14.06/15.44 YES 14.06/15.44 14.06/15.44 Problem 1: 14.06/15.44 14.06/15.44 (VAR x y z) 14.06/15.44 (THEORY 14.06/15.44 (AC plus times)) 14.06/15.44 (RULES 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 ) 14.06/15.44 14.06/15.44 Problem 1: 14.06/15.44 14.06/15.44 Dependency Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.06/15.44 TIMES(x3,x4) = TIMES(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y)) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> ZERO(plus(x,plus(y,un(S)))) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y)) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 PLUS(un(x),un(y)) -> ZERO(plus(x,plus(y,un(S)))) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(un(y),z)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,zero(y)) -> TIMES(x,y) 14.06/15.44 TIMES(x,zero(y)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(x,un(y)) -> TIMES(x,y) 14.06/15.44 TIMES(x,un(y)) -> ZERO(times(x,y)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.06/15.44 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.06/15.44 14.06/15.44 Problem 1: 14.06/15.44 14.06/15.44 SCC Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.06/15.44 TIMES(x3,x4) = TIMES(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y)) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> ZERO(plus(x,plus(y,un(S)))) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y)) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 PLUS(un(x),un(y)) -> ZERO(plus(x,plus(y,un(S)))) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(un(y),z)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,zero(y)) -> TIMES(x,y) 14.06/15.44 TIMES(x,zero(y)) -> ZERO(times(x,y)) 14.06/15.44 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y))) 14.06/15.44 TIMES(x,un(y)) -> TIMES(x,y) 14.06/15.44 TIMES(x,un(y)) -> ZERO(times(x,y)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.06/15.44 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.06/15.44 ->Strongly Connected Components: 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.06/15.44 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.06/15.44 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.06/15.44 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.06/15.44 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.06/15.44 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.06/15.44 TIMES(x,zero(y)) -> TIMES(x,y) 14.06/15.44 TIMES(x,un(y)) -> TIMES(x,y) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.06/15.44 TIMES(x3,x4) -> TIMES(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.06/15.44 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.06/15.44 14.06/15.44 14.06/15.44 The problem is decomposed in 2 subproblems. 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 Reduction Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Usable Equations: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> Usable Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Interpretation type: 14.06/15.44 Linear 14.06/15.44 ->Coefficients: 14.06/15.44 Natural Numbers 14.06/15.44 ->Dimension: 14.06/15.44 1 14.06/15.44 ->Bound: 14.06/15.44 2 14.06/15.44 ->Interpretation: 14.06/15.44 14.06/15.44 [plus](X1,X2) = X1 + X2 14.06/15.44 [times](X1,X2) = 0 14.06/15.44 [zero](X) = X + 1 14.06/15.44 [S] = 0 14.06/15.44 [un](X) = X + 2 14.06/15.44 [PLUS](X1,X2) = 2.X1 + 2.X2 14.06/15.44 [TIMES](X1,X2) = 0 14.06/15.44 [ZERO](X) = 0 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 SCC Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Strongly Connected Components: 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 Reduction Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Usable Equations: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> Usable Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Interpretation type: 14.06/15.44 Linear 14.06/15.44 ->Coefficients: 14.06/15.44 Natural Numbers 14.06/15.44 ->Dimension: 14.06/15.44 1 14.06/15.44 ->Bound: 14.06/15.44 2 14.06/15.44 ->Interpretation: 14.06/15.44 14.06/15.44 [plus](X1,X2) = X1 + X2 + 2 14.06/15.44 [times](X1,X2) = 0 14.06/15.44 [zero](X) = X 14.06/15.44 [S] = 0 14.06/15.44 [un](X) = X + 2 14.06/15.44 [PLUS](X1,X2) = 2.X1 + 2.X2 14.06/15.44 [TIMES](X1,X2) = 0 14.06/15.44 [ZERO](X) = 0 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 SCC Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Strongly Connected Components: 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 Reduction Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3) 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Usable Equations: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> Usable Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Interpretation type: 14.06/15.44 Linear 14.06/15.44 ->Coefficients: 14.06/15.44 Natural Numbers 14.06/15.44 ->Dimension: 14.06/15.44 1 14.06/15.44 ->Bound: 14.06/15.44 2 14.06/15.44 ->Interpretation: 14.06/15.44 14.06/15.44 [plus](X1,X2) = X1 + X2 + 1 14.06/15.44 [times](X1,X2) = 0 14.06/15.44 [zero](X) = X + 1 14.06/15.44 [S] = 0 14.06/15.44 [un](X) = X + 2 14.06/15.44 [PLUS](X1,X2) = 2.X1 + 2.X2 14.06/15.44 [TIMES](X1,X2) = 0 14.06/15.44 [ZERO](X) = 0 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 SCC Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Strongly Connected Components: 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 Reduction Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Usable Equations: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> Usable Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Interpretation type: 14.06/15.44 Linear 14.06/15.44 ->Coefficients: 14.06/15.44 Natural Numbers 14.06/15.44 ->Dimension: 14.06/15.44 1 14.06/15.44 ->Bound: 14.06/15.44 2 14.06/15.44 ->Interpretation: 14.06/15.44 14.06/15.44 [plus](X1,X2) = X1 + X2 + 2 14.06/15.44 [times](X1,X2) = 0 14.06/15.44 [zero](X) = X 14.06/15.44 [S] = 0 14.06/15.44 [un](X) = X + 2 14.06/15.44 [PLUS](X1,X2) = 2.X1 + 2.X2 14.06/15.44 [TIMES](X1,X2) = 0 14.06/15.44 [ZERO](X) = 0 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 SCC Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Strongly Connected Components: 14.06/15.44 ->->Cycle: 14.06/15.44 ->->-> Pairs: 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> FAxioms: 14.06/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) -> plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) -> times(x4,x3) 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 ->->-> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 14.06/15.44 Problem 1.1: 14.06/15.44 14.06/15.44 Reduction Pairs Processor: 14.06/15.44 -> FAxioms: 14.06/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.06/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.06/15.44 -> Pairs: 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.06/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.06/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.06/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.06/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.06/15.44 -> EAxioms: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.06/15.44 times(x3,x4) = times(x4,x3) 14.06/15.44 -> Usable Equations: 14.06/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.06/15.44 plus(x3,x4) = plus(x4,x3) 14.06/15.44 -> Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.06/15.44 times(x,times(S,z)) -> times(S,z) 14.06/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.06/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.06/15.44 times(x,S) -> S 14.06/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.06/15.44 zero(S) -> S 14.06/15.44 -> Usable Rules: 14.06/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.06/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.06/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.06/15.44 plus(x,S) -> x 14.06/15.44 zero(S) -> S 14.06/15.44 -> SRules: 14.06/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.06/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.06/15.44 ->Interpretation type: 14.06/15.44 Linear 14.06/15.44 ->Coefficients: 14.06/15.44 Natural Numbers 14.06/15.44 ->Dimension: 14.06/15.44 1 14.06/15.44 ->Bound: 14.06/15.44 2 14.06/15.44 ->Interpretation: 14.06/15.44 14.06/15.44 [plus](X1,X2) = X1 + X2 + 1 14.06/15.44 [times](X1,X2) = 0 14.06/15.44 [zero](X) = X 14.10/15.44 [S] = 0 14.10/15.44 [un](X) = X + 2 14.10/15.44 [PLUS](X1,X2) = 2.X1 + 2.X2 14.10/15.44 [TIMES](X1,X2) = 0 14.10/15.44 [ZERO](X) = 0 14.10/15.44 14.10/15.44 Problem 1.1: 14.10/15.44 14.10/15.44 SCC Processor: 14.10/15.44 -> FAxioms: 14.10/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.44 -> Pairs: 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.44 -> EAxioms: 14.10/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) = plus(x4,x3) 14.10/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.44 times(x3,x4) = times(x4,x3) 14.10/15.44 -> Rules: 14.10/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.44 plus(x,S) -> x 14.10/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.44 times(x,times(S,z)) -> times(S,z) 14.10/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.44 times(x,S) -> S 14.10/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.44 zero(S) -> S 14.10/15.44 -> SRules: 14.10/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.44 ->Strongly Connected Components: 14.10/15.44 ->->Cycle: 14.10/15.44 ->->-> Pairs: 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.44 -> FAxioms: 14.10/15.44 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) -> plus(x4,x3) 14.10/15.44 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.44 times(x3,x4) -> times(x4,x3) 14.10/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.44 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.44 -> EAxioms: 14.10/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) = plus(x4,x3) 14.10/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.44 times(x3,x4) = times(x4,x3) 14.10/15.44 ->->-> Rules: 14.10/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.44 plus(x,S) -> x 14.10/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.44 times(x,times(S,z)) -> times(S,z) 14.10/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.44 times(x,S) -> S 14.10/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.44 zero(S) -> S 14.10/15.44 -> SRules: 14.10/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.44 14.10/15.44 Problem 1.1: 14.10/15.44 14.10/15.44 Reduction Pairs Processor: 14.10/15.44 -> FAxioms: 14.10/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.44 -> Pairs: 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.44 -> EAxioms: 14.10/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) = plus(x4,x3) 14.10/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.44 times(x3,x4) = times(x4,x3) 14.10/15.44 -> Usable Equations: 14.10/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) = plus(x4,x3) 14.10/15.44 -> Rules: 14.10/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.44 plus(x,S) -> x 14.10/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.44 times(x,times(S,z)) -> times(S,z) 14.10/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.44 times(x,S) -> S 14.10/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.44 zero(S) -> S 14.10/15.44 -> Usable Rules: 14.10/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.44 plus(x,S) -> x 14.10/15.44 zero(S) -> S 14.10/15.44 -> SRules: 14.10/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.44 ->Interpretation type: 14.10/15.44 Linear 14.10/15.44 ->Coefficients: 14.10/15.44 Natural Numbers 14.10/15.44 ->Dimension: 14.10/15.44 1 14.10/15.44 ->Bound: 14.10/15.44 2 14.10/15.44 ->Interpretation: 14.10/15.44 14.10/15.44 [plus](X1,X2) = X1 + X2 + 1 14.10/15.44 [times](X1,X2) = 0 14.10/15.44 [zero](X) = X 14.10/15.44 [S] = 1 14.10/15.44 [un](X) = X + 2 14.10/15.44 [PLUS](X1,X2) = X1 + X2 14.10/15.44 [TIMES](X1,X2) = 0 14.10/15.44 [ZERO](X) = 0 14.10/15.44 14.10/15.44 Problem 1.1: 14.10/15.44 14.10/15.44 SCC Processor: 14.10/15.44 -> FAxioms: 14.10/15.44 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.44 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.44 -> Pairs: 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.44 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.44 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.44 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.44 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.44 -> EAxioms: 14.10/15.44 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.44 plus(x3,x4) = plus(x4,x3) 14.10/15.44 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.44 times(x3,x4) = times(x4,x3) 14.10/15.44 -> Rules: 14.10/15.44 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.44 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.44 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.44 plus(x,S) -> x 14.10/15.44 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.44 times(x,times(S,z)) -> times(S,z) 14.10/15.44 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.44 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.44 times(x,S) -> S 14.10/15.44 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.44 zero(S) -> S 14.10/15.44 -> SRules: 14.10/15.44 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.44 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.44 ->Strongly Connected Components: 14.10/15.44 ->->Cycle: 14.10/15.44 ->->-> Pairs: 14.10/15.44 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.45 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S)) 14.10/15.45 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 + 1 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 2 14.10/15.45 [PLUS](X1,X2) = X1 + X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(plus(x,S),x3) -> PLUS(x,x3) 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 + 1 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 2 14.10/15.45 [PLUS](X1,X2) = X1 + X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4) 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 + 1 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 2 14.10/15.45 [PLUS](X1,X2) = X1 + X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),zero(y)) -> PLUS(x,y) 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = X + 1 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 2 14.10/15.45 [PLUS](X1,X2) = X1 + X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(zero(x),un(y)) -> PLUS(x,y) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = X + 1 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 2 14.10/15.45 [PLUS](X1,X2) = X1 + X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S))) 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = 2.X 14.10/15.45 [S] = 0 14.10/15.45 [un](X) = 2.X + 2 14.10/15.45 [PLUS](X1,X2) = 2.X1 + 2.X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) -> PLUS(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 PLUS(un(x),un(y)) -> PLUS(y,un(S)) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Linear 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 2 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = 0 14.10/15.45 [zero](X) = 2.X 14.10/15.45 [S] = 0 14.10/15.45 [un](X) = 2.X + 2 14.10/15.45 [PLUS](X1,X2) = 2.X1 + 2.X2 14.10/15.45 [TIMES](X1,X2) = 0 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.1: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5)) 14.10/15.45 PLUS(x3,x4) = PLUS(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 Empty 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 There is no strongly connected component 14.10/15.45 14.10/15.45 The problem is finite. 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X + 1 14.10/15.45 [S] = 0 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X + 1 14.10/15.45 [S] = 0 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 1 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(times(x,un(y)),x3) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.45 Simple mixed 14.10/15.45 ->Coefficients: 14.10/15.45 Natural Numbers 14.10/15.45 ->Dimension: 14.10/15.45 1 14.10/15.45 ->Bound: 14.10/15.45 1 14.10/15.45 ->Interpretation: 14.10/15.45 14.10/15.45 [plus](X1,X2) = X1 + X2 + 1 14.10/15.45 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [zero](X) = X 14.10/15.45 [S] = 0 14.10/15.45 [un](X) = X + 1 14.10/15.45 [PLUS](X1,X2) = 0 14.10/15.45 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.45 [ZERO](X) = 0 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 SCC Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Strongly Connected Components: 14.10/15.45 ->->Cycle: 14.10/15.45 ->->-> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> FAxioms: 14.10/15.45 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) -> plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) -> times(x4,x3) 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 ->->-> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 14.10/15.45 Problem 1.2: 14.10/15.45 14.10/15.45 Reduction Pairs Processor: 14.10/15.45 -> FAxioms: 14.10/15.45 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.45 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.45 -> Pairs: 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.45 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.45 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.45 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.45 TIMES(x,times(zero(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.45 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.45 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.45 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.45 -> EAxioms: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Usable Equations: 14.10/15.45 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.45 plus(x3,x4) = plus(x4,x3) 14.10/15.45 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.45 times(x3,x4) = times(x4,x3) 14.10/15.45 -> Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> Usable Rules: 14.10/15.45 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.45 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.45 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.45 plus(x,S) -> x 14.10/15.45 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.45 times(x,times(S,z)) -> times(S,z) 14.10/15.45 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.45 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.45 times(x,S) -> S 14.10/15.45 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.45 zero(S) -> S 14.10/15.45 -> SRules: 14.10/15.45 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.45 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.45 ->Interpretation type: 14.10/15.46 Simple mixed 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 1 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [zero](X) = X + 1 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = X + 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Simple mixed 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 1 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [zero](X) = X 14.10/15.46 [S] = 1 14.10/15.46 [un](X) = X + 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,times(un(y),z)) -> TIMES(x,y) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Simple mixed 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 1 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [zero](X) = X + 1 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = X + 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,zero(y)) -> TIMES(x,y) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Simple mixed 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 1 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [zero](X) = X + 1 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = X + 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(x,un(y)) -> TIMES(x,y) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Simple mixed 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 1 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [zero](X) = X 14.10/15.46 [S] = 1 14.10/15.46 [un](X) = X + 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = X1.X2 + X1 + X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3) 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1 + X2 + 1 14.10/15.46 [zero](X) = 0 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = 0 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z) 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 + 2 14.10/15.46 [times](X1,X2) = X1 + X2 + 2 14.10/15.46 [zero](X) = 0 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3) 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 + 1 14.10/15.46 [times](X1,X2) = X1 + X2 + 2 14.10/15.46 [zero](X) = 2 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = 2 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3) 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 14.10/15.46 [times](X1,X2) = X1 + X2 + 2 14.10/15.46 [zero](X) = 0 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = 0 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(times(x,S),x3) -> TIMES(S,x3) 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 + 1 14.10/15.46 [times](X1,X2) = X1 + X2 + 1 14.10/15.46 [zero](X) = 0 14.10/15.46 [S] = 0 14.10/15.46 [un](X) = 0 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,x4) 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 + 1 14.10/15.46 [times](X1,X2) = X1 + X2 + 2 14.10/15.46 [zero](X) = 1 14.10/15.46 [S] = 1 14.10/15.46 [un](X) = 0 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 ->->Cycle: 14.10/15.46 ->->-> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> FAxioms: 14.10/15.46 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) -> plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) -> times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) -> times(x4,x3) 14.10/15.46 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) -> TIMES(x4,x3) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 ->->-> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 Reduction Pairs Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z) 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Usable Equations: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> Usable Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Interpretation type: 14.10/15.46 Linear 14.10/15.46 ->Coefficients: 14.10/15.46 Natural Numbers 14.10/15.46 ->Dimension: 14.10/15.46 1 14.10/15.46 ->Bound: 14.10/15.46 2 14.10/15.46 ->Interpretation: 14.10/15.46 14.10/15.46 [plus](X1,X2) = X1 + X2 + 1 14.10/15.46 [times](X1,X2) = X1 + X2 + 2 14.10/15.46 [zero](X) = 2 14.10/15.46 [S] = 2 14.10/15.46 [un](X) = 1 14.10/15.46 [PLUS](X1,X2) = 0 14.10/15.46 [TIMES](X1,X2) = 2.X1 + 2.X2 14.10/15.46 [ZERO](X) = 0 14.10/15.46 14.10/15.46 Problem 1.2: 14.10/15.46 14.10/15.46 SCC Processor: 14.10/15.46 -> FAxioms: 14.10/15.46 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5)) 14.10/15.46 TIMES(x3,x4) = TIMES(x4,x3) 14.10/15.46 -> Pairs: 14.10/15.46 Empty 14.10/15.46 -> EAxioms: 14.10/15.46 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5)) 14.10/15.46 plus(x3,x4) = plus(x4,x3) 14.10/15.46 times(times(x3,x4),x5) = times(x3,times(x4,x5)) 14.10/15.46 times(x3,x4) = times(x4,x3) 14.10/15.46 -> Rules: 14.10/15.46 plus(zero(x),zero(y)) -> zero(plus(x,y)) 14.10/15.46 plus(zero(x),un(y)) -> un(plus(x,y)) 14.10/15.46 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S)))) 14.10/15.46 plus(x,S) -> x 14.10/15.46 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z) 14.10/15.46 times(x,times(S,z)) -> times(S,z) 14.10/15.46 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z) 14.10/15.46 times(x,zero(y)) -> zero(times(x,y)) 14.10/15.46 times(x,S) -> S 14.10/15.46 times(x,un(y)) -> plus(x,zero(times(x,y))) 14.10/15.46 zero(S) -> S 14.10/15.46 -> SRules: 14.10/15.46 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5) 14.10/15.46 ->Strongly Connected Components: 14.10/15.46 There is no strongly connected component 14.10/15.46 14.10/15.46 The problem is finite. 14.10/15.46 EOF