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