2.93/3.02 YES 2.93/3.02 2.93/3.02 Problem 1: 2.93/3.02 2.93/3.02 (VAR b x y) 2.93/3.02 (THEORY 2.93/3.02 (AC * + U)) 2.93/3.02 (RULES 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 ) 2.93/3.02 2.93/3.02 Problem 1: 2.93/3.02 2.93/3.02 Dependency Pairs Processor: 2.93/3.02 -> FAxioms: 2.93/3.02 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.93/3.02 *#(x3,x4) = *#(x4,x3) 2.93/3.02 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.02 +#(x3,x4) = +#(x4,x3) 2.93/3.02 U#(U(x3,x4),x5) = U#(x3,U(x4,x5)) 2.93/3.02 U#(x3,x4) = U#(x4,x3) 2.93/3.02 -> Pairs: 2.93/3.02 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.93/3.02 *#(*(0(x),y),x3) -> *#(x,y) 2.93/3.02 *#(*(0(x),y),x3) -> 0#(*(x,y)) 2.93/3.02 *#(*(#,x),x3) -> *#(#,x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(x,y) 2.93/3.02 *#(*(1(x),y),x3) -> +#(0(*(x,y)),y) 2.93/3.02 *#(*(1(x),y),x3) -> 0#(*(x,y)) 2.93/3.02 *#(0(x),y) -> *#(x,y) 2.93/3.02 *#(0(x),y) -> 0#(*(x,y)) 2.93/3.02 *#(1(x),y) -> *#(x,y) 2.93/3.02 *#(1(x),y) -> +#(0(*(x,y)),y) 2.93/3.02 *#(1(x),y) -> 0#(*(x,y)) 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(0(+(x,y)),x3) 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(0(x),0(y)),x3) -> 0#(+(x,y)) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(1(x),1(y)),x3) -> 0#(+(1(#),+(x,y))) 2.93/3.02 +#(0(x),0(y)) -> +#(x,y) 2.93/3.02 +#(0(x),0(y)) -> 0#(+(x,y)) 2.93/3.02 +#(0(x),1(y)) -> +#(x,y) 2.93/3.02 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.02 +#(1(x),1(y)) -> +#(x,y) 2.93/3.02 +#(1(x),1(y)) -> 0#(+(1(#),+(x,y))) 2.93/3.02 U#(U(empty,b),x3) -> U#(b,x3) 2.93/3.02 PROD(U(x,y)) -> *#(prod(x),prod(y)) 2.93/3.02 PROD(U(x,y)) -> PROD(x) 2.93/3.02 PROD(U(x,y)) -> PROD(y) 2.93/3.02 SUM(U(x,y)) -> +#(sum(x),sum(y)) 2.93/3.02 SUM(U(x,y)) -> SUM(x) 2.93/3.02 SUM(U(x,y)) -> SUM(y) 2.93/3.02 SUM(empty) -> 0#(#) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 -> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 *#(*(x3,x4),x5) -> *#(x3,x4) 2.93/3.02 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.93/3.02 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.02 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.02 U#(U(x3,x4),x5) -> U#(x3,x4) 2.93/3.02 U#(x3,U(x4,x5)) -> U#(x4,x5) 2.93/3.02 2.93/3.02 Problem 1: 2.93/3.02 2.93/3.02 SCC Processor: 2.93/3.02 -> FAxioms: 2.93/3.02 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.93/3.02 *#(x3,x4) = *#(x4,x3) 2.93/3.02 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.02 +#(x3,x4) = +#(x4,x3) 2.93/3.02 U#(U(x3,x4),x5) = U#(x3,U(x4,x5)) 2.93/3.02 U#(x3,x4) = U#(x4,x3) 2.93/3.02 -> Pairs: 2.93/3.02 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.93/3.02 *#(*(0(x),y),x3) -> *#(x,y) 2.93/3.02 *#(*(0(x),y),x3) -> 0#(*(x,y)) 2.93/3.02 *#(*(#,x),x3) -> *#(#,x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(x,y) 2.93/3.02 *#(*(1(x),y),x3) -> +#(0(*(x,y)),y) 2.93/3.02 *#(*(1(x),y),x3) -> 0#(*(x,y)) 2.93/3.02 *#(0(x),y) -> *#(x,y) 2.93/3.02 *#(0(x),y) -> 0#(*(x,y)) 2.93/3.02 *#(1(x),y) -> *#(x,y) 2.93/3.02 *#(1(x),y) -> +#(0(*(x,y)),y) 2.93/3.02 *#(1(x),y) -> 0#(*(x,y)) 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(0(+(x,y)),x3) 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(0(x),0(y)),x3) -> 0#(+(x,y)) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(1(x),1(y)),x3) -> 0#(+(1(#),+(x,y))) 2.93/3.02 +#(0(x),0(y)) -> +#(x,y) 2.93/3.02 +#(0(x),0(y)) -> 0#(+(x,y)) 2.93/3.02 +#(0(x),1(y)) -> +#(x,y) 2.93/3.02 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.02 +#(1(x),1(y)) -> +#(x,y) 2.93/3.02 +#(1(x),1(y)) -> 0#(+(1(#),+(x,y))) 2.93/3.02 U#(U(empty,b),x3) -> U#(b,x3) 2.93/3.02 PROD(U(x,y)) -> *#(prod(x),prod(y)) 2.93/3.02 PROD(U(x,y)) -> PROD(x) 2.93/3.02 PROD(U(x,y)) -> PROD(y) 2.93/3.02 SUM(U(x,y)) -> +#(sum(x),sum(y)) 2.93/3.02 SUM(U(x,y)) -> SUM(x) 2.93/3.02 SUM(U(x,y)) -> SUM(y) 2.93/3.02 SUM(empty) -> 0#(#) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 -> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 *#(*(x3,x4),x5) -> *#(x3,x4) 2.93/3.02 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.93/3.02 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.02 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.02 U#(U(x3,x4),x5) -> U#(x3,x4) 2.93/3.02 U#(x3,U(x4,x5)) -> U#(x4,x5) 2.93/3.02 ->Strongly Connected Components: 2.93/3.02 ->->Cycle: 2.93/3.02 ->->-> Pairs: 2.93/3.02 U#(U(empty,b),x3) -> U#(b,x3) 2.93/3.02 -> FAxioms: 2.93/3.02 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) -> *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) -> +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) -> U(x4,x3) 2.93/3.02 U#(U(x3,x4),x5) -> U#(x3,U(x4,x5)) 2.93/3.02 U#(x3,x4) -> U#(x4,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 ->->-> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 U#(U(x3,x4),x5) -> U#(x3,x4) 2.93/3.02 U#(x3,U(x4,x5)) -> U#(x4,x5) 2.93/3.02 ->->Cycle: 2.93/3.02 ->->-> Pairs: 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(0(+(x,y)),x3) 2.93/3.02 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.02 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.02 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.02 +#(0(x),0(y)) -> +#(x,y) 2.93/3.02 +#(0(x),1(y)) -> +#(x,y) 2.93/3.02 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.02 +#(1(x),1(y)) -> +#(x,y) 2.93/3.02 -> FAxioms: 2.93/3.02 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) -> *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) -> +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) -> U(x4,x3) 2.93/3.02 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.02 +#(x3,x4) -> +#(x4,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 ->->-> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.02 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.02 ->->Cycle: 2.93/3.02 ->->-> Pairs: 2.93/3.02 SUM(U(x,y)) -> SUM(x) 2.93/3.02 SUM(U(x,y)) -> SUM(y) 2.93/3.02 -> FAxioms: 2.93/3.02 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) -> *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) -> +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) -> U(x4,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 ->->-> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 Empty 2.93/3.02 ->->Cycle: 2.93/3.02 ->->-> Pairs: 2.93/3.02 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.93/3.02 *#(*(0(x),y),x3) -> *#(x,y) 2.93/3.02 *#(*(#,x),x3) -> *#(#,x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.93/3.02 *#(*(1(x),y),x3) -> *#(x,y) 2.93/3.02 *#(0(x),y) -> *#(x,y) 2.93/3.02 *#(1(x),y) -> *#(x,y) 2.93/3.02 -> FAxioms: 2.93/3.02 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) -> *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) -> +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) -> U(x4,x3) 2.93/3.02 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.93/3.02 *#(x3,x4) -> *#(x4,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 ->->-> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 *#(*(x3,x4),x5) -> *#(x3,x4) 2.93/3.02 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.93/3.02 ->->Cycle: 2.93/3.02 ->->-> Pairs: 2.93/3.02 PROD(U(x,y)) -> PROD(x) 2.93/3.02 PROD(U(x,y)) -> PROD(y) 2.93/3.02 -> FAxioms: 2.93/3.02 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) -> *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) -> +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) -> U(x4,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 ->->-> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> SRules: 2.93/3.02 Empty 2.93/3.02 2.93/3.02 2.93/3.02 The problem is decomposed in 5 subproblems. 2.93/3.02 2.93/3.02 Problem 1.1: 2.93/3.02 2.93/3.02 Reduction Pairs Processor: 2.93/3.02 -> FAxioms: 2.93/3.02 U#(U(x3,x4),x5) = U#(x3,U(x4,x5)) 2.93/3.02 U#(x3,x4) = U#(x4,x3) 2.93/3.02 -> Pairs: 2.93/3.02 U#(U(empty,b),x3) -> U#(b,x3) 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 -> Usable Equations: 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 -> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.02 U(empty,b) -> b 2.93/3.02 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.02 prod(empty) -> 1(#) 2.93/3.02 prod(singl(x)) -> x 2.93/3.02 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.02 sum(empty) -> 0(#) 2.93/3.02 sum(singl(x)) -> x 2.93/3.02 -> Usable Rules: 2.93/3.02 U(empty,b) -> b 2.93/3.02 -> SRules: 2.93/3.02 U#(U(x3,x4),x5) -> U#(x3,x4) 2.93/3.02 U#(x3,U(x4,x5)) -> U#(x4,x5) 2.93/3.02 ->Interpretation type: 2.93/3.02 Linear 2.93/3.02 ->Coefficients: 2.93/3.02 Natural Numbers 2.93/3.02 ->Dimension: 2.93/3.02 1 2.93/3.02 ->Bound: 2.93/3.02 2 2.93/3.02 ->Interpretation: 2.93/3.02 2.93/3.02 [*](X1,X2) = 0 2.93/3.02 [+](X1,X2) = 0 2.93/3.02 [0](X) = 0 2.93/3.02 [U](X1,X2) = X1 + X2 2.93/3.02 [prod](X) = 0 2.93/3.02 [sum](X) = 0 2.93/3.02 [#] = 0 2.93/3.02 [1](X) = 0 2.93/3.02 [empty] = 2 2.93/3.02 [singl](X) = 0 2.93/3.02 [*#](X1,X2) = 0 2.93/3.02 [+#](X1,X2) = 0 2.93/3.02 [0#](X) = 0 2.93/3.02 [U#](X1,X2) = 2.X1 + 2.X2 2.93/3.02 [PROD](X) = 0 2.93/3.02 [SUM](X) = 0 2.93/3.02 2.93/3.02 Problem 1.1: 2.93/3.02 2.93/3.02 SCC Processor: 2.93/3.02 -> FAxioms: 2.93/3.02 U#(U(x3,x4),x5) = U#(x3,U(x4,x5)) 2.93/3.02 U#(x3,x4) = U#(x4,x3) 2.93/3.02 -> Pairs: 2.93/3.02 Empty 2.93/3.02 -> EAxioms: 2.93/3.02 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.02 *(x3,x4) = *(x4,x3) 2.93/3.02 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.02 +(x3,x4) = +(x4,x3) 2.93/3.02 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.02 U(x3,x4) = U(x4,x3) 2.93/3.02 -> Rules: 2.93/3.02 *(0(x),y) -> 0(*(x,y)) 2.93/3.02 *(#,x) -> # 2.93/3.02 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.02 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.02 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.02 +(#,x) -> x 2.93/3.02 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.02 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 U#(U(x3,x4),x5) -> U#(x3,x4) 2.93/3.03 U#(x3,U(x4,x5)) -> U#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 There is no strongly connected component 2.93/3.03 2.93/3.03 The problem is finite. 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),0(y)),x3) -> +#(0(+(x,y)),x3) 2.93/3.03 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Interpretation type: 2.93/3.03 Linear 2.93/3.03 ->Coefficients: 2.93/3.03 Natural Numbers 2.93/3.03 ->Dimension: 2.93/3.03 1 2.93/3.03 ->Bound: 2.93/3.03 2 2.93/3.03 ->Interpretation: 2.93/3.03 2.93/3.03 [*](X1,X2) = 0 2.93/3.03 [+](X1,X2) = X1 + X2 + 1 2.93/3.03 [0](X) = X + 1 2.93/3.03 [U](X1,X2) = 0 2.93/3.03 [prod](X) = 0 2.93/3.03 [sum](X) = 0 2.93/3.03 [#] = 0 2.93/3.03 [1](X) = X + 2 2.93/3.03 [empty] = 0 2.93/3.03 [singl](X) = 0 2.93/3.03 [*#](X1,X2) = 0 2.93/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.93/3.03 [0#](X) = 0 2.93/3.03 [U#](X1,X2) = 0 2.93/3.03 [PROD](X) = 0 2.93/3.03 [SUM](X) = 0 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 SCC Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 ->->Cycle: 2.93/3.03 ->->-> Pairs: 2.93/3.03 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> FAxioms: 2.93/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) -> *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) -> +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) -> U(x4,x3) 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) -> +#(x4,x3) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 ->->-> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),0(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Interpretation type: 2.93/3.03 Linear 2.93/3.03 ->Coefficients: 2.93/3.03 Natural Numbers 2.93/3.03 ->Dimension: 2.93/3.03 1 2.93/3.03 ->Bound: 2.93/3.03 2 2.93/3.03 ->Interpretation: 2.93/3.03 2.93/3.03 [*](X1,X2) = 0 2.93/3.03 [+](X1,X2) = X1 + X2 2.93/3.03 [0](X) = X + 1 2.93/3.03 [U](X1,X2) = 0 2.93/3.03 [prod](X) = 0 2.93/3.03 [sum](X) = 0 2.93/3.03 [#] = 1 2.93/3.03 [1](X) = X + 2 2.93/3.03 [empty] = 0 2.93/3.03 [singl](X) = 0 2.93/3.03 [*#](X1,X2) = 0 2.93/3.03 [+#](X1,X2) = X1 + X2 2.93/3.03 [0#](X) = 0 2.93/3.03 [U#](X1,X2) = 0 2.93/3.03 [PROD](X) = 0 2.93/3.03 [SUM](X) = 0 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 SCC Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 ->->Cycle: 2.93/3.03 ->->-> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> FAxioms: 2.93/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) -> *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) -> +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) -> U(x4,x3) 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) -> +#(x4,x3) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 ->->-> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(1(+(x,y)),x3) 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Interpretation type: 2.93/3.03 Linear 2.93/3.03 ->Coefficients: 2.93/3.03 Natural Numbers 2.93/3.03 ->Dimension: 2.93/3.03 1 2.93/3.03 ->Bound: 2.93/3.03 2 2.93/3.03 ->Interpretation: 2.93/3.03 2.93/3.03 [*](X1,X2) = 0 2.93/3.03 [+](X1,X2) = X1 + X2 2.93/3.03 [0](X) = X + 1 2.93/3.03 [U](X1,X2) = 0 2.93/3.03 [prod](X) = 0 2.93/3.03 [sum](X) = 0 2.93/3.03 [#] = 1 2.93/3.03 [1](X) = X + 2 2.93/3.03 [empty] = 0 2.93/3.03 [singl](X) = 0 2.93/3.03 [*#](X1,X2) = 0 2.93/3.03 [+#](X1,X2) = X1 + X2 2.93/3.03 [0#](X) = 0 2.93/3.03 [U#](X1,X2) = 0 2.93/3.03 [PROD](X) = 0 2.93/3.03 [SUM](X) = 0 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 SCC Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 ->->Cycle: 2.93/3.03 ->->-> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> FAxioms: 2.93/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) -> *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) -> +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) -> U(x4,x3) 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) -> +#(x4,x3) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 ->->-> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(0(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Interpretation type: 2.93/3.03 Linear 2.93/3.03 ->Coefficients: 2.93/3.03 Natural Numbers 2.93/3.03 ->Dimension: 2.93/3.03 1 2.93/3.03 ->Bound: 2.93/3.03 2 2.93/3.03 ->Interpretation: 2.93/3.03 2.93/3.03 [*](X1,X2) = 0 2.93/3.03 [+](X1,X2) = X1 + X2 + 2 2.93/3.03 [0](X) = X 2.93/3.03 [U](X1,X2) = 0 2.93/3.03 [prod](X) = 0 2.93/3.03 [sum](X) = 0 2.93/3.03 [#] = 0 2.93/3.03 [1](X) = X + 2 2.93/3.03 [empty] = 0 2.93/3.03 [singl](X) = 0 2.93/3.03 [*#](X1,X2) = 0 2.93/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.93/3.03 [0#](X) = 0 2.93/3.03 [U#](X1,X2) = 0 2.93/3.03 [PROD](X) = 0 2.93/3.03 [SUM](X) = 0 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 SCC Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 ->->Cycle: 2.93/3.03 ->->-> Pairs: 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> FAxioms: 2.93/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) -> *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) -> +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) -> U(x4,x3) 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) -> +#(x4,x3) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 ->->-> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(#,x),x3) -> +#(x,x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Interpretation type: 2.93/3.03 Linear 2.93/3.03 ->Coefficients: 2.93/3.03 Natural Numbers 2.93/3.03 ->Dimension: 2.93/3.03 1 2.93/3.03 ->Bound: 2.93/3.03 2 2.93/3.03 ->Interpretation: 2.93/3.03 2.93/3.03 [*](X1,X2) = 0 2.93/3.03 [+](X1,X2) = X1 + X2 + 1 2.93/3.03 [0](X) = X 2.93/3.03 [U](X1,X2) = 0 2.93/3.03 [prod](X) = 0 2.93/3.03 [sum](X) = 0 2.93/3.03 [#] = 1 2.93/3.03 [1](X) = X + 2 2.93/3.03 [empty] = 0 2.93/3.03 [singl](X) = 0 2.93/3.03 [*#](X1,X2) = 0 2.93/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.93/3.03 [0#](X) = 0 2.93/3.03 [U#](X1,X2) = 0 2.93/3.03 [PROD](X) = 0 2.93/3.03 [SUM](X) = 0 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 SCC Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 ->Strongly Connected Components: 2.93/3.03 ->->Cycle: 2.93/3.03 ->->-> Pairs: 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> FAxioms: 2.93/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) -> *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) -> +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) -> U(x4,x3) 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) -> +#(x4,x3) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 ->->-> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> SRules: 2.93/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.93/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.93/3.03 2.93/3.03 Problem 1.2: 2.93/3.03 2.93/3.03 Reduction Pairs Processor: 2.93/3.03 -> FAxioms: 2.93/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.93/3.03 +#(x3,x4) = +#(x4,x3) 2.93/3.03 -> Pairs: 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(0(+(1(#),+(x,y))),x3) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.93/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.93/3.03 +#(0(x),0(y)) -> +#(x,y) 2.93/3.03 +#(0(x),1(y)) -> +#(x,y) 2.93/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.93/3.03 +#(1(x),1(y)) -> +#(x,y) 2.93/3.03 -> EAxioms: 2.93/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.93/3.03 *(x3,x4) = *(x4,x3) 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.93/3.03 U(x3,x4) = U(x4,x3) 2.93/3.03 -> Usable Equations: 2.93/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.93/3.03 +(x3,x4) = +(x4,x3) 2.93/3.03 -> Rules: 2.93/3.03 *(0(x),y) -> 0(*(x,y)) 2.93/3.03 *(#,x) -> # 2.93/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.93/3.03 +(#,x) -> x 2.93/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.93/3.03 0(#) -> # 2.93/3.03 U(empty,b) -> b 2.93/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.93/3.03 prod(empty) -> 1(#) 2.93/3.03 prod(singl(x)) -> x 2.93/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.93/3.03 sum(empty) -> 0(#) 2.93/3.03 sum(singl(x)) -> x 2.93/3.03 -> Usable Rules: 2.93/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.93/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 2.97/3.03 [0](X) = X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 1 2.97/3.03 [1](X) = X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = X1 + X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(1(#),+(x,y)) 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 + 1 2.97/3.03 [0](X) = X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 1 2.97/3.03 [1](X) = X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(+(1(x),1(y)),x3) -> +#(x,y) 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 2.97/3.03 [0](X) = 2.X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = 2.X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,x4) 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 + 1 2.97/3.03 [0](X) = X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = X + 1 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),0(y)) -> +#(x,y) 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 + 1 2.97/3.03 [0](X) = X + 1 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(0(x),1(y)) -> +#(x,y) 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 + 2 2.97/3.03 [0](X) = X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = X1 + X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(1(#),+(x,y)) 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 2.97/3.03 [0](X) = X + 1 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = X + 2 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 ->->Cycle: 2.97/3.03 ->->-> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> FAxioms: 2.97/3.03 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) -> *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) -> +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) -> U(x4,x3) 2.97/3.03 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) -> +#(x4,x3) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 ->->-> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 +#(1(x),1(y)) -> +#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Linear 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 2 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = 0 2.97/3.03 [+](X1,X2) = X1 + X2 2.97/3.03 [0](X) = X 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 1 2.97/3.03 [1](X) = X + 1 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = 0 2.97/3.03 [+#](X1,X2) = 2.X1 + 2.X2 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.2: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 +#(+(x3,x4),x5) = +#(x3,+(x4,x5)) 2.97/3.03 +#(x3,x4) = +#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 Empty 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 +#(x3,+(x4,x5)) -> +#(x4,x5) 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 There is no strongly connected component 2.97/3.03 2.97/3.03 The problem is finite. 2.97/3.03 2.97/3.03 Problem 1.3: 2.97/3.03 2.97/3.03 Subterm Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 Empty 2.97/3.03 -> Pairs: 2.97/3.03 SUM(U(x,y)) -> SUM(x) 2.97/3.03 SUM(U(x,y)) -> SUM(y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 Empty 2.97/3.03 ->Projection: 2.97/3.03 pi(SUM) = [1] 2.97/3.03 2.97/3.03 Problem 1.3: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 Empty 2.97/3.03 -> Pairs: 2.97/3.03 Empty 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> SRules: 2.97/3.03 Empty 2.97/3.03 ->Strongly Connected Components: 2.97/3.03 There is no strongly connected component 2.97/3.03 2.97/3.03 The problem is finite. 2.97/3.03 2.97/3.03 Problem 1.4: 2.97/3.03 2.97/3.03 Reduction Pairs Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.03 *#(x3,x4) = *#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.03 *#(*(0(x),y),x3) -> *#(x,y) 2.97/3.03 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.03 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.97/3.03 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.03 *#(0(x),y) -> *#(x,y) 2.97/3.03 *#(1(x),y) -> *#(x,y) 2.97/3.03 -> EAxioms: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.03 U(x3,x4) = U(x4,x3) 2.97/3.03 -> Usable Equations: 2.97/3.03 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.03 *(x3,x4) = *(x4,x3) 2.97/3.03 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.03 +(x3,x4) = +(x4,x3) 2.97/3.03 -> Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 U(empty,b) -> b 2.97/3.03 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.03 prod(empty) -> 1(#) 2.97/3.03 prod(singl(x)) -> x 2.97/3.03 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.03 sum(empty) -> 0(#) 2.97/3.03 sum(singl(x)) -> x 2.97/3.03 -> Usable Rules: 2.97/3.03 *(0(x),y) -> 0(*(x,y)) 2.97/3.03 *(#,x) -> # 2.97/3.03 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.03 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.03 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.03 +(#,x) -> x 2.97/3.03 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.03 0(#) -> # 2.97/3.03 -> SRules: 2.97/3.03 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.03 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.03 ->Interpretation type: 2.97/3.03 Simple mixed 2.97/3.03 ->Coefficients: 2.97/3.03 Natural Numbers 2.97/3.03 ->Dimension: 2.97/3.03 1 2.97/3.03 ->Bound: 2.97/3.03 1 2.97/3.03 ->Interpretation: 2.97/3.03 2.97/3.03 [*](X1,X2) = X1.X2 + X1 + X2 2.97/3.03 [+](X1,X2) = X1 + X2 2.97/3.03 [0](X) = X + 1 2.97/3.03 [U](X1,X2) = 0 2.97/3.03 [prod](X) = 0 2.97/3.03 [sum](X) = 0 2.97/3.03 [#] = 0 2.97/3.03 [1](X) = X + 1 2.97/3.03 [empty] = 0 2.97/3.03 [singl](X) = 0 2.97/3.03 [*#](X1,X2) = X1.X2 + X1 + X2 2.97/3.03 [+#](X1,X2) = 0 2.97/3.03 [0#](X) = 0 2.97/3.03 [U#](X1,X2) = 0 2.97/3.03 [PROD](X) = 0 2.97/3.03 [SUM](X) = 0 2.97/3.03 2.97/3.03 Problem 1.4: 2.97/3.03 2.97/3.03 SCC Processor: 2.97/3.03 -> FAxioms: 2.97/3.03 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.03 *#(x3,x4) = *#(x4,x3) 2.97/3.03 -> Pairs: 2.97/3.03 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.03 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.03 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(+(0(*(x,y)),y),x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Simple mixed 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 1 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+](X1,X2) = X1 + X2 2.97/3.04 [0](X) = X 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 1 2.97/3.04 [1](X) = X + 1 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(*(1(x),y),x3) -> *#(x,y) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Simple mixed 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 1 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+](X1,X2) = X1 + X2 2.97/3.04 [0](X) = X 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 1 2.97/3.04 [1](X) = X + 1 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(0(x),y) -> *#(x,y) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Simple mixed 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 1 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+](X1,X2) = X1 + X2 2.97/3.04 [0](X) = X + 1 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 0 2.97/3.04 [1](X) = X + 1 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 *#(1(x),y) -> *#(x,y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Simple mixed 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 1 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+](X1,X2) = X1 + X2 2.97/3.04 [0](X) = X 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 1 2.97/3.04 [1](X) = X + 1 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = X1.X2 + X1 + X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(0(x),y),x3) -> *#(0(*(x,y)),x3) 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Linear 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 2 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1 + X2 + 2 2.97/3.04 [+](X1,X2) = X1 + X2 + 1 2.97/3.04 [0](X) = 2 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 2 2.97/3.04 [1](X) = 1 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = 2.X1 + 2.X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 ->->Cycle: 2.97/3.04 ->->-> Pairs: 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> FAxioms: 2.97/3.04 *(*(x3,x4),x5) -> *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) -> *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) -> +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) -> +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) -> U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) -> U(x4,x3) 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) -> *#(x4,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 ->->-> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 Reduction Pairs Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 *#(*(#,x),x3) -> *#(#,x3) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Usable Equations: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> Usable Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Interpretation type: 2.97/3.04 Linear 2.97/3.04 ->Coefficients: 2.97/3.04 Natural Numbers 2.97/3.04 ->Dimension: 2.97/3.04 1 2.97/3.04 ->Bound: 2.97/3.04 2 2.97/3.04 ->Interpretation: 2.97/3.04 2.97/3.04 [*](X1,X2) = X1 + X2 + 2 2.97/3.04 [+](X1,X2) = X1 + X2 2.97/3.04 [0](X) = 0 2.97/3.04 [U](X1,X2) = 0 2.97/3.04 [prod](X) = 0 2.97/3.04 [sum](X) = 0 2.97/3.04 [#] = 0 2.97/3.04 [1](X) = 0 2.97/3.04 [empty] = 0 2.97/3.04 [singl](X) = 0 2.97/3.04 [*#](X1,X2) = 2.X1 + 2.X2 2.97/3.04 [+#](X1,X2) = 0 2.97/3.04 [0#](X) = 0 2.97/3.04 [U#](X1,X2) = 0 2.97/3.04 [PROD](X) = 0 2.97/3.04 [SUM](X) = 0 2.97/3.04 2.97/3.04 Problem 1.4: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 *#(*(x3,x4),x5) = *#(x3,*(x4,x5)) 2.97/3.04 *#(x3,x4) = *#(x4,x3) 2.97/3.04 -> Pairs: 2.97/3.04 Empty 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 *#(*(x3,x4),x5) -> *#(x3,x4) 2.97/3.04 *#(x3,*(x4,x5)) -> *#(x4,x5) 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 There is no strongly connected component 2.97/3.04 2.97/3.04 The problem is finite. 2.97/3.04 2.97/3.04 Problem 1.5: 2.97/3.04 2.97/3.04 Subterm Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 Empty 2.97/3.04 -> Pairs: 2.97/3.04 PROD(U(x,y)) -> PROD(x) 2.97/3.04 PROD(U(x,y)) -> PROD(y) 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 Empty 2.97/3.04 ->Projection: 2.97/3.04 pi(PROD) = [1] 2.97/3.04 2.97/3.04 Problem 1.5: 2.97/3.04 2.97/3.04 SCC Processor: 2.97/3.04 -> FAxioms: 2.97/3.04 Empty 2.97/3.04 -> Pairs: 2.97/3.04 Empty 2.97/3.04 -> EAxioms: 2.97/3.04 *(*(x3,x4),x5) = *(x3,*(x4,x5)) 2.97/3.04 *(x3,x4) = *(x4,x3) 2.97/3.04 +(+(x3,x4),x5) = +(x3,+(x4,x5)) 2.97/3.04 +(x3,x4) = +(x4,x3) 2.97/3.04 U(U(x3,x4),x5) = U(x3,U(x4,x5)) 2.97/3.04 U(x3,x4) = U(x4,x3) 2.97/3.04 -> Rules: 2.97/3.04 *(0(x),y) -> 0(*(x,y)) 2.97/3.04 *(#,x) -> # 2.97/3.04 *(1(x),y) -> +(0(*(x,y)),y) 2.97/3.04 +(0(x),0(y)) -> 0(+(x,y)) 2.97/3.04 +(0(x),1(y)) -> 1(+(x,y)) 2.97/3.04 +(#,x) -> x 2.97/3.04 +(1(x),1(y)) -> 0(+(1(#),+(x,y))) 2.97/3.04 0(#) -> # 2.97/3.04 U(empty,b) -> b 2.97/3.04 prod(U(x,y)) -> *(prod(x),prod(y)) 2.97/3.04 prod(empty) -> 1(#) 2.97/3.04 prod(singl(x)) -> x 2.97/3.04 sum(U(x,y)) -> +(sum(x),sum(y)) 2.97/3.04 sum(empty) -> 0(#) 2.97/3.04 sum(singl(x)) -> x 2.97/3.04 -> SRules: 2.97/3.04 Empty 2.97/3.04 ->Strongly Connected Components: 2.97/3.04 There is no strongly connected component 2.97/3.04 2.97/3.04 The problem is finite. 2.97/3.04 EOF