4.12/4.74 YES 4.12/4.74 4.12/4.74 Problem 1: 4.12/4.74 4.12/4.74 (VAR x y z) 4.12/4.74 (THEORY 4.12/4.74 (AC plus times)) 4.12/4.74 (RULES 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 ) 4.12/4.74 4.12/4.74 Problem 1: 4.12/4.74 4.12/4.74 Dependency Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 TIMES(times(s(x),y),x3) -> PLUS(times(x,y),y) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> PLUS(times(x,y),y) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 TIMES(times(s(x),y),x3) -> PLUS(times(x,y),y) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> PLUS(times(x,y),times(x,z)) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> PLUS(times(x,y),y) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> PLUS(times(x,y),times(x,z)) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) -> PLUS(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 4.12/4.74 The problem is decomposed in 2 subproblems. 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(x,y) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Linear 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 2 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 0 4.12/4.74 [0] = 0 4.12/4.74 [s](X) = X + 2 4.12/4.74 [PLUS](X1,X2) = 2.X1 + 2.X2 4.12/4.74 [TIMES](X1,X2) = 0 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) -> PLUS(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(plus(x,0),x3) -> PLUS(x,x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Linear 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 2 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 0 4.12/4.74 [0] = 2 4.12/4.74 [s](X) = X + 2 4.12/4.74 [PLUS](X1,X2) = 2.X1 + 2.X2 4.12/4.74 [TIMES](X1,X2) = 0 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) -> PLUS(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 -> SRules: 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Linear 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 2 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = 0 4.12/4.74 [0] = 0 4.12/4.74 [s](X) = X + 2 4.12/4.74 [PLUS](X1,X2) = 2.X1 + 2.X2 4.12/4.74 [TIMES](X1,X2) = 0 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) -> PLUS(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 PLUS(s(x),y) -> PLUS(x,y) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Linear 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 2 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 0 4.12/4.74 [0] = 0 4.12/4.74 [s](X) = X + 2 4.12/4.74 [PLUS](X1,X2) = 2.X1 + 2.X2 4.12/4.74 [TIMES](X1,X2) = 0 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) -> PLUS(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 PLUS(plus(s(x),y),x3) -> PLUS(s(plus(x,y)),x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Linear 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 2 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 0 4.12/4.74 [0] = 0 4.12/4.74 [s](X) = 0 4.12/4.74 [PLUS](X1,X2) = 2.X1 + 2.X2 4.12/4.74 [TIMES](X1,X2) = 0 4.12/4.74 4.12/4.74 Problem 1.1: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 4.12/4.74 PLUS(x3,x4) = PLUS(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 Empty 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 There is no strongly connected component 4.12/4.74 4.12/4.74 The problem is finite. 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,y) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(x,z) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(s(x),y) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,y) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 TIMES(x,plus(y,z)) -> TIMES(x,z) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 Natural Numbers 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 1 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 1 4.12/4.74 [times](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 [0] = 1 4.12/4.74 [s](X) = X + 1 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = X1.X2 + X1 + X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(s(x),y),x3) -> TIMES(plus(times(x,y),y),x3) 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 All rationals 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 3 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 1/2.X1.X2 + X1 + X2 4.12/4.74 [0] = 0 4.12/4.74 [s](X) = X + 3 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = 1/3.X1.X2 + 2/3.X1 + 2/3.X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(x,plus(y,z)),x3) -> TIMES(plus(times(x,y),times(x,z)),x3) 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 All rationals 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 3 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 2 4.12/4.74 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 4.12/4.74 [0] = 3/2 4.12/4.74 [s](X) = X + 3 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 ->->Cycle: 4.12/4.74 ->->-> Pairs: 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> FAxioms: 4.12/4.74 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) -> plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) -> times(x4,x3) 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) -> TIMES(x4,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 ->->-> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 Reduction Pairs Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 TIMES(times(x,0),x3) -> TIMES(0,x3) 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Usable Equations: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> Usable Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Interpretation type: 4.12/4.74 Simple mixed 4.12/4.74 ->Coefficients: 4.12/4.74 All rationals 4.12/4.74 ->Dimension: 4.12/4.74 1 4.12/4.74 ->Bound: 4.12/4.74 3 4.12/4.74 ->Interpretation: 4.12/4.74 4.12/4.74 [plus](X1,X2) = X1 + X2 + 3/2 4.12/4.74 [times](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1 4.12/4.74 [0] = 3/2 4.12/4.74 [s](X) = X + 3/2 4.12/4.74 [PLUS](X1,X2) = 0 4.12/4.74 [TIMES](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 4.12/4.74 4.12/4.74 Problem 1.2: 4.12/4.74 4.12/4.74 SCC Processor: 4.12/4.74 -> FAxioms: 4.12/4.74 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 4.12/4.74 TIMES(x3,x4) = TIMES(x4,x3) 4.12/4.74 -> Pairs: 4.12/4.74 Empty 4.12/4.74 -> EAxioms: 4.12/4.74 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 4.12/4.74 plus(x3,x4) = plus(x4,x3) 4.12/4.74 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 4.12/4.74 times(x3,x4) = times(x4,x3) 4.12/4.74 -> Rules: 4.12/4.74 plus(s(x),y) -> s(plus(x,y)) 4.12/4.74 plus(x,0) -> x 4.12/4.74 times(s(x),y) -> plus(times(x,y),y) 4.12/4.74 times(x,plus(y,z)) -> plus(times(x,y),times(x,z)) 4.12/4.74 times(x,0) -> 0 4.12/4.74 -> SRules: 4.12/4.74 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 4.12/4.74 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 4.12/4.74 ->Strongly Connected Components: 4.12/4.74 There is no strongly connected component 4.12/4.74 4.12/4.74 The problem is finite. 4.12/4.74 EOF