0.00/0.27 YES 0.00/0.27 0.00/0.27 Problem 1: 0.00/0.27 0.00/0.27 (VAR v_NonEmpty:S M:S N:S X:S X1:S X2:S X3:S) 0.00/0.27 (RULES 0.00/0.27 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.27 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.27 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.27 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.27 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.27 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.27 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.27 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.27 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.27 active(plus(N:S,0)) -> mark(N:S) 0.00/0.27 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.27 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.27 active(s(X:S)) -> s(active(X:S)) 0.00/0.27 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.27 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.27 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.27 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.27 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.27 proper(0) -> ok(0) 0.00/0.27 proper(tt) -> ok(tt) 0.00/0.27 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.27 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.27 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.27 top(ok(X:S)) -> top(active(X:S)) 0.00/0.27 ) 0.00/0.27 (STRATEGY INNERMOST) 0.00/0.27 0.00/0.27 Problem 1: 0.00/0.27 0.00/0.27 Dependency Pairs Processor: 0.00/0.27 -> Pairs: 0.00/0.27 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 0.00/0.27 U11#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U11#(X1:S,X2:S,X3:S) 0.00/0.27 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 0.00/0.27 U12#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U12#(X1:S,X2:S,X3:S) 0.00/0.27 ACTIVE(U11(X1:S,X2:S,X3:S)) -> U11#(active(X1:S),X2:S,X3:S) 0.00/0.27 ACTIVE(U11(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(U12(tt,M:S,N:S)) -> PLUS(N:S,M:S) 0.00/0.27 ACTIVE(U12(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 0.00/0.27 ACTIVE(U12(X1:S,X2:S,X3:S)) -> U12#(active(X1:S),X2:S,X3:S) 0.00/0.27 ACTIVE(U12(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.27 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.27 ACTIVE(plus(X1:S,X2:S)) -> PLUS(active(X1:S),X2:S) 0.00/0.27 ACTIVE(plus(X1:S,X2:S)) -> PLUS(X1:S,active(X2:S)) 0.00/0.27 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.00/0.27 ACTIVE(s(X:S)) -> S(active(X:S)) 0.00/0.27 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 0.00/0.27 PLUS(ok(X1:S),ok(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.27 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.27 PROPER(U11(X1:S,X2:S,X3:S)) -> U11#(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.27 PROPER(U12(X1:S,X2:S,X3:S)) -> U12#(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.27 PROPER(plus(X1:S,X2:S)) -> PLUS(proper(X1:S),proper(X2:S)) 0.00/0.27 PROPER(plus(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.27 PROPER(plus(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.27 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.27 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.27 S(mark(X:S)) -> S(X:S) 0.00/0.27 S(ok(X:S)) -> S(X:S) 0.00/0.27 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.27 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.27 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.27 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.27 -> Rules: 0.00/0.27 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.27 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.27 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.27 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.27 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.27 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.27 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.27 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.27 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.27 active(plus(N:S,0)) -> mark(N:S) 0.00/0.27 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.27 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.27 active(s(X:S)) -> s(active(X:S)) 0.00/0.27 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.27 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.27 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.27 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.27 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 0.00/0.28 Problem 1: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 U11#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 U12#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 ACTIVE(U11(X1:S,X2:S,X3:S)) -> U11#(active(X1:S),X2:S,X3:S) 0.00/0.28 ACTIVE(U11(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(U12(tt,M:S,N:S)) -> PLUS(N:S,M:S) 0.00/0.28 ACTIVE(U12(tt,M:S,N:S)) -> S(plus(N:S,M:S)) 0.00/0.28 ACTIVE(U12(X1:S,X2:S,X3:S)) -> U12#(active(X1:S),X2:S,X3:S) 0.00/0.28 ACTIVE(U12(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> PLUS(active(X1:S),X2:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> PLUS(X1:S,active(X2:S)) 0.00/0.28 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.00/0.28 ACTIVE(s(X:S)) -> S(active(X:S)) 0.00/0.28 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(ok(X1:S),ok(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> U11#(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> U12#(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.28 PROPER(plus(X1:S,X2:S)) -> PLUS(proper(X1:S),proper(X2:S)) 0.00/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.28 PROPER(s(X:S)) -> S(proper(X:S)) 0.00/0.28 S(mark(X:S)) -> S(X:S) 0.00/0.28 S(ok(X:S)) -> S(X:S) 0.00/0.28 TOP(mark(X:S)) -> PROPER(X:S) 0.00/0.28 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.28 TOP(ok(X:S)) -> ACTIVE(X:S) 0.00/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 S(mark(X:S)) -> S(X:S) 0.00/0.28 S(ok(X:S)) -> S(X:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(ok(X1:S),ok(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 U12#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 U11#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 ACTIVE(U11(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(U12(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.28 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.00/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X1:S) 0.00/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X2:S) 0.00/0.28 PROPER(s(X:S)) -> PROPER(X:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.00/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 0.00/0.28 0.00/0.28 The problem is decomposed in 7 subproblems. 0.00/0.28 0.00/0.28 Problem 1.1: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 S(mark(X:S)) -> S(X:S) 0.00/0.28 S(ok(X:S)) -> S(X:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Projection: 0.00/0.28 pi(S) = 1 0.00/0.28 0.00/0.28 Problem 1.1: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 Empty 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(ok(X1:S),ok(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Projection: 0.00/0.28 pi(PLUS) = 1 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 ->->Cycle: 0.00/0.28 ->->-> Pairs: 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 ->->-> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Projection: 0.00/0.28 pi(PLUS) = 2 0.00/0.28 0.00/0.28 Problem 1.2: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 Empty 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 0.00/0.28 Problem 1.3: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 U12#(mark(X1:S),X2:S,X3:S) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 U12#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U12#(X1:S,X2:S,X3:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Projection: 0.00/0.28 pi(U12#) = 1 0.00/0.28 0.00/0.28 Problem 1.3: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 Empty 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 0.00/0.28 Problem 1.4: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 U11#(ok(X1:S),ok(X2:S),ok(X3:S)) -> U11#(X1:S,X2:S,X3:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Projection: 0.00/0.28 pi(U11#) = 1 0.00/0.28 0.00/0.28 Problem 1.4: 0.00/0.28 0.00/0.28 SCC Processor: 0.00/0.28 -> Pairs: 0.00/0.28 Empty 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.00/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.00/0.28 proper(0) -> ok(0) 0.00/0.28 proper(tt) -> ok(tt) 0.00/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.00/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.00/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.00/0.28 top(ok(X:S)) -> top(active(X:S)) 0.00/0.28 ->Strongly Connected Components: 0.00/0.28 There is no strongly connected component 0.00/0.28 0.00/0.28 The problem is finite. 0.00/0.28 0.00/0.28 Problem 1.5: 0.00/0.28 0.00/0.28 Subterm Processor: 0.00/0.28 -> Pairs: 0.00/0.28 ACTIVE(U11(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(U12(X1:S,X2:S,X3:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X1:S) 0.00/0.28 ACTIVE(plus(X1:S,X2:S)) -> ACTIVE(X2:S) 0.00/0.28 ACTIVE(s(X:S)) -> ACTIVE(X:S) 0.00/0.28 -> Rules: 0.00/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.00/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.00/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.00/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.00/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.00/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.00/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.00/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.00/0.28 active(plus(N:S,0)) -> mark(N:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.00/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.00/0.28 active(s(X:S)) -> s(active(X:S)) 0.00/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.00/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.00/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.00/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.00/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Projection: 0.27/0.28 pi(ACTIVE) = 1 0.27/0.28 0.27/0.28 Problem 1.5: 0.27/0.28 0.27/0.28 SCC Processor: 0.27/0.28 -> Pairs: 0.27/0.28 Empty 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Strongly Connected Components: 0.27/0.28 There is no strongly connected component 0.27/0.28 0.27/0.28 The problem is finite. 0.27/0.28 0.27/0.28 Problem 1.6: 0.27/0.28 0.27/0.28 Subterm Processor: 0.27/0.28 -> Pairs: 0.27/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.27/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.27/0.28 PROPER(U11(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.27/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X1:S) 0.27/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X2:S) 0.27/0.28 PROPER(U12(X1:S,X2:S,X3:S)) -> PROPER(X3:S) 0.27/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X1:S) 0.27/0.28 PROPER(plus(X1:S,X2:S)) -> PROPER(X2:S) 0.27/0.28 PROPER(s(X:S)) -> PROPER(X:S) 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Projection: 0.27/0.28 pi(PROPER) = 1 0.27/0.28 0.27/0.28 Problem 1.6: 0.27/0.28 0.27/0.28 SCC Processor: 0.27/0.28 -> Pairs: 0.27/0.28 Empty 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Strongly Connected Components: 0.27/0.28 There is no strongly connected component 0.27/0.28 0.27/0.28 The problem is finite. 0.27/0.28 0.27/0.28 Problem 1.7: 0.27/0.28 0.27/0.28 Reduction Pairs Processor: 0.27/0.28 -> Pairs: 0.27/0.28 TOP(mark(X:S)) -> TOP(proper(X:S)) 0.27/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 -> Usable rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 ->Interpretation type: 0.27/0.28 Linear 0.27/0.28 ->Coefficients: 0.27/0.28 All rationals 0.27/0.28 ->Dimension: 0.27/0.28 1 0.27/0.28 ->Bound: 0.27/0.28 2 0.27/0.28 ->Interpretation: 0.27/0.28 0.27/0.28 [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.27/0.28 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1/2 0.27/0.28 [active](X) = X 0.27/0.28 [plus](X1,X2) = 2.X1 + 2.X2 0.27/0.28 [proper](X) = X 0.27/0.28 [s](X) = X + 2 0.27/0.28 [top](X) = 0 0.27/0.28 [0] = 1 0.27/0.28 [fSNonEmpty] = 0 0.27/0.28 [mark](X) = X + 1/2 0.27/0.28 [ok](X) = X 0.27/0.28 [tt] = 1 0.27/0.28 [U11#](X1,X2,X3) = 0 0.27/0.28 [U12#](X1,X2,X3) = 0 0.27/0.28 [ACTIVE](X) = 0 0.27/0.28 [PLUS](X1,X2) = 0 0.27/0.28 [PROPER](X) = 0 0.27/0.28 [S](X) = 0 0.27/0.28 [TOP](X) = 1/2.X 0.27/0.28 0.27/0.28 Problem 1.7: 0.27/0.28 0.27/0.28 SCC Processor: 0.27/0.28 -> Pairs: 0.27/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Strongly Connected Components: 0.27/0.28 ->->Cycle: 0.27/0.28 ->->-> Pairs: 0.27/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.27/0.28 ->->-> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 0.27/0.28 Problem 1.7: 0.27/0.28 0.27/0.28 Reduction Pairs Processor: 0.27/0.28 -> Pairs: 0.27/0.28 TOP(ok(X:S)) -> TOP(active(X:S)) 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 -> Usable rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 ->Interpretation type: 0.27/0.28 Linear 0.27/0.28 ->Coefficients: 0.27/0.28 Natural Numbers 0.27/0.28 ->Dimension: 0.27/0.28 1 0.27/0.28 ->Bound: 0.27/0.28 2 0.27/0.28 ->Interpretation: 0.27/0.28 0.27/0.28 [U11](X1,X2,X3) = 2.X1 + 2.X2 + X3 0.27/0.28 [U12](X1,X2,X3) = 2.X1 + 2.X3 + 1 0.27/0.28 [active](X) = 2.X 0.27/0.28 [plus](X1,X2) = 2.X2 + 2 0.27/0.28 [proper](X) = 0 0.27/0.28 [s](X) = 2.X 0.27/0.28 [top](X) = 0 0.27/0.28 [0] = 2 0.27/0.28 [fSNonEmpty] = 0 0.27/0.28 [mark](X) = 2 0.27/0.28 [ok](X) = 2.X + 2 0.27/0.28 [tt] = 1 0.27/0.28 [U11#](X1,X2,X3) = 0 0.27/0.28 [U12#](X1,X2,X3) = 0 0.27/0.28 [ACTIVE](X) = 0 0.27/0.28 [PLUS](X1,X2) = 0 0.27/0.28 [PROPER](X) = 0 0.27/0.28 [S](X) = 0 0.27/0.28 [TOP](X) = 2.X 0.27/0.28 0.27/0.28 Problem 1.7: 0.27/0.28 0.27/0.28 SCC Processor: 0.27/0.28 -> Pairs: 0.27/0.28 Empty 0.27/0.28 -> Rules: 0.27/0.28 U11(mark(X1:S),X2:S,X3:S) -> mark(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U11(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U11(X1:S,X2:S,X3:S)) 0.27/0.28 U12(mark(X1:S),X2:S,X3:S) -> mark(U12(X1:S,X2:S,X3:S)) 0.27/0.28 U12(ok(X1:S),ok(X2:S),ok(X3:S)) -> ok(U12(X1:S,X2:S,X3:S)) 0.27/0.28 active(U11(tt,M:S,N:S)) -> mark(U12(tt,M:S,N:S)) 0.27/0.28 active(U11(X1:S,X2:S,X3:S)) -> U11(active(X1:S),X2:S,X3:S) 0.27/0.28 active(U12(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S))) 0.27/0.28 active(U12(X1:S,X2:S,X3:S)) -> U12(active(X1:S),X2:S,X3:S) 0.27/0.28 active(plus(N:S,s(M:S))) -> mark(U11(tt,M:S,N:S)) 0.27/0.28 active(plus(N:S,0)) -> mark(N:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(active(X1:S),X2:S) 0.27/0.28 active(plus(X1:S,X2:S)) -> plus(X1:S,active(X2:S)) 0.27/0.28 active(s(X:S)) -> s(active(X:S)) 0.27/0.28 plus(mark(X1:S),X2:S) -> mark(plus(X1:S,X2:S)) 0.27/0.28 plus(ok(X1:S),ok(X2:S)) -> ok(plus(X1:S,X2:S)) 0.27/0.28 plus(X1:S,mark(X2:S)) -> mark(plus(X1:S,X2:S)) 0.27/0.28 proper(U11(X1:S,X2:S,X3:S)) -> U11(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(U12(X1:S,X2:S,X3:S)) -> U12(proper(X1:S),proper(X2:S),proper(X3:S)) 0.27/0.28 proper(plus(X1:S,X2:S)) -> plus(proper(X1:S),proper(X2:S)) 0.27/0.28 proper(s(X:S)) -> s(proper(X:S)) 0.27/0.28 proper(0) -> ok(0) 0.27/0.28 proper(tt) -> ok(tt) 0.27/0.28 s(mark(X:S)) -> mark(s(X:S)) 0.27/0.28 s(ok(X:S)) -> ok(s(X:S)) 0.27/0.28 top(mark(X:S)) -> top(proper(X:S)) 0.27/0.28 top(ok(X:S)) -> top(active(X:S)) 0.27/0.28 ->Strongly Connected Components: 0.27/0.28 There is no strongly connected component 0.27/0.28 0.27/0.28 The problem is finite. 0.27/0.28 EOF