- : unit = () - : unit = () h : heuristic = - : unit = () APPLY CRITERIA (Marked dependency pairs) TRS termination of: [1] a__nats -> cons(0,incr(nats)) [2] a__pairs -> cons(0,incr(odds)) [3] a__odds -> a__incr(a__pairs) [4] a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) [5] a__head(cons(X,XS)) -> mark(X) [6] a__tail(cons(X,XS)) -> mark(XS) [7] mark(nats) -> a__nats [8] mark(pairs) -> a__pairs [9] mark(odds) -> a__odds [10] mark(incr(X)) -> a__incr(mark(X)) [11] mark(head(X)) -> a__head(mark(X)) [12] mark(tail(X)) -> a__tail(mark(X)) [13] mark(0) -> 0 [14] mark(s(X)) -> s(mark(X)) [15] mark(nil) -> nil [16] mark(cons(X1,X2)) -> cons(mark(X1),X2) [17] a__nats -> nats [18] a__pairs -> pairs [19] a__odds -> odds [20] a__incr(X) -> incr(X) [21] a__head(X) -> head(X) [22] a__tail(X) -> tail(X) Sub problem: guided: DP termination of: END GUIDED APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__nats >= cons(0,incr(nats)) ; a__nats >= nats ; a__pairs >= cons(0,incr(odds)) ; a__pairs >= pairs ; a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) ; a__incr(X) >= incr(X) ; a__odds >= odds ; a__odds >= a__incr(a__pairs) ; mark(cons(X1,X2)) >= cons(mark(X1),X2) ; mark(0) >= 0 ; mark(incr(X)) >= a__incr(mark(X)) ; mark(nats) >= a__nats ; mark(odds) >= a__odds ; mark(s(X)) >= s(mark(X)) ; mark(pairs) >= a__pairs ; mark(head(X)) >= a__head(mark(X)) ; mark(tail(X)) >= a__tail(mark(X)) ; mark(nil) >= nil ; a__head(cons(X,XS)) >= mark(X) ; a__head(X) >= head(X) ; a__tail(cons(X,XS)) >= mark(XS) ; a__tail(X) >= tail(X) ; Marked_a__tail(cons(X,XS)) >= Marked_mark(XS) ; Marked_a__head(cons(X,XS)) >= Marked_mark(X) ; Marked_a__incr(cons(X,XS)) >= Marked_mark(X) ; Marked_a__odds >= Marked_a__incr(a__pairs) ; Marked_mark(cons(X1,X2)) >= Marked_mark(X1) ; Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(odds) >= Marked_a__odds ; Marked_mark(s(X)) >= Marked_mark(X) ; Marked_mark(head(X)) >= Marked_a__head(mark(X)) ; Marked_mark(head(X)) >= Marked_mark(X) ; Marked_mark(tail(X)) >= Marked_a__tail(mark(X)) ; Marked_mark(tail(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__tail(cons(X,XS)) > Marked_mark(XS) ; } { Marked_a__head(cons(X,XS)) > Marked_mark(X) ; } { Marked_a__incr(cons(X,XS)) > Marked_mark(X) ; } { Marked_a__odds > Marked_a__incr(a__pairs) ; } { Marked_mark(cons(X1,X2)) > Marked_mark(X1) ; } { Marked_mark(incr(X)) > Marked_a__incr(mark(X)) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(odds) > Marked_a__odds ; } { Marked_mark(s(X)) > Marked_mark(X) ; } { Marked_mark(head(X)) > Marked_a__head(mark(X)) ; } { Marked_mark(head(X)) > Marked_mark(X) ; } { Marked_mark(tail(X)) > Marked_a__tail(mark(X)) ; } { Marked_mark(tail(X)) > Marked_mark(X) ; } } === TIMER virtual : 10.000000 === Entering poly_solver Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 10.000000 === === STOPING TIMER real === Sat solver returned Sat solver result read === STOPING TIMER real === === STOPING TIMER virtual === constraint: a__nats >= cons(0,incr(nats)) constraint: a__nats >= nats constraint: a__pairs >= cons(0,incr(odds)) constraint: a__pairs >= pairs constraint: a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) constraint: a__incr(X) >= incr(X) constraint: a__odds >= odds constraint: a__odds >= a__incr(a__pairs) constraint: mark(cons(X1,X2)) >= cons(mark(X1),X2) constraint: mark(0) >= 0 constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(odds) >= a__odds constraint: mark(s(X)) >= s(mark(X)) constraint: mark(pairs) >= a__pairs constraint: mark(head(X)) >= a__head(mark(X)) constraint: mark(tail(X)) >= a__tail(mark(X)) constraint: mark(nil) >= nil constraint: a__head(cons(X,XS)) >= mark(X) constraint: a__head(X) >= head(X) constraint: a__tail(cons(X,XS)) >= mark(XS) constraint: a__tail(X) >= tail(X) constraint: Marked_a__tail(cons(X,XS)) >= Marked_mark(XS) constraint: Marked_a__head(cons(X,XS)) >= Marked_mark(X) constraint: Marked_a__incr(cons(X,XS)) >= Marked_mark(X) constraint: Marked_a__odds >= Marked_a__incr(a__pairs) constraint: Marked_mark(cons(X1,X2)) >= Marked_mark(X1) constraint: Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(odds) >= Marked_a__odds constraint: Marked_mark(s(X)) >= Marked_mark(X) constraint: Marked_mark(head(X)) >= Marked_a__head(mark(X)) constraint: Marked_mark(head(X)) >= Marked_mark(X) constraint: Marked_mark(tail(X)) >= Marked_a__tail(mark(X)) constraint: Marked_mark(tail(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__nats >= cons(0,incr(nats)) ; a__nats >= nats ; a__pairs >= cons(0,incr(odds)) ; a__pairs >= pairs ; a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) ; a__incr(X) >= incr(X) ; a__odds >= odds ; a__odds >= a__incr(a__pairs) ; mark(cons(X1,X2)) >= cons(mark(X1),X2) ; mark(0) >= 0 ; mark(incr(X)) >= a__incr(mark(X)) ; mark(nats) >= a__nats ; mark(odds) >= a__odds ; mark(s(X)) >= s(mark(X)) ; mark(pairs) >= a__pairs ; mark(head(X)) >= a__head(mark(X)) ; mark(tail(X)) >= a__tail(mark(X)) ; mark(nil) >= nil ; a__head(cons(X,XS)) >= mark(X) ; a__head(X) >= head(X) ; a__tail(cons(X,XS)) >= mark(XS) ; a__tail(X) >= tail(X) ; Marked_a__head(cons(X,XS)) >= Marked_mark(X) ; Marked_a__incr(cons(X,XS)) >= Marked_mark(X) ; Marked_a__odds >= Marked_a__incr(a__pairs) ; Marked_mark(cons(X1,X2)) >= Marked_mark(X1) ; Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(odds) >= Marked_a__odds ; Marked_mark(s(X)) >= Marked_mark(X) ; Marked_mark(head(X)) >= Marked_a__head(mark(X)) ; Marked_mark(head(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__head(cons(X,XS)) > Marked_mark(X) ; } { Marked_a__incr(cons(X,XS)) > Marked_mark(X) ; } { Marked_a__odds > Marked_a__incr(a__pairs) ; } { Marked_mark(cons(X1,X2)) > Marked_mark(X1) ; } { Marked_mark(incr(X)) > Marked_a__incr(mark(X)) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(odds) > Marked_a__odds ; } { Marked_mark(s(X)) > Marked_mark(X) ; } { Marked_mark(head(X)) > Marked_a__head(mark(X)) ; } { Marked_mark(head(X)) > Marked_mark(X) ; } } === TIMER virtual : 10.000000 === Entering poly_solver Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 10.000000 === === STOPING TIMER real === Sat solver returned Sat solver result read === STOPING TIMER real === === STOPING TIMER virtual === constraint: a__nats >= cons(0,incr(nats)) constraint: a__nats >= nats constraint: a__pairs >= cons(0,incr(odds)) constraint: a__pairs >= pairs constraint: a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) constraint: a__incr(X) >= incr(X) constraint: a__odds >= odds constraint: a__odds >= a__incr(a__pairs) constraint: mark(cons(X1,X2)) >= cons(mark(X1),X2) constraint: mark(0) >= 0 constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(odds) >= a__odds constraint: mark(s(X)) >= s(mark(X)) constraint: mark(pairs) >= a__pairs constraint: mark(head(X)) >= a__head(mark(X)) constraint: mark(tail(X)) >= a__tail(mark(X)) constraint: mark(nil) >= nil constraint: a__head(cons(X,XS)) >= mark(X) constraint: a__head(X) >= head(X) constraint: a__tail(cons(X,XS)) >= mark(XS) constraint: a__tail(X) >= tail(X) constraint: Marked_a__head(cons(X,XS)) >= Marked_mark(X) constraint: Marked_a__incr(cons(X,XS)) >= Marked_mark(X) constraint: Marked_a__odds >= Marked_a__incr(a__pairs) constraint: Marked_mark(cons(X1,X2)) >= Marked_mark(X1) constraint: Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(odds) >= Marked_a__odds constraint: Marked_mark(s(X)) >= Marked_mark(X) constraint: Marked_mark(head(X)) >= Marked_a__head(mark(X)) constraint: Marked_mark(head(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__nats >= cons(0,incr(nats)) ; a__nats >= nats ; a__pairs >= cons(0,incr(odds)) ; a__pairs >= pairs ; a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) ; a__incr(X) >= incr(X) ; a__odds >= odds ; a__odds >= a__incr(a__pairs) ; mark(cons(X1,X2)) >= cons(mark(X1),X2) ; mark(0) >= 0 ; mark(incr(X)) >= a__incr(mark(X)) ; mark(nats) >= a__nats ; mark(odds) >= a__odds ; mark(s(X)) >= s(mark(X)) ; mark(pairs) >= a__pairs ; mark(head(X)) >= a__head(mark(X)) ; mark(tail(X)) >= a__tail(mark(X)) ; mark(nil) >= nil ; a__head(cons(X,XS)) >= mark(X) ; a__head(X) >= head(X) ; a__tail(cons(X,XS)) >= mark(XS) ; a__tail(X) >= tail(X) ; Marked_a__incr(cons(X,XS)) >= Marked_mark(X) ; Marked_a__odds >= Marked_a__incr(a__pairs) ; Marked_mark(cons(X1,X2)) >= Marked_mark(X1) ; Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(odds) >= Marked_a__odds ; Marked_mark(s(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__incr(cons(X,XS)) > Marked_mark(X) ; } { Marked_a__odds > Marked_a__incr(a__pairs) ; } { Marked_mark(cons(X1,X2)) > Marked_mark(X1) ; } { Marked_mark(incr(X)) > Marked_a__incr(mark(X)) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(odds) > Marked_a__odds ; } { Marked_mark(s(X)) > Marked_mark(X) ; } } === TIMER virtual : 10.000000 === Entering poly_solver Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 10.000000 === === STOPING TIMER real === Sat solver returned Sat solver result read === STOPING TIMER real === === STOPING TIMER virtual === constraint: a__nats >= cons(0,incr(nats)) constraint: a__nats >= nats constraint: a__pairs >= cons(0,incr(odds)) constraint: a__pairs >= pairs constraint: a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) constraint: a__incr(X) >= incr(X) constraint: a__odds >= odds constraint: a__odds >= a__incr(a__pairs) constraint: mark(cons(X1,X2)) >= cons(mark(X1),X2) constraint: mark(0) >= 0 constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(odds) >= a__odds constraint: mark(s(X)) >= s(mark(X)) constraint: mark(pairs) >= a__pairs constraint: mark(head(X)) >= a__head(mark(X)) constraint: mark(tail(X)) >= a__tail(mark(X)) constraint: mark(nil) >= nil constraint: a__head(cons(X,XS)) >= mark(X) constraint: a__head(X) >= head(X) constraint: a__tail(cons(X,XS)) >= mark(XS) constraint: a__tail(X) >= tail(X) constraint: Marked_a__incr(cons(X,XS)) >= Marked_mark(X) constraint: Marked_a__odds >= Marked_a__incr(a__pairs) constraint: Marked_mark(cons(X1,X2)) >= Marked_mark(X1) constraint: Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(odds) >= Marked_a__odds constraint: Marked_mark(s(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__nats >= cons(0,incr(nats)) ; a__nats >= nats ; a__pairs >= cons(0,incr(odds)) ; a__pairs >= pairs ; a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) ; a__incr(X) >= incr(X) ; a__odds >= odds ; a__odds >= a__incr(a__pairs) ; mark(cons(X1,X2)) >= cons(mark(X1),X2) ; mark(0) >= 0 ; mark(incr(X)) >= a__incr(mark(X)) ; mark(nats) >= a__nats ; mark(odds) >= a__odds ; mark(s(X)) >= s(mark(X)) ; mark(pairs) >= a__pairs ; mark(head(X)) >= a__head(mark(X)) ; mark(tail(X)) >= a__tail(mark(X)) ; mark(nil) >= nil ; a__head(cons(X,XS)) >= mark(X) ; a__head(X) >= head(X) ; a__tail(cons(X,XS)) >= mark(XS) ; a__tail(X) >= tail(X) ; Marked_a__incr(cons(X,XS)) >= Marked_mark(X) ; Marked_mark(cons(X1,X2)) >= Marked_mark(X1) ; Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(s(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__incr(cons(X,XS)) > Marked_mark(X) ; } { Marked_mark(cons(X1,X2)) > Marked_mark(X1) ; } { Marked_mark(incr(X)) > Marked_a__incr(mark(X)) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(s(X)) > Marked_mark(X) ; } } === TIMER virtual : 10.000000 === Entering poly_solver Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 10.000000 === === STOPING TIMER real === Sat solver returned === STOPING TIMER real === === STOPING TIMER virtual === No solution found for these parameters. Entering rpo_solver === TIMER virtual : 25.000000 === Search parameters: AFS type: 2 ; time limit: 25.. === STOPING TIMER virtual === === TIMER virtual : 25.000000 === Search parameters: AFS type: 2 ; time limit: 25.. === STOPING TIMER virtual === === TIMER virtual : 25.000000 === Search parameters: AFS type: 2 ; time limit: 25.. === STOPING TIMER virtual === === TIMER virtual : 25.000000 === Search parameters: AFS type: 2 ; time limit: 25.. === STOPING TIMER virtual === === TIMER virtual : 25.000000 === Search parameters: AFS type: 2 ; time limit: 25.. === STOPING TIMER virtual === === TIMER virtual : 15.000000 === Entering poly_solver Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 15.000000 === === STOPING TIMER real === Sat solver returned === STOPING TIMER real === === STOPING TIMER virtual === No solution found for these parameters. === TIMER virtual : 50.000000 === trying sub matrices of size: 1 Matrix interpretation constraints generated. Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. Termination constraints generated. Starting Sat solver initialization Calling Sat solver... === STOPING TIMER virtual === === TIMER real : 50.000000 === === STOPING TIMER real === Sat solver returned Sat solver result read === STOPING TIMER real === === STOPING TIMER virtual === constraint: a__nats >= cons(0,incr(nats)) constraint: a__nats >= nats constraint: a__pairs >= cons(0,incr(odds)) constraint: a__pairs >= pairs constraint: a__incr(cons(X,XS)) >= cons(s(mark(X)),incr(XS)) constraint: a__incr(X) >= incr(X) constraint: a__odds >= odds constraint: a__odds >= a__incr(a__pairs) constraint: mark(cons(X1,X2)) >= cons(mark(X1),X2) constraint: mark(0) >= 0 constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(odds) >= a__odds constraint: mark(s(X)) >= s(mark(X)) constraint: mark(pairs) >= a__pairs constraint: mark(head(X)) >= a__head(mark(X)) constraint: mark(tail(X)) >= a__tail(mark(X)) constraint: mark(nil) >= nil constraint: a__head(cons(X,XS)) >= mark(X) constraint: a__head(X) >= head(X) constraint: a__tail(cons(X,XS)) >= mark(XS) constraint: a__tail(X) >= tail(X) constraint: Marked_a__incr(cons(X,XS)) >= Marked_mark(X) constraint: Marked_mark(cons(X1,X2)) >= Marked_mark(X1) constraint: Marked_mark(incr(X)) >= Marked_a__incr(mark(X)) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(s(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> } APPLY CRITERIA (Subterm criterion) ST: Marked_mark -> 1 APPLY CRITERIA (Graph splitting) Found 0 components: SOLVED { TRS termination of: [1] a__nats -> cons(0,incr(nats)) [2] a__pairs -> cons(0,incr(odds)) [3] a__odds -> a__incr(a__pairs) [4] a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) [5] a__head(cons(X,XS)) -> mark(X) [6] a__tail(cons(X,XS)) -> mark(XS) [7] mark(nats) -> a__nats [8] mark(pairs) -> a__pairs [9] mark(odds) -> a__odds [10] mark(incr(X)) -> a__incr(mark(X)) [11] mark(head(X)) -> a__head(mark(X)) [12] mark(tail(X)) -> a__tail(mark(X)) [13] mark(0) -> 0 [14] mark(s(X)) -> s(mark(X)) [15] mark(nil) -> nil [16] mark(cons(X1,X2)) -> cons(mark(X1),X2) [17] a__nats -> nats [18] a__pairs -> pairs [19] a__odds -> odds [20] a__incr(X) -> incr(X) [21] a__head(X) -> head(X) [22] a__tail(X) -> tail(X) , CRITERION: MDP [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: CG using polynomial interpretation = [ cons ] (X0,X1) = 2*X1 + 3*X0; [ nil ] () = 0; [ a__odds ] () = 0; [ a__nats ] () = 0; [ Marked_a__odds ] () = 2; [ a__tail ] (X0) = 1*X0 + 2; [ incr ] (X0) = 1*X0; [ Marked_a__head ] (X0) = 1*X0 + 2; [ mark ] (X0) = 1*X0; [ a__pairs ] () = 0; [ head ] (X0) = 3*X0; [ 0 ] () = 0; [ Marked_a__tail ] (X0) = 2*X0 + 2; [ s ] (X0) = 1*X0; [ odds ] () = 0; [ pairs ] () = 0; [ nats ] () = 0; [ Marked_a__incr ] (X0) = 1*X0 + 2; [ a__head ] (X0) = 3*X0; [ a__incr ] (X0) = 1*X0; [ Marked_mark ] (X0) = 3*X0 + 2; [ tail ] (X0) = 1*X0 + 2; removing [ { DP termination of: , CRITERION: SG [ { DP termination of: