0.60/0.70 YES 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 (VAR x xs y) 0.60/0.70 (THEORY 0.60/0.70 (AC plus)) 0.60/0.70 (RULES 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 S(nil) -> 0 0.60/0.70 int(0,0) -> cons(0,nil) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),0) -> nil 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 sum(x,y) -> S(int(x,y)) 0.60/0.70 ) 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 Reduction Order Processor: 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 S(nil) -> 0 0.60/0.70 int(0,0) -> cons(0,nil) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),0) -> nil 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 sum(x,y) -> S(int(x,y)) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = X 0.60/0.70 [int](X1,X2) = 2.X1 + 2.X2 + 2 0.60/0.70 [intlist](X) = X 0.60/0.70 [plus](X1,X2) = X1 + X2 0.60/0.70 [sum](X1,X2) = 2.X1 + 2.X2 + 2 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 2.X1 + X2 0.60/0.70 [nil] = 2 0.60/0.70 [s](X) = X 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 Reduction Order Processor: 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,0) -> cons(0,nil) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),0) -> nil 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 sum(x,y) -> S(int(x,y)) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = X + 1 0.60/0.70 [int](X1,X2) = 2.X1 + 2.X2 + 1 0.60/0.70 [intlist](X) = X 0.60/0.70 [plus](X1,X2) = X1 + X2 0.60/0.70 [sum](X1,X2) = 2.X1 + 2.X2 + 2 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 2.X1 + X2 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 Reduction Order Processor: 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),0) -> nil 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 sum(x,y) -> S(int(x,y)) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = X 0.60/0.70 [int](X1,X2) = 2.X1 + 2.X2 + 1 0.60/0.70 [intlist](X) = X 0.60/0.70 [plus](X1,X2) = X1 + X2 0.60/0.70 [sum](X1,X2) = 2.X1 + 2.X2 + 2 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 2.X1 + X2 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 Reduction Order Processor: 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 sum(x,y) -> S(int(x,y)) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = X 0.60/0.70 [int](X1,X2) = 2.X1 + 2.X2 0.60/0.70 [intlist](X) = X 0.60/0.70 [plus](X1,X2) = X1 + X2 0.60/0.70 [sum](X1,X2) = 2.X1 + 2.X2 + 1 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 2.X1 + X2 0.60/0.70 [nil] = 1 0.60/0.70 [s](X) = X 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 Dependency Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 S#(cons(x,xs)) -> S#(xs) 0.60/0.70 S#(cons(x,xs)) -> PLUS(x,S(xs)) 0.60/0.70 INT(0,s(y)) -> INT(s(0),s(y)) 0.60/0.70 INT(s(x),s(y)) -> INT(x,y) 0.60/0.70 INT(s(x),s(y)) -> INTLIST(int(x,y)) 0.60/0.70 INTLIST(cons(x,y)) -> INTLIST(y) 0.60/0.70 PLUS(plus(x,0),x3) -> PLUS(x,x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 0.60/0.70 Problem 1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 S#(cons(x,xs)) -> S#(xs) 0.60/0.70 S#(cons(x,xs)) -> PLUS(x,S(xs)) 0.60/0.70 INT(0,s(y)) -> INT(s(0),s(y)) 0.60/0.70 INT(s(x),s(y)) -> INT(x,y) 0.60/0.70 INT(s(x),s(y)) -> INTLIST(int(x,y)) 0.60/0.70 INTLIST(cons(x,y)) -> INTLIST(y) 0.60/0.70 PLUS(plus(x,0),x3) -> PLUS(x,x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 PLUS(plus(x,0),x3) -> PLUS(x,x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) -> PLUS(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 INTLIST(cons(x,y)) -> INTLIST(y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 Empty 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 INT(0,s(y)) -> INT(s(0),s(y)) 0.60/0.70 INT(s(x),s(y)) -> INT(x,y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 Empty 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 S#(cons(x,xs)) -> S#(xs) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 Empty 0.60/0.70 0.60/0.70 0.60/0.70 The problem is decomposed in 4 subproblems. 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 Reduction Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,0),x3) -> PLUS(x,x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Usable Equations: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> Usable Rules: 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = 0 0.60/0.70 [int](X1,X2) = 0 0.60/0.70 [intlist](X) = 0 0.60/0.70 [plus](X1,X2) = X1 + X2 + 2 0.60/0.70 [sum](X1,X2) = 0 0.60/0.70 [0] = 2 0.60/0.70 [cons](X1,X2) = 0 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X + 2 0.60/0.70 [S#](X) = 0 0.60/0.70 [INT](X1,X2) = 0 0.60/0.70 [INTLIST](X) = 0 0.60/0.70 [PLUS](X1,X2) = 2.X1 + 2.X2 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) -> PLUS(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 Reduction Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(x,y) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Usable Equations: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> Usable Rules: 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = 0 0.60/0.70 [int](X1,X2) = 0 0.60/0.70 [intlist](X) = 0 0.60/0.70 [plus](X1,X2) = X1 + X2 + 2 0.60/0.70 [sum](X1,X2) = 0 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 0 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X + 2 0.60/0.70 [S#](X) = 0 0.60/0.70 [INT](X1,X2) = 0 0.60/0.70 [INTLIST](X) = 0 0.60/0.70 [PLUS](X1,X2) = X1 + X2 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) -> PLUS(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 Reduction Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Usable Equations: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> Usable Rules: 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = 0 0.60/0.70 [int](X1,X2) = 0 0.60/0.70 [intlist](X) = 0 0.60/0.70 [plus](X1,X2) = X1 + X2 + 2 0.60/0.70 [sum](X1,X2) = 0 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 0 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X + 2 0.60/0.70 [S#](X) = 0 0.60/0.70 [INT](X1,X2) = 0 0.60/0.70 [INTLIST](X) = 0 0.60/0.70 [PLUS](X1,X2) = 2.X1 + 2.X2 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) -> PLUS(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 Reduction Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 PLUS(x,s(y)) -> PLUS(x,y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Usable Equations: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> Usable Rules: 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = 0 0.60/0.70 [int](X1,X2) = 0 0.60/0.70 [intlist](X) = 0 0.60/0.70 [plus](X1,X2) = X1 + X2 + 2 0.60/0.70 [sum](X1,X2) = 0 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 0 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = X + 2 0.60/0.70 [S#](X) = 0 0.60/0.70 [INT](X1,X2) = 0 0.60/0.70 [INTLIST](X) = 0 0.60/0.70 [PLUS](X1,X2) = 2.X1 + 2.X2 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 ->->Cycle: 0.60/0.70 ->->-> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 -> FAxioms: 0.60/0.70 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) -> plus(x4,x3) 0.60/0.70 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) -> PLUS(x4,x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 ->->-> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 Reduction Pairs Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 PLUS(plus(x,s(y)),x3) -> PLUS(s(plus(x,y)),x3) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Usable Equations: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> Usable Rules: 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Interpretation type: 0.60/0.70 Linear 0.60/0.70 ->Coefficients: 0.60/0.70 Natural Numbers 0.60/0.70 ->Dimension: 0.60/0.70 1 0.60/0.70 ->Bound: 0.60/0.70 2 0.60/0.70 ->Interpretation: 0.60/0.70 0.60/0.70 [S](X) = 0 0.60/0.70 [int](X1,X2) = 0 0.60/0.70 [intlist](X) = 0 0.60/0.70 [plus](X1,X2) = X1 + X2 + 2 0.60/0.70 [sum](X1,X2) = 0 0.60/0.70 [0] = 0 0.60/0.70 [cons](X1,X2) = 0 0.60/0.70 [nil] = 0 0.60/0.70 [s](X) = 2 0.60/0.70 [S#](X) = 0 0.60/0.70 [INT](X1,X2) = 0 0.60/0.70 [INTLIST](X) = 0 0.60/0.70 [PLUS](X1,X2) = 2.X1 + 2.X2 0.60/0.70 0.60/0.70 Problem 1.1: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 0.60/0.70 PLUS(x3,x4) = PLUS(x4,x3) 0.60/0.70 -> Pairs: 0.60/0.70 Empty 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 0.60/0.70 ->Strongly Connected Components: 0.60/0.70 There is no strongly connected component 0.60/0.70 0.60/0.70 The problem is finite. 0.60/0.70 0.60/0.70 Problem 1.2: 0.60/0.70 0.60/0.70 Subterm Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 Empty 0.60/0.70 -> Pairs: 0.60/0.70 INTLIST(cons(x,y)) -> INTLIST(y) 0.60/0.70 -> EAxioms: 0.60/0.70 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.70 plus(x3,x4) = plus(x4,x3) 0.60/0.70 -> Rules: 0.60/0.70 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.70 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.70 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.70 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.70 intlist(nil) -> nil 0.60/0.70 plus(x,0) -> x 0.60/0.70 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.70 -> SRules: 0.60/0.70 Empty 0.60/0.70 ->Projection: 0.60/0.70 pi(INTLIST) = [1] 0.60/0.70 0.60/0.70 Problem 1.2: 0.60/0.70 0.60/0.70 SCC Processor: 0.60/0.70 -> FAxioms: 0.60/0.70 Empty 0.60/0.70 -> Pairs: 0.60/0.70 Empty 0.60/0.70 -> EAxioms: 0.60/0.71 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.71 plus(x3,x4) = plus(x4,x3) 0.60/0.71 -> Rules: 0.60/0.71 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.71 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.71 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.71 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.71 intlist(nil) -> nil 0.60/0.71 plus(x,0) -> x 0.60/0.71 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.71 -> SRules: 0.60/0.71 Empty 0.60/0.71 ->Strongly Connected Components: 0.60/0.71 There is no strongly connected component 0.60/0.71 0.60/0.71 The problem is finite. 0.60/0.71 0.60/0.71 Problem 1.3: 0.60/0.71 0.60/0.71 Subterm Processor: 0.60/0.71 -> FAxioms: 0.60/0.71 Empty 0.60/0.71 -> Pairs: 0.60/0.71 INT(0,s(y)) -> INT(s(0),s(y)) 0.60/0.71 INT(s(x),s(y)) -> INT(x,y) 0.60/0.71 -> EAxioms: 0.60/0.71 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.71 plus(x3,x4) = plus(x4,x3) 0.60/0.71 -> Rules: 0.60/0.71 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.71 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.71 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.71 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.71 intlist(nil) -> nil 0.60/0.71 plus(x,0) -> x 0.60/0.71 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.71 -> SRules: 0.60/0.71 Empty 0.60/0.71 ->Projection: 0.60/0.71 pi(INT) = [2] 0.60/0.71 0.60/0.71 Problem 1.3: 0.60/0.71 0.60/0.71 SCC Processor: 0.60/0.71 -> FAxioms: 0.60/0.71 Empty 0.60/0.71 -> Pairs: 0.60/0.71 INT(0,s(y)) -> INT(s(0),s(y)) 0.60/0.71 -> EAxioms: 0.60/0.71 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.71 plus(x3,x4) = plus(x4,x3) 0.60/0.71 -> Rules: 0.60/0.71 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.71 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.71 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.71 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.71 intlist(nil) -> nil 0.60/0.71 plus(x,0) -> x 0.60/0.71 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.71 -> SRules: 0.60/0.71 Empty 0.60/0.71 ->Strongly Connected Components: 0.60/0.71 There is no strongly connected component 0.60/0.71 0.60/0.71 The problem is finite. 0.60/0.71 0.60/0.71 Problem 1.4: 0.60/0.71 0.60/0.71 Subterm Processor: 0.60/0.71 -> FAxioms: 0.60/0.71 Empty 0.60/0.71 -> Pairs: 0.60/0.71 S#(cons(x,xs)) -> S#(xs) 0.60/0.71 -> EAxioms: 0.60/0.71 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.71 plus(x3,x4) = plus(x4,x3) 0.60/0.71 -> Rules: 0.60/0.71 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.71 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.71 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.71 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.71 intlist(nil) -> nil 0.60/0.71 plus(x,0) -> x 0.60/0.71 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.71 -> SRules: 0.60/0.71 Empty 0.60/0.71 ->Projection: 0.60/0.71 pi(S#) = [1] 0.60/0.71 0.60/0.71 Problem 1.4: 0.60/0.71 0.60/0.71 SCC Processor: 0.60/0.71 -> FAxioms: 0.60/0.71 Empty 0.60/0.71 -> Pairs: 0.60/0.71 Empty 0.60/0.71 -> EAxioms: 0.60/0.71 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 0.60/0.71 plus(x3,x4) = plus(x4,x3) 0.60/0.71 -> Rules: 0.60/0.71 S(cons(x,xs)) -> plus(x,S(xs)) 0.60/0.71 int(0,s(y)) -> cons(0,int(s(0),s(y))) 0.60/0.71 int(s(x),s(y)) -> intlist(int(x,y)) 0.60/0.71 intlist(cons(x,y)) -> cons(s(x),intlist(y)) 0.60/0.71 intlist(nil) -> nil 0.60/0.71 plus(x,0) -> x 0.60/0.71 plus(x,s(y)) -> s(plus(x,y)) 0.60/0.71 -> SRules: 0.60/0.71 Empty 0.60/0.71 ->Strongly Connected Components: 0.60/0.71 There is no strongly connected component 0.60/0.71 0.60/0.71 The problem is finite. 0.60/0.71 EOF