2.11/2.35 YES 2.11/2.35 2.11/2.35 Problem 1: 2.11/2.35 2.11/2.35 (VAR x y) 2.11/2.35 (THEORY 2.11/2.35 (AC plus times)) 2.11/2.35 (RULES 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 ) 2.11/2.35 2.11/2.35 Problem 1: 2.11/2.35 2.11/2.35 Dependency Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.35 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(0(x),0(y)) -> 0#(plus(x,y)) 2.11/2.35 PLUS(0(x),0(y)) -> PLUS(x,y) 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y)) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> 0#(plus(x,plus(y,1(S)))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> 0#(plus(x,plus(y,1(S)))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 TIMES(0(x),y) -> 0#(times(x,y)) 2.11/2.35 TIMES(0(x),y) -> TIMES(x,y) 2.11/2.35 TIMES(times(0(x),y),x2) -> 0#(times(x,y)) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(1(x),y),x2) -> 0#(times(x,y)) 2.11/2.35 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.35 TIMES(1(x),y) -> 0#(times(x,y)) 2.11/2.35 TIMES(1(x),y) -> PLUS(0(times(x,y)),y) 2.11/2.35 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.35 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.35 2.11/2.35 Problem 1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.35 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(0(x),0(y)) -> 0#(plus(x,y)) 2.11/2.35 PLUS(0(x),0(y)) -> PLUS(x,y) 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y)) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> 0#(plus(x,plus(y,1(S)))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> 0#(plus(x,plus(y,1(S)))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 TIMES(0(x),y) -> 0#(times(x,y)) 2.11/2.35 TIMES(0(x),y) -> TIMES(x,y) 2.11/2.35 TIMES(times(0(x),y),x2) -> 0#(times(x,y)) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(1(x),y),x2) -> 0#(times(x,y)) 2.11/2.35 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.35 TIMES(1(x),y) -> 0#(times(x,y)) 2.11/2.35 TIMES(1(x),y) -> PLUS(0(times(x,y)),y) 2.11/2.35 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.35 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(0(x),0(y)) -> PLUS(x,y) 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 TIMES(0(x),y) -> TIMES(x,y) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.35 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.35 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.35 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.35 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.35 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.35 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.35 2.11/2.35 2.11/2.35 The problem is decomposed in 2 subproblems. 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 Reduction Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(0(x),0(y)) -> PLUS(x,y) 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Usable Equations: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> Usable Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Interpretation type: 2.11/2.35 Linear 2.11/2.35 ->Coefficients: 2.11/2.35 Natural Numbers 2.11/2.35 ->Dimension: 2.11/2.35 1 2.11/2.35 ->Bound: 2.11/2.35 2 2.11/2.35 ->Interpretation: 2.11/2.35 2.11/2.35 [0](X) = X + 1 2.11/2.35 [plus](X1,X2) = X1 + X2 + 1 2.11/2.35 [times](X1,X2) = 0 2.11/2.35 [1](X) = X + 2 2.11/2.35 [S] = 0 2.11/2.35 [0#](X) = 0 2.11/2.35 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.35 [TIMES](X1,X2) = 0 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 Reduction Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(0(x),1(y)) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Usable Equations: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> Usable Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Interpretation type: 2.11/2.35 Linear 2.11/2.35 ->Coefficients: 2.11/2.35 Natural Numbers 2.11/2.35 ->Dimension: 2.11/2.35 1 2.11/2.35 ->Bound: 2.11/2.35 2 2.11/2.35 ->Interpretation: 2.11/2.35 2.11/2.35 [0](X) = X 2.11/2.35 [plus](X1,X2) = X1 + X2 + 2 2.11/2.35 [times](X1,X2) = 0 2.11/2.35 [1](X) = X + 2 2.11/2.35 [S] = 0 2.11/2.35 [0#](X) = 0 2.11/2.35 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.35 [TIMES](X1,X2) = 0 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 Reduction Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Usable Equations: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> Usable Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Interpretation type: 2.11/2.35 Linear 2.11/2.35 ->Coefficients: 2.11/2.35 Natural Numbers 2.11/2.35 ->Dimension: 2.11/2.35 1 2.11/2.35 ->Bound: 2.11/2.35 2 2.11/2.35 ->Interpretation: 2.11/2.35 2.11/2.35 [0](X) = X + 1 2.11/2.35 [plus](X1,X2) = X1 + X2 + 1 2.11/2.35 [times](X1,X2) = 0 2.11/2.35 [1](X) = X + 2 2.11/2.35 [S] = 0 2.11/2.35 [0#](X) = 0 2.11/2.35 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.35 [TIMES](X1,X2) = 0 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 Reduction Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Usable Equations: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> Usable Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Interpretation type: 2.11/2.35 Linear 2.11/2.35 ->Coefficients: 2.11/2.35 Natural Numbers 2.11/2.35 ->Dimension: 2.11/2.35 1 2.11/2.35 ->Bound: 2.11/2.35 2 2.11/2.35 ->Interpretation: 2.11/2.35 2.11/2.35 [0](X) = X 2.11/2.35 [plus](X1,X2) = X1 + X2 + 2 2.11/2.35 [times](X1,X2) = 0 2.11/2.35 [1](X) = X + 2 2.11/2.35 [S] = 0 2.11/2.35 [0#](X) = 0 2.11/2.35 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.35 [TIMES](X1,X2) = 0 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) -> plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) -> times(x3,x2) 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 ->->-> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 Reduction Pairs Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2) 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Usable Equations: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> Usable Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Interpretation type: 2.11/2.35 Linear 2.11/2.35 ->Coefficients: 2.11/2.35 Natural Numbers 2.11/2.35 ->Dimension: 2.11/2.35 1 2.11/2.35 ->Bound: 2.11/2.35 2 2.11/2.35 ->Interpretation: 2.11/2.35 2.11/2.35 [0](X) = X + 1 2.11/2.35 [plus](X1,X2) = X1 + X2 + 1 2.11/2.35 [times](X1,X2) = 0 2.11/2.35 [1](X) = X + 2 2.11/2.35 [S] = 0 2.11/2.35 [0#](X) = 0 2.11/2.35 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.35 [TIMES](X1,X2) = 0 2.11/2.35 2.11/2.35 Problem 1.1: 2.11/2.35 2.11/2.35 SCC Processor: 2.11/2.35 -> FAxioms: 2.11/2.35 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.35 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.35 -> Pairs: 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> EAxioms: 2.11/2.35 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.35 plus(x2,x3) = plus(x3,x2) 2.11/2.35 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.35 times(x2,x3) = times(x3,x2) 2.11/2.35 -> Rules: 2.11/2.35 0(S) -> S 2.11/2.35 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.35 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.35 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.35 plus(S,x) -> x 2.11/2.35 times(0(x),y) -> 0(times(x,y)) 2.11/2.35 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.35 times(S,x) -> S 2.11/2.35 -> SRules: 2.11/2.35 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.35 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.35 ->Strongly Connected Components: 2.11/2.35 ->->Cycle: 2.11/2.35 ->->-> Pairs: 2.11/2.35 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.35 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.35 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.35 -> FAxioms: 2.11/2.35 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 2 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(0(plus(x,plus(y,1(S)))),x2) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = 2.X 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = 2.X + 2 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(1(x),1(y)),x2) -> PLUS(y,1(S)) 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 1 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = X1 + X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(plus(S,x),x2) -> PLUS(x,x2) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 1 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = X1 + X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3) 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 1 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = X1 + X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(x,plus(y,1(S))) 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = 2.X 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = 2.X + 1 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) -> PLUS(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 PLUS(1(x),1(y)) -> PLUS(y,1(S)) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = 0 2.11/2.36 [1](X) = X + 2 2.11/2.36 [S] = 1 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 2.X1 + 2.X2 2.11/2.36 [TIMES](X1,X2) = 0 2.11/2.36 2.11/2.36 Problem 1.1: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4)) 2.11/2.36 PLUS(x2,x3) = PLUS(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 Empty 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 There is no strongly connected component 2.11/2.36 2.11/2.36 The problem is finite. 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(0(x),y) -> TIMES(x,y) 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Simple mixed 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 1 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X + 1 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 [1](X) = X + 1 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Simple mixed 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 1 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X + 1 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 [1](X) = X + 1 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Simple mixed 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 1 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 [1](X) = X + 1 2.11/2.36 [S] = 1 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(1(x),y),x2) -> TIMES(x,y) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Simple mixed 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 1 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X + 1 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 [1](X) = X + 1 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 TIMES(1(x),y) -> TIMES(x,y) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Simple mixed 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 1 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = X 2.11/2.36 [plus](X1,X2) = X1 + X2 + 1 2.11/2.36 [times](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 [1](X) = X + 1 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1.X2 + X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2) 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = 2 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1 + X2 + 1 2.11/2.36 [1](X) = 2 2.11/2.36 [S] = 2 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = X1 + X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 ->->Cycle: 2.11/2.36 ->->-> Pairs: 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> FAxioms: 2.11/2.36 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) -> plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) -> times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) -> times(x3,x2) 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) -> TIMES(x3,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 ->->-> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 Reduction Pairs Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 TIMES(times(S,x),x2) -> TIMES(S,x2) 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Usable Equations: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> Usable Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Interpretation type: 2.11/2.36 Linear 2.11/2.36 ->Coefficients: 2.11/2.36 Natural Numbers 2.11/2.36 ->Dimension: 2.11/2.36 1 2.11/2.36 ->Bound: 2.11/2.36 2 2.11/2.36 ->Interpretation: 2.11/2.36 2.11/2.36 [0](X) = 0 2.11/2.36 [plus](X1,X2) = X1 + X2 2.11/2.36 [times](X1,X2) = X1 + X2 + 1 2.11/2.36 [1](X) = 0 2.11/2.36 [S] = 0 2.11/2.36 [0#](X) = 0 2.11/2.36 [PLUS](X1,X2) = 0 2.11/2.36 [TIMES](X1,X2) = 2.X1 + 2.X2 2.11/2.36 2.11/2.36 Problem 1.2: 2.11/2.36 2.11/2.36 SCC Processor: 2.11/2.36 -> FAxioms: 2.11/2.36 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4)) 2.11/2.36 TIMES(x2,x3) = TIMES(x3,x2) 2.11/2.36 -> Pairs: 2.11/2.36 Empty 2.11/2.36 -> EAxioms: 2.11/2.36 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4)) 2.11/2.36 plus(x2,x3) = plus(x3,x2) 2.11/2.36 times(times(x2,x3),x4) = times(x2,times(x3,x4)) 2.11/2.36 times(x2,x3) = times(x3,x2) 2.11/2.36 -> Rules: 2.11/2.36 0(S) -> S 2.11/2.36 plus(0(x),0(y)) -> 0(plus(x,y)) 2.11/2.36 plus(0(x),1(y)) -> 1(plus(x,y)) 2.11/2.36 plus(1(x),1(y)) -> 0(plus(x,plus(y,1(S)))) 2.11/2.36 plus(S,x) -> x 2.11/2.36 times(0(x),y) -> 0(times(x,y)) 2.11/2.36 times(1(x),y) -> plus(0(times(x,y)),y) 2.11/2.36 times(S,x) -> S 2.11/2.36 -> SRules: 2.11/2.36 TIMES(times(x2,x3),x4) -> TIMES(x2,x3) 2.11/2.36 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4) 2.11/2.36 ->Strongly Connected Components: 2.11/2.36 There is no strongly connected component 2.11/2.36 2.11/2.36 The problem is finite. 2.11/2.36 EOF