x5)) -> OR(x4,x5) 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 3.68/6.43 Problem 1: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 OR(or(x3,x4),x5) = OR(x3,or(x4,x5)) 3.68/6.43 OR(x3,x4) = OR(x4,x3) 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(x,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> XOR(and(x,z),and(y,z)) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 AND(xor(x,y),z) -> XOR(and(x,z),and(y,z)) 3.68/6.43 EQUIV(x,y) -> XOR(xor(T,y),x) 3.68/6.43 EQUIV(x,y) -> XOR(T,y) 3.68/6.43 IMPL(x,y) -> AND(x,y) 3.68/6.43 IMPL(x,y) -> XOR(and(x,y),xor(T,x)) 3.68/6.43 IMPL(x,y) -> XOR(T,x) 3.68/6.43 NEG(x) -> XOR(T,x) 3.68/6.43 OR(or(x,y),x3) -> AND(x,y) 3.68/6.43 OR(or(x,y),x3) -> OR(xor(and(x,y),xor(x,y)),x3) 3.68/6.43 OR(or(x,y),x3) -> XOR(and(x,y),xor(x,y)) 3.68/6.43 OR(or(x,y),x3) -> XOR(x,y) 3.68/6.43 OR(x,y) -> AND(x,y) 3.68/6.43 OR(x,y) -> XOR(and(x,y),xor(x,y)) 3.68/6.43 OR(x,y) -> XOR(x,y) 3.68/6.43 XOR(xor(neg(x),x),x3) -> XOR(F,x3) 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,x4) 3.68/6.43 OR(x3,or(x4,x5)) -> OR(x4,x5) 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 XOR(xor(neg(x),x),x3) -> XOR(F,x3) 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) -> XOR(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(x,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 OR(or(x,y),x3) -> OR(xor(and(x,y),xor(x,y)),x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,or(x4,x5)) 3.68/6.43 OR(x3,x4) -> OR(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,x4) 3.68/6.43 OR(x3,or(x4,x5)) -> OR(x4,x5) 3.68/6.43 3.68/6.43 3.68/6.43 The problem is decomposed in 3 subproblems. 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 XOR(xor(neg(x),x),x3) -> XOR(F,x3) 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Linear 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 2 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 0 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 2 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 2 3.68/6.43 [F] = 2 3.68/6.43 [T] = 0 3.68/6.43 [AND](X1,X2) = 0 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 2.X1 + 2.X2 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) -> XOR(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 XOR(xor(F,x),x3) -> XOR(x,x3) 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Linear 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 2 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 0 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 2 3.68/6.43 [F] = 2 3.68/6.43 [T] = 0 3.68/6.43 [AND](X1,X2) = 0 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 2.X1 + 2.X2 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) -> XOR(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 XOR(xor(x,x),x3) -> XOR(F,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Linear 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 2 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 0 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 2 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 0 3.68/6.43 [T] = 0 3.68/6.43 [AND](X1,X2) = 0 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 2.X1 + 2.X2 3.68/6.43 3.68/6.43 Problem 1.1: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 XOR(xor(x3,x4),x5) = XOR(x3,xor(x4,x5)) 3.68/6.43 XOR(x3,x4) = XOR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 Empty 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 XOR(xor(x3,x4),x5) -> XOR(x3,x4) 3.68/6.43 XOR(x3,xor(x4,x5)) -> XOR(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 There is no strongly connected component 3.68/6.43 3.68/6.43 The problem is finite. 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(x,z) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 1 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 1 3.68/6.43 [T] = 1 3.68/6.43 [AND](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(y,z) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 1 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 1 3.68/6.43 [T] = 1 3.68/6.43 [AND](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(T,x),x3) -> AND(x,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 1 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 1 3.68/6.43 [T] = 1 3.68/6.43 [AND](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(x,z) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 1 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 1 3.68/6.43 [T] = 1 3.68/6.43 [AND](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 AND(xor(x,y),z) -> AND(y,z) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 Natural Numbers 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 1 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 1 3.68/6.43 [T] = 1 3.68/6.43 [AND](X1,X2) = X1.X2 + X1 + X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(xor(x,y),z),x3) -> AND(xor(and(x,z),and(y,z)),x3) 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 All rationals 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 3 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1/3 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 3 3.68/6.43 [F] = 0 3.68/6.43 [T] = 1/3 3.68/6.43 [AND](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(F,x),x3) -> AND(F,x3) 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 All rationals 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 3 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1/3 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 2 3.68/6.43 [F] = 1 3.68/6.43 [T] = 3/2 3.68/6.43 [AND](X1,X2) = 2/3.X1.X2 + 2/3.X1 + 2/3.X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 ->->Cycle: 3.68/6.43 ->->-> Pairs: 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> FAxioms: 3.68/6.43 and(and(x3,x4),x5) -> and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) -> and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) -> or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) -> or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) -> xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) -> xor(x4,x3) 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) -> AND(x4,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 ->->-> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 AND(and(x,x),x3) -> AND(x,x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 All rationals 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 3 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1/2 3.68/6.43 [or](X1,X2) = 0 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1 3.68/6.43 [F] = 0 3.68/6.43 [T] = 3/2 3.68/6.43 [AND](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 0 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.2: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 AND(and(x3,x4),x5) = AND(x3,and(x4,x5)) 3.68/6.43 AND(x3,x4) = AND(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 Empty 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 AND(and(x3,x4),x5) -> AND(x3,x4) 3.68/6.43 AND(x3,and(x4,x5)) -> AND(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 There is no strongly connected component 3.68/6.43 3.68/6.43 The problem is finite. 3.68/6.43 3.68/6.43 Problem 1.3: 3.68/6.43 3.68/6.43 Reduction Pairs Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 OR(or(x3,x4),x5) = OR(x3,or(x4,x5)) 3.68/6.43 OR(x3,x4) = OR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 OR(or(x,y),x3) -> OR(xor(and(x,y),xor(x,y)),x3) 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Usable Equations: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> Usable Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,x4) 3.68/6.43 OR(x3,or(x4,x5)) -> OR(x4,x5) 3.68/6.43 ->Interpretation type: 3.68/6.43 Simple mixed 3.68/6.43 ->Coefficients: 3.68/6.43 All rationals 3.68/6.43 ->Dimension: 3.68/6.43 1 3.68/6.43 ->Bound: 3.68/6.43 3 3.68/6.43 ->Interpretation: 3.68/6.43 3.68/6.43 [and](X1,X2) = 3.X1.X2 + X1 + X2 3.68/6.43 [equiv](X1,X2) = 0 3.68/6.43 [impl](X1,X2) = 0 3.68/6.43 [neg](X) = 1/3 3.68/6.43 [or](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 3.68/6.43 [xor](X1,X2) = X1 + X2 + 1/3 3.68/6.43 [F] = 0 3.68/6.43 [T] = 1/3 3.68/6.43 [AND](X1,X2) = 0 3.68/6.43 [EQUIV](X1,X2) = 0 3.68/6.43 [IMPL](X1,X2) = 0 3.68/6.43 [NEG](X) = 0 3.68/6.43 [OR](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 3.68/6.43 [XOR](X1,X2) = 0 3.68/6.43 3.68/6.43 Problem 1.3: 3.68/6.43 3.68/6.43 SCC Processor: 3.68/6.43 -> FAxioms: 3.68/6.43 OR(or(x3,x4),x5) = OR(x3,or(x4,x5)) 3.68/6.43 OR(x3,x4) = OR(x4,x3) 3.68/6.43 -> Pairs: 3.68/6.43 Empty 3.68/6.43 -> EAxioms: 3.68/6.43 and(and(x3,x4),x5) = and(x3,and(x4,x5)) 3.68/6.43 and(x3,x4) = and(x4,x3) 3.68/6.43 or(or(x3,x4),x5) = or(x3,or(x4,x5)) 3.68/6.43 or(x3,x4) = or(x4,x3) 3.68/6.43 xor(xor(x3,x4),x5) = xor(x3,xor(x4,x5)) 3.68/6.43 xor(x3,x4) = xor(x4,x3) 3.68/6.43 -> Rules: 3.68/6.43 and(xor(x,y),z) -> xor(and(x,z),and(y,z)) 3.68/6.43 and(F,x) -> F 3.68/6.43 and(T,x) -> x 3.68/6.43 and(x,x) -> x 3.68/6.43 equiv(x,y) -> xor(xor(T,y),x) 3.68/6.43 impl(x,y) -> xor(and(x,y),xor(T,x)) 3.68/6.43 neg(x) -> xor(T,x) 3.68/6.43 or(x,y) -> xor(and(x,y),xor(x,y)) 3.68/6.43 xor(neg(x),x) -> F 3.68/6.43 xor(F,x) -> x 3.68/6.43 xor(x,x) -> F 3.68/6.43 -> SRules: 3.68/6.43 OR(or(x3,x4),x5) -> OR(x3,x4) 3.68/6.43 OR(x3,or(x4,x5)) -> OR(x4,x5) 3.68/6.43 ->Strongly Connected Components: 3.68/6.43 There is no strongly connected component 3.68/6.43 3.68/6.43 The problem is finite. 3.68/6.44 EOF