YES Problem 1: (VAR v_NonEmpty:S x1:S) (RULES 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ) Problem 1: Dependency Pairs Processor: -> Pairs: 0#(ql(x1:S)) -> 0#(x1:S) 0#(qr(x1:S)) -> 0#(x1:S) 1#(ql(x1:S)) -> 1#(x1:S) 1#(qr(x1:S)) -> 1#(x1:S) B(ql(0(x1:S))) -> 0#(b(r0(x1:S))) B(ql(0(x1:S))) -> B(r0(x1:S)) B(ql(0(x1:S))) -> R0(x1:S) B(ql(1(x1:S))) -> 1#(b(r1(x1:S))) B(ql(1(x1:S))) -> B(r1(x1:S)) B(ql(1(x1:S))) -> R1(x1:S) M(qr(x1:S)) -> M(x1:S) R0(0(x1:S)) -> 0#(r0(x1:S)) R0(0(x1:S)) -> R0(x1:S) R0(1(x1:S)) -> 1#(r0(x1:S)) R0(1(x1:S)) -> R0(x1:S) R0(b(x1:S)) -> 0#(b(x1:S)) R0(m(x1:S)) -> M(r0(x1:S)) R0(m(x1:S)) -> R0(x1:S) R1(0(x1:S)) -> 0#(r1(x1:S)) R1(0(x1:S)) -> R1(x1:S) R1(1(x1:S)) -> 1#(r1(x1:S)) R1(1(x1:S)) -> R1(x1:S) R1(b(x1:S)) -> 1#(b(x1:S)) R1(m(x1:S)) -> M(r1(x1:S)) R1(m(x1:S)) -> R1(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) Problem 1: SCC Processor: -> Pairs: 0#(ql(x1:S)) -> 0#(x1:S) 0#(qr(x1:S)) -> 0#(x1:S) 1#(ql(x1:S)) -> 1#(x1:S) 1#(qr(x1:S)) -> 1#(x1:S) B(ql(0(x1:S))) -> 0#(b(r0(x1:S))) B(ql(0(x1:S))) -> B(r0(x1:S)) B(ql(0(x1:S))) -> R0(x1:S) B(ql(1(x1:S))) -> 1#(b(r1(x1:S))) B(ql(1(x1:S))) -> B(r1(x1:S)) B(ql(1(x1:S))) -> R1(x1:S) M(qr(x1:S)) -> M(x1:S) R0(0(x1:S)) -> 0#(r0(x1:S)) R0(0(x1:S)) -> R0(x1:S) R0(1(x1:S)) -> 1#(r0(x1:S)) R0(1(x1:S)) -> R0(x1:S) R0(b(x1:S)) -> 0#(b(x1:S)) R0(m(x1:S)) -> M(r0(x1:S)) R0(m(x1:S)) -> R0(x1:S) R1(0(x1:S)) -> 0#(r1(x1:S)) R1(0(x1:S)) -> R1(x1:S) R1(1(x1:S)) -> 1#(r1(x1:S)) R1(1(x1:S)) -> R1(x1:S) R1(b(x1:S)) -> 1#(b(x1:S)) R1(m(x1:S)) -> M(r1(x1:S)) R1(m(x1:S)) -> R1(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: M(qr(x1:S)) -> M(x1:S) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->->Cycle: ->->-> Pairs: 1#(ql(x1:S)) -> 1#(x1:S) 1#(qr(x1:S)) -> 1#(x1:S) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->->Cycle: ->->-> Pairs: 0#(ql(x1:S)) -> 0#(x1:S) 0#(qr(x1:S)) -> 0#(x1:S) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->->Cycle: ->->-> Pairs: R0(0(x1:S)) -> R0(x1:S) R0(1(x1:S)) -> R0(x1:S) R0(m(x1:S)) -> R0(x1:S) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->->Cycle: ->->-> Pairs: R1(0(x1:S)) -> R1(x1:S) R1(1(x1:S)) -> R1(x1:S) R1(m(x1:S)) -> R1(x1:S) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->->Cycle: ->->-> Pairs: B(ql(0(x1:S))) -> B(r0(x1:S)) B(ql(1(x1:S))) -> B(r1(x1:S)) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) The problem is decomposed in 6 subproblems. Problem 1.1: Subterm Processor: -> Pairs: M(qr(x1:S)) -> M(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Projection: pi(M) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: 1#(ql(x1:S)) -> 1#(x1:S) 1#(qr(x1:S)) -> 1#(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Projection: pi(1#) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.3: Subterm Processor: -> Pairs: 0#(ql(x1:S)) -> 0#(x1:S) 0#(qr(x1:S)) -> 0#(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Projection: pi(0#) = 1 Problem 1.3: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.4: Subterm Processor: -> Pairs: R0(0(x1:S)) -> R0(x1:S) R0(1(x1:S)) -> R0(x1:S) R0(m(x1:S)) -> R0(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Projection: pi(R0) = 1 Problem 1.4: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.5: Subterm Processor: -> Pairs: R1(0(x1:S)) -> R1(x1:S) R1(1(x1:S)) -> R1(x1:S) R1(m(x1:S)) -> R1(x1:S) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Projection: pi(R1) = 1 Problem 1.5: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.6: Reduction Pair Processor: -> Pairs: B(ql(0(x1:S))) -> B(r0(x1:S)) B(ql(1(x1:S))) -> B(r1(x1:S)) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) -> Usable rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = X + 2 [1](X) = X + 2 [b](X) = 2.X + 2 [m](X) = 2 [r0](X) = X [r1](X) = X + 1 [ql](X) = X [qr](X) = 2 [B](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: B(ql(1(x1:S))) -> B(r1(x1:S)) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: B(ql(1(x1:S))) -> B(r1(x1:S)) ->->-> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) Problem 1.6: Reduction Pair Processor: -> Pairs: B(ql(1(x1:S))) -> B(r1(x1:S)) -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) -> Usable rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [0](X) = 2.X + 2 [1](X) = 2.X + 2 [b](X) = 2.X + 2 [m](X) = 2 [r0](X) = X [r1](X) = X [ql](X) = X [qr](X) = 2 [B](X) = 2.X Problem 1.6: SCC Processor: -> Pairs: Empty -> Rules: 0(ql(x1:S)) -> ql(0(x1:S)) 0(qr(x1:S)) -> qr(0(x1:S)) 1(ql(x1:S)) -> ql(1(x1:S)) 1(qr(x1:S)) -> qr(1(x1:S)) b(ql(0(x1:S))) -> 0(b(r0(x1:S))) b(ql(1(x1:S))) -> 1(b(r1(x1:S))) m(qr(x1:S)) -> ql(m(x1:S)) r0(0(x1:S)) -> 0(r0(x1:S)) r0(1(x1:S)) -> 1(r0(x1:S)) r0(b(x1:S)) -> qr(0(b(x1:S))) r0(m(x1:S)) -> m(r0(x1:S)) r1(0(x1:S)) -> 0(r1(x1:S)) r1(1(x1:S)) -> 1(r1(x1:S)) r1(b(x1:S)) -> qr(1(b(x1:S))) r1(m(x1:S)) -> m(r1(x1:S)) ->Strongly Connected Components: There is no strongly connected component The problem is finite.