0.57/0.94 YES 0.57/0.94 0.57/0.94 Problem 1: 0.57/0.94 0.57/0.94 (VAR v_NonEmpty:S fun:S x:S xs:S y:S) 0.57/0.94 (RULES 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ) 0.57/0.94 (STRATEGY INNERMOST) 0.57/0.94 0.57/0.94 Problem 1: 0.57/0.94 0.57/0.94 Dependency Pairs Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(if,app(f,x:S)),app(s,x:S)) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(f,x:S) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(if,app(f,x:S)) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(f,app(s,x:S)) -> APP(f,x:S) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 0.57/0.94 Problem 1: 0.57/0.94 0.57/0.94 SCC Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(filter2,app(fun:S,x:S)),fun:S),x:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(filter2,app(fun:S,x:S)),fun:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(filter2,app(fun:S,x:S)) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(app(if,app(f,x:S)),app(s,x:S)) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(f,x:S) 0.57/0.94 APP(app(g,app(s,x:S)),app(s,y:S)) -> APP(if,app(f,x:S)) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(cons,app(fun:S,x:S)) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(f,app(s,x:S)) -> APP(f,x:S) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Strongly Connected Components: 0.57/0.94 ->->Cycle: 0.57/0.94 ->->-> Pairs: 0.57/0.94 APP(f,app(s,x:S)) -> APP(f,x:S) 0.57/0.94 ->->-> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->->Cycle: 0.57/0.94 ->->-> Pairs: 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 ->->-> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->->Cycle: 0.57/0.94 ->->-> Pairs: 0.57/0.94 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 ->->-> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 0.57/0.94 0.57/0.94 The problem is decomposed in 3 subproblems. 0.57/0.94 0.57/0.94 Problem 1.1: 0.57/0.94 0.57/0.94 Subterm Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(f,app(s,x:S)) -> APP(f,x:S) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Projection: 0.57/0.94 pi(APP) = 2 0.57/0.94 0.57/0.94 Problem 1.1: 0.57/0.94 0.57/0.94 SCC Processor: 0.57/0.94 -> Pairs: 0.57/0.94 Empty 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Strongly Connected Components: 0.57/0.94 There is no strongly connected component 0.57/0.94 0.57/0.94 The problem is finite. 0.57/0.94 0.57/0.94 Problem 1.2: 0.57/0.94 0.57/0.94 Reduction Pairs Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,app(s,app(c,y:S))),y:S) 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 -> Usable rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Interpretation type: 0.57/0.94 Simple mixed 0.57/0.94 ->Coefficients: 0.57/0.94 Natural Numbers 0.57/0.94 ->Dimension: 0.57/0.94 1 0.57/0.94 ->Bound: 0.57/0.94 2 0.57/0.94 ->Interpretation: 0.57/0.94 0.57/0.94 [app](X1,X2) = 2.X1.X2 + X1 0.57/0.94 [0] = 2 0.57/0.94 [1] = 0 0.57/0.94 [c] = 2 0.57/0.94 [cons] = 0 0.57/0.94 [f] = 0 0.57/0.94 [fSNonEmpty] = 0 0.57/0.94 [false] = 0 0.57/0.94 [filter] = 0 0.57/0.94 [filter2] = 0 0.57/0.94 [g] = 2 0.57/0.94 [if] = 2 0.57/0.94 [map] = 2 0.57/0.94 [nil] = 0 0.57/0.94 [s] = 0 0.57/0.94 [true] = 0 0.57/0.94 [APP](X1,X2) = X1.X2 + X1 + 2.X2 0.57/0.94 0.57/0.94 Problem 1.2: 0.57/0.94 0.57/0.94 SCC Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Strongly Connected Components: 0.57/0.94 ->->Cycle: 0.57/0.94 ->->-> Pairs: 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 ->->-> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 0.57/0.94 Problem 1.2: 0.57/0.94 0.57/0.94 Reduction Pairs Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(g,x:S),app(c,y:S)) -> APP(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 -> Usable rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Interpretation type: 0.57/0.94 Simple mixed 0.57/0.94 ->Coefficients: 0.57/0.94 Natural Numbers 0.57/0.94 ->Dimension: 0.57/0.94 1 0.57/0.94 ->Bound: 0.57/0.94 2 0.57/0.94 ->Interpretation: 0.57/0.94 0.57/0.94 [app](X1,X2) = 2.X1.X2 + X1 0.57/0.94 [0] = 2 0.57/0.94 [1] = 0 0.57/0.94 [c] = 2 0.57/0.94 [cons] = 0 0.57/0.94 [f] = 0 0.57/0.94 [fSNonEmpty] = 0 0.57/0.94 [false] = 0 0.57/0.94 [filter] = 0 0.57/0.94 [filter2] = 0 0.57/0.94 [g] = 1 0.57/0.94 [if] = 1 0.57/0.94 [map] = 1 0.57/0.94 [nil] = 0 0.57/0.94 [s] = 0 0.57/0.94 [true] = 0 0.57/0.94 [APP](X1,X2) = X1.X2 + 2.X2 0.57/0.94 0.57/0.94 Problem 1.2: 0.57/0.94 0.57/0.94 SCC Processor: 0.57/0.94 -> Pairs: 0.57/0.94 Empty 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Strongly Connected Components: 0.57/0.94 There is no strongly connected component 0.57/0.94 0.57/0.94 The problem is finite. 0.57/0.94 0.57/0.94 Problem 1.3: 0.57/0.94 0.57/0.94 Subterm Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 APP(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(app(map,fun:S),xs:S) 0.57/0.94 APP(app(map,fun:S),app(app(cons,x:S),xs:S)) -> APP(fun:S,x:S) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Projection: 0.57/0.94 pi(APP) = 2 0.57/0.94 0.57/0.94 Problem 1.3: 0.57/0.94 0.57/0.94 SCC Processor: 0.57/0.94 -> Pairs: 0.57/0.94 APP(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 APP(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> APP(app(filter,fun:S),xs:S) 0.57/0.94 -> Rules: 0.57/0.94 app(app(app(app(filter2,ffalse),fun:S),x:S),xs:S) -> app(app(filter,fun:S),xs:S) 0.57/0.94 app(app(app(app(filter2,ttrue),fun:S),x:S),xs:S) -> app(app(cons,x:S),app(app(filter,fun:S),xs:S)) 0.57/0.94 app(app(app(if,ffalse),x:S),y:S) -> y:S 0.57/0.94 app(app(app(if,ttrue),x:S),y:S) -> x:S 0.57/0.94 app(app(filter,fun:S),app(app(cons,x:S),xs:S)) -> app(app(app(app(filter2,app(fun:S,x:S)),fun:S),x:S),xs:S) 0.57/0.94 app(app(filter,fun:S),nil) -> nil 0.57/0.94 app(app(g,app(s,x:S)),app(s,y:S)) -> app(app(app(if,app(f,x:S)),app(s,x:S)),app(s,y:S)) 0.57/0.94 app(app(g,x:S),app(c,y:S)) -> app(app(g,x:S),app(app(g,app(s,app(c,y:S))),y:S)) 0.57/0.94 app(app(map,fun:S),app(app(cons,x:S),xs:S)) -> app(app(cons,app(fun:S,x:S)),app(app(map,fun:S),xs:S)) 0.57/0.94 app(app(map,fun:S),nil) -> nil 0.57/0.94 app(f,app(s,x:S)) -> app(f,x:S) 0.57/0.94 app(f,0) -> ttrue 0.57/0.94 app(f,1) -> ffalse 0.57/0.94 ->Strongly Connected Components: 0.57/0.94 There is no strongly connected component 0.57/0.94 0.57/0.94 The problem is finite. 0.57/0.94 EOF