4.35/5.21 YES 4.35/5.21 4.35/5.21 Problem 1: 4.35/5.21 4.35/5.21 (VAR x y z) 4.35/5.21 (THEORY 4.35/5.21 (AC plus times)) 4.35/5.21 (RULES 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 ) 4.35/5.21 4.35/5.21 Problem 1: 4.35/5.21 4.35/5.21 Dependency Pairs Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.21 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(x,times(s(y),z)) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.21 TIMES(x,s(y)) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.21 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.21 4.35/5.21 Problem 1: 4.35/5.21 4.35/5.21 SCC Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.21 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(x,times(s(y),z)) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.21 TIMES(x,s(y)) -> PLUS(times(x,y),x) 4.35/5.21 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.21 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.21 ->Strongly Connected Components: 4.35/5.21 ->->Cycle: 4.35/5.21 ->->-> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> FAxioms: 4.35/5.21 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) -> plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) -> times(x4,x3) 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 ->->-> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 ->->Cycle: 4.35/5.21 ->->-> Pairs: 4.35/5.21 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.21 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.21 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.21 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.21 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.21 -> FAxioms: 4.35/5.21 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) -> plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) -> times(x4,x3) 4.35/5.21 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.21 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 ->->-> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.21 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.21 4.35/5.21 4.35/5.21 The problem is decomposed in 2 subproblems. 4.35/5.21 4.35/5.21 Problem 1.1: 4.35/5.21 4.35/5.21 Reduction Pairs Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Usable Equations: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> Usable Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 ->Interpretation type: 4.35/5.21 Linear 4.35/5.21 ->Coefficients: 4.35/5.21 Natural Numbers 4.35/5.21 ->Dimension: 4.35/5.21 1 4.35/5.21 ->Bound: 4.35/5.21 2 4.35/5.21 ->Interpretation: 4.35/5.21 4.35/5.21 [plus](X1,X2) = X1 + X2 + 2 4.35/5.21 [times](X1,X2) = 0 4.35/5.21 [0] = 0 4.35/5.21 [s](X) = X 4.35/5.21 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.21 [TIMES](X1,X2) = 0 4.35/5.21 4.35/5.21 Problem 1.1: 4.35/5.21 4.35/5.21 SCC Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 ->Strongly Connected Components: 4.35/5.21 ->->Cycle: 4.35/5.21 ->->-> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> FAxioms: 4.35/5.21 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) -> plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) -> times(x4,x3) 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 ->->-> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 4.35/5.21 Problem 1.1: 4.35/5.21 4.35/5.21 Reduction Pairs Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Usable Equations: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> Usable Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.21 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.21 ->Interpretation type: 4.35/5.21 Linear 4.35/5.21 ->Coefficients: 4.35/5.21 Natural Numbers 4.35/5.21 ->Dimension: 4.35/5.21 1 4.35/5.21 ->Bound: 4.35/5.21 2 4.35/5.21 ->Interpretation: 4.35/5.21 4.35/5.21 [plus](X1,X2) = X1 + X2 4.35/5.21 [times](X1,X2) = 0 4.35/5.21 [0] = 0 4.35/5.21 [s](X) = X + 2 4.35/5.21 [PLUS](X1,X2) = X1 + X2 4.35/5.21 [TIMES](X1,X2) = 0 4.35/5.21 4.35/5.21 Problem 1.1: 4.35/5.21 4.35/5.21 SCC Processor: 4.35/5.21 -> FAxioms: 4.35/5.21 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.21 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.21 -> Pairs: 4.35/5.21 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.21 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.21 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.21 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.21 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.21 -> EAxioms: 4.35/5.21 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.21 plus(x3,x4) = plus(x4,x3) 4.35/5.21 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.21 times(x3,x4) = times(x4,x3) 4.35/5.21 -> Rules: 4.35/5.21 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.21 plus(x,0) -> x 4.35/5.21 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.21 times(x,times(0,z)) -> times(0,z) 4.35/5.21 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.21 times(x,0) -> 0 4.35/5.21 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.21 -> SRules: 4.35/5.21 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X 4.35/5.22 [PLUS](X1,X2) = X1 + X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X + 2 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(x,y) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 PLUS(x,s(y)) -> PLUS(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X + 2 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,plus(s(y),z)),x3) -> PLUS(plus(s(plus(x,y)),z),x3) 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = 0 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = 0 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) -> PLUS(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 PLUS(x,plus(s(y),z)) -> PLUS(s(plus(x,y)),z) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Linear 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 2 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 2 4.35/5.22 [times](X1,X2) = 0 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = 0 4.35/5.22 [PLUS](X1,X2) = 2.X1 + 2.X2 4.35/5.22 [TIMES](X1,X2) = 0 4.35/5.22 4.35/5.22 Problem 1.1: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.35/5.22 PLUS(x3,x4) = PLUS(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 Empty 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 There is no strongly connected component 4.35/5.22 4.35/5.22 The problem is finite. 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(times(plus(times(x,y),x),z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,times(s(y),z)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(plus(times(x,y),x),x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(times(x,s(y)),x3) -> TIMES(x,y) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(plus(times(x,y),x),z) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,times(s(y),z)) -> TIMES(x,y) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 TIMES(x,s(y)) -> TIMES(x,y) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 Natural Numbers 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 1 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 1 4.35/5.22 [times](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 [0] = 1 4.35/5.22 [s](X) = X + 1 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3) 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 All rationals 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 3 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 4.35/5.22 [times](X1,X2) = 2.X1.X2 + 3.X1 + 3.X2 + 3 4.35/5.22 [0] = 0 4.35/5.22 [s](X) = X + 2 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = 1/3.X1.X2 + 1/2.X1 + 1/2.X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 ->->Cycle: 4.35/5.22 ->->-> Pairs: 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> FAxioms: 4.35/5.22 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) -> plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) -> times(x4,x3) 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) -> TIMES(x4,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 ->->-> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 Reduction Pairs Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Usable Equations: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> Usable Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Interpretation type: 4.35/5.22 Simple mixed 4.35/5.22 ->Coefficients: 4.35/5.22 All rationals 4.35/5.22 ->Dimension: 4.35/5.22 1 4.35/5.22 ->Bound: 4.35/5.22 3 4.35/5.22 ->Interpretation: 4.35/5.22 4.35/5.22 [plus](X1,X2) = X1 + X2 + 3/2 4.35/5.22 [times](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2 4.35/5.22 [0] = 3 4.35/5.22 [s](X) = X + 2 4.35/5.22 [PLUS](X1,X2) = 0 4.35/5.22 [TIMES](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 4.35/5.22 4.35/5.22 Problem 1.2: 4.35/5.22 4.35/5.22 SCC Processor: 4.35/5.22 -> FAxioms: 4.35/5.22 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.35/5.22 TIMES(x3,x4) = TIMES(x4,x3) 4.35/5.22 -> Pairs: 4.35/5.22 Empty 4.35/5.22 -> EAxioms: 4.35/5.22 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.35/5.22 plus(x3,x4) = plus(x4,x3) 4.35/5.22 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.35/5.22 times(x3,x4) = times(x4,x3) 4.35/5.22 -> Rules: 4.35/5.22 plus(x,plus(s(y),z)) -> plus(s(plus(x,y)),z) 4.35/5.22 plus(x,0) -> x 4.35/5.22 plus(x,s(y)) -> s(plus(x,y)) 4.35/5.22 times(x,times(0,z)) -> times(0,z) 4.35/5.22 times(x,times(s(y),z)) -> times(plus(times(x,y),x),z) 4.35/5.22 times(x,0) -> 0 4.35/5.22 times(x,s(y)) -> plus(times(x,y),x) 4.35/5.22 -> SRules: 4.35/5.22 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.35/5.22 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.35/5.22 ->Strongly Connected Components: 4.35/5.22 There is no strongly connected component 4.35/5.22 4.35/5.22 The problem is finite. 4.35/5.22 EOF