122.85/123.49 YES 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 (VAR v_NonEmpty:S u:S x:S y:S z:S) 122.85/123.49 (RULES 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ) 122.85/123.49 (STRATEGY INNERMOST) 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 Dependency Pairs Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> A(b) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> F(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 SCC Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> A(b) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> F(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ->Strongly Connected Components: 122.85/123.49 ->->Cycle: 122.85/123.49 ->->-> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> F(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 ->->-> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 Subterm Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> F(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ->Projection: 122.85/123.49 pi(F) = 3 122.85/123.49 pi(G) = 3 122.85/123.49 pi(H) = 3 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 SCC Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ->Strongly Connected Components: 122.85/123.49 ->->Cycle: 122.85/123.49 ->->-> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 ->->-> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 Forward Instantiation Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ->Instantiated Pairs: 122.85/123.49 ->->Original Pair: 122.85/123.49 G(x:S,y:S,z:S,u:S) -> H(x:S,y:S,z:S,u:S) 122.85/123.49 ->-> Instantiated pairs: 122.85/123.49 G(b,y:S,z:S,u:S) -> H(b,y:S,z:S,u:S) 122.85/123.49 ->->Original Pair: 122.85/123.49 H(b,y:S,z:S,u:S) -> F(y:S,y:S,z:S,u:S) 122.85/123.49 ->-> Instantiated pairs: 122.85/123.49 H(b,a(x9:S),s(x11:S),u:S) -> F(a(x9:S),a(x9:S),s(x11:S),u:S) 122.85/123.49 122.85/123.49 Problem 1: 122.85/123.49 122.85/123.49 SCC Processor: 122.85/123.49 -> Pairs: 122.85/123.49 F(a(x:S),y:S,s(z:S),u:S) -> G(x:S,y:S,s(z:S),u:S) 122.85/123.49 G(b,y:S,z:S,u:S) -> H(b,y:S,z:S,u:S) 122.85/123.49 H(b,a(x9:S),s(x11:S),u:S) -> F(a(x9:S),a(x9:S),s(x11:S),u:S) 122.85/123.49 -> Rules: 122.85/123.49 a(b) -> c 122.85/123.49 f(a(x:S),y:S,s(z:S),u:S) -> f(a(b),y:S,z:S,g(x:S,y:S,s(z:S),u:S)) 122.85/123.49 g(x:S,y:S,z:S,u:S) -> h(x:S,y:S,z:S,u:S) 122.85/123.49 h(b,y:S,z:S,u:S) -> f(y:S,y:S,z:S,u:S) 122.85/123.49 ->Strongly Connected Components: 122.85/123.49 There is no strongly connected component 122.85/123.49 122.85/123.49 The problem is finite. 122.85/123.49 EOF