YES Problem 1: (VAR v_NonEmpty:S L:S X:S X1:S X2:S) (RULES active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S) ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S)) ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S))) ACTIVE(adx(X:S)) -> ACTIVE(X:S) ACTIVE(adx(X:S)) -> ADX(active(X:S)) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) ACTIVE(head(X:S)) -> ACTIVE(X:S) ACTIVE(head(X:S)) -> HEAD(active(X:S)) ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S)) ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S) ACTIVE(incr(cons(X:S,L:S))) -> S(X:S) ACTIVE(incr(X:S)) -> ACTIVE(X:S) ACTIVE(incr(X:S)) -> INCR(active(X:S)) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X:S)) ACTIVE(tail(X:S)) -> ACTIVE(X:S) ACTIVE(tail(X:S)) -> TAIL(active(X:S)) ADX(mark(X:S)) -> ADX(X:S) ADX(ok(X:S)) -> ADX(X:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) HEAD(mark(X:S)) -> HEAD(X:S) HEAD(ok(X:S)) -> HEAD(X:S) INCR(mark(X:S)) -> INCR(X:S) INCR(ok(X:S)) -> INCR(X:S) PROPER(adx(X:S)) -> ADX(proper(X:S)) PROPER(adx(X:S)) -> PROPER(X:S) PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(head(X:S)) -> HEAD(proper(X:S)) PROPER(head(X:S)) -> PROPER(X:S) PROPER(incr(X:S)) -> INCR(proper(X:S)) PROPER(incr(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(tail(X:S)) -> PROPER(X:S) PROPER(tail(X:S)) -> TAIL(proper(X:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TAIL(mark(X:S)) -> TAIL(X:S) TAIL(ok(X:S)) -> TAIL(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: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1: SCC Processor: -> Pairs: ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S) ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S)) ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S))) ACTIVE(adx(X:S)) -> ACTIVE(X:S) ACTIVE(adx(X:S)) -> ADX(active(X:S)) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S) ACTIVE(head(X:S)) -> ACTIVE(X:S) ACTIVE(head(X:S)) -> HEAD(active(X:S)) ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S)) ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S) ACTIVE(incr(cons(X:S,L:S))) -> S(X:S) ACTIVE(incr(X:S)) -> ACTIVE(X:S) ACTIVE(incr(X:S)) -> INCR(active(X:S)) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> S(active(X:S)) ACTIVE(tail(X:S)) -> ACTIVE(X:S) ACTIVE(tail(X:S)) -> TAIL(active(X:S)) ADX(mark(X:S)) -> ADX(X:S) ADX(ok(X:S)) -> ADX(X:S) CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) HEAD(mark(X:S)) -> HEAD(X:S) HEAD(ok(X:S)) -> HEAD(X:S) INCR(mark(X:S)) -> INCR(X:S) INCR(ok(X:S)) -> INCR(X:S) PROPER(adx(X:S)) -> ADX(proper(X:S)) PROPER(adx(X:S)) -> PROPER(X:S) PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S)) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(head(X:S)) -> HEAD(proper(X:S)) PROPER(head(X:S)) -> PROPER(X:S) PROPER(incr(X:S)) -> INCR(proper(X:S)) PROPER(incr(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> S(proper(X:S)) PROPER(tail(X:S)) -> PROPER(X:S) PROPER(tail(X:S)) -> TAIL(proper(X:S)) S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) TAIL(mark(X:S)) -> TAIL(X:S) TAIL(ok(X:S)) -> TAIL(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: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: TAIL(mark(X:S)) -> TAIL(X:S) TAIL(ok(X:S)) -> TAIL(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: INCR(mark(X:S)) -> INCR(X:S) INCR(ok(X:S)) -> INCR(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: HEAD(mark(X:S)) -> HEAD(X:S) HEAD(ok(X:S)) -> HEAD(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ADX(mark(X:S)) -> ADX(X:S) ADX(ok(X:S)) -> ADX(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: PROPER(adx(X:S)) -> PROPER(X:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(head(X:S)) -> PROPER(X:S) PROPER(incr(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(tail(X:S)) -> PROPER(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->->Cycle: ->->-> Pairs: ACTIVE(adx(X:S)) -> ACTIVE(X:S) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(head(X:S)) -> ACTIVE(X:S) ACTIVE(incr(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(tail(X:S)) -> ACTIVE(X:S) ->->-> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) The problem is decomposed in 9 subproblems. Problem 1.1: Subterm Processor: -> Pairs: TAIL(mark(X:S)) -> TAIL(X:S) TAIL(ok(X:S)) -> TAIL(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(TAIL) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: S(mark(X:S)) -> S(X:S) S(ok(X:S)) -> S(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(S) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: INCR(mark(X:S)) -> INCR(X:S) INCR(ok(X:S)) -> INCR(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(INCR) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: HEAD(mark(X:S)) -> HEAD(X:S) HEAD(ok(X:S)) -> HEAD(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(HEAD) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S) CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(CONS) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: ADX(mark(X:S)) -> ADX(X:S) ADX(ok(X:S)) -> ADX(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ADX) = 1 Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: Subterm Processor: -> Pairs: PROPER(adx(X:S)) -> PROPER(X:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S) PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S) PROPER(head(X:S)) -> PROPER(X:S) PROPER(incr(X:S)) -> PROPER(X:S) PROPER(s(X:S)) -> PROPER(X:S) PROPER(tail(X:S)) -> PROPER(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(PROPER) = 1 Problem 1.7: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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 1.8: Subterm Processor: -> Pairs: ACTIVE(adx(X:S)) -> ACTIVE(X:S) ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S) ACTIVE(head(X:S)) -> ACTIVE(X:S) ACTIVE(incr(X:S)) -> ACTIVE(X:S) ACTIVE(s(X:S)) -> ACTIVE(X:S) ACTIVE(tail(X:S)) -> ACTIVE(X:S) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) ->Projection: pi(ACTIVE) = 1 Problem 1.8: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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 1.9: Reduction Pairs Processor: -> Pairs: TOP(mark(X:S)) -> TOP(proper(X:S)) TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) ->Interpretation type: Linear ->Coefficients: All rationals ->Dimension: 1 ->Bound: 4 ->Interpretation: [active](X) = X [adx](X) = X + 4/3 [cons](X1,X2) = 4/3.X1 + 1/4.X2 [head](X) = 4.X + 1/3 [incr](X) = X + 2/3 [proper](X) = X [s](X) = X [tail](X) = 4.X + 1/3 [top](X) = 0 [0] = 0 [fSNonEmpty] = 0 [mark](X) = X + 1/3 [nats] = 4 [nil] = 1/4 [ok](X) = X [zeros] = 2 [ACTIVE](X) = 0 [ADX](X) = 0 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TOP](X) = 4.X Problem 1.9: SCC Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) Problem 1.9: Reduction Pairs Processor: -> Pairs: TOP(ok(X:S)) -> TOP(active(X:S)) -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) top(mark(X:S)) -> top(proper(X:S)) top(ok(X:S)) -> top(active(X:S)) -> Usable rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(X:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [active](X) = 2.X [adx](X) = 2.X [cons](X1,X2) = 2.X1 + 2 [head](X) = 2.X + 1 [incr](X) = 2.X [proper](X) = 0 [s](X) = 2.X + 2 [tail](X) = 2.X + 2 [top](X) = 0 [0] = 2 [fSNonEmpty] = 0 [mark](X) = 2 [nats] = 2 [nil] = 2 [ok](X) = 2.X + 2 [zeros] = 2 [ACTIVE](X) = 0 [ADX](X) = 0 [CONS](X1,X2) = 0 [HEAD](X) = 0 [INCR](X) = 0 [PROPER](X) = 0 [S](X) = 0 [TAIL](X) = 0 [TOP](X) = X Problem 1.9: SCC Processor: -> Pairs: Empty -> Rules: active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S)))) active(adx(nil)) -> mark(nil) active(adx(X:S)) -> adx(active(X:S)) active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S) active(head(cons(X:S,L:S))) -> mark(X:S) active(head(X:S)) -> head(active(X:S)) active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S))) active(incr(nil)) -> mark(nil) active(incr(X:S)) -> incr(active(X:S)) active(s(X:S)) -> s(active(X:S)) active(tail(cons(X:S,L:S))) -> mark(L:S) active(tail(X:S)) -> tail(active(X:S)) active(nats) -> mark(adx(zeros)) active(zeros) -> mark(cons(0,zeros)) adx(mark(X:S)) -> mark(adx(X:S)) adx(ok(X:S)) -> ok(adx(X:S)) cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S)) cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S)) head(mark(X:S)) -> mark(head(X:S)) head(ok(X:S)) -> ok(head(X:S)) incr(mark(X:S)) -> mark(incr(X:S)) incr(ok(X:S)) -> ok(incr(X:S)) proper(adx(X:S)) -> adx(proper(X:S)) proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S)) proper(head(X:S)) -> head(proper(X:S)) proper(incr(X:S)) -> incr(proper(X:S)) proper(s(X:S)) -> s(proper(X:S)) proper(tail(X:S)) -> tail(proper(X:S)) proper(0) -> ok(0) proper(nats) -> ok(nats) proper(nil) -> ok(nil) proper(zeros) -> ok(zeros) s(mark(X:S)) -> mark(s(X:S)) s(ok(X:S)) -> ok(s(X:S)) tail(mark(X:S)) -> mark(tail(X:S)) tail(ok(X:S)) -> ok(tail(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.