- : unit = () h : heuristic = - : unit = () APPLY CRITERIA (Marked dependency pairs) TRS termination of: [1] a__nats -> a__adx(a__zeros) [2] a__zeros -> cons(0,zeros) [3] a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) [4] a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) [5] a__hd(cons(X,Y)) -> mark(X) [6] a__tl(cons(X,Y)) -> mark(Y) [7] mark(nats) -> a__nats [8] mark(adx(X)) -> a__adx(mark(X)) [9] mark(zeros) -> a__zeros [10] mark(incr(X)) -> a__incr(mark(X)) [11] mark(hd(X)) -> a__hd(mark(X)) [12] mark(tl(X)) -> a__tl(mark(X)) [13] mark(cons(X1,X2)) -> cons(X1,X2) [14] mark(0) -> 0 [15] mark(s(X)) -> s(X) [16] a__nats -> nats [17] a__adx(X) -> adx(X) [18] a__zeros -> zeros [19] a__incr(X) -> incr(X) [20] a__hd(X) -> hd(X) [21] a__tl(X) -> tl(X) Sub problem: guided: DP termination of: END GUIDED APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) ; a__adx(X) >= adx(X) ; a__zeros >= cons(0,zeros) ; a__zeros >= zeros ; a__nats >= a__adx(a__zeros) ; a__nats >= nats ; a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) ; a__incr(X) >= incr(X) ; mark(cons(X1,X2)) >= cons(X1,X2) ; mark(0) >= 0 ; mark(zeros) >= a__zeros ; mark(s(X)) >= s(X) ; mark(incr(X)) >= a__incr(mark(X)) ; mark(adx(X)) >= a__adx(mark(X)) ; mark(nats) >= a__nats ; mark(hd(X)) >= a__hd(mark(X)) ; mark(tl(X)) >= a__tl(mark(X)) ; a__hd(cons(X,Y)) >= mark(X) ; a__hd(X) >= hd(X) ; a__tl(cons(X,Y)) >= mark(Y) ; a__tl(X) >= tl(X) ; Marked_a__tl(cons(X,Y)) >= Marked_mark(Y) ; Marked_a__hd(cons(X,Y)) >= Marked_mark(X) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(adx(X)) >= Marked_mark(X) ; Marked_mark(hd(X)) >= Marked_a__hd(mark(X)) ; Marked_mark(hd(X)) >= Marked_mark(X) ; Marked_mark(tl(X)) >= Marked_a__tl(mark(X)) ; Marked_mark(tl(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__tl(cons(X,Y)) > Marked_mark(Y) ; } { Marked_a__hd(cons(X,Y)) > Marked_mark(X) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(adx(X)) > Marked_mark(X) ; } { Marked_mark(hd(X)) > Marked_a__hd(mark(X)) ; } { Marked_mark(hd(X)) > Marked_mark(X) ; } { Marked_mark(tl(X)) > Marked_a__tl(mark(X)) ; } { Marked_mark(tl(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__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) constraint: a__adx(X) >= adx(X) constraint: a__zeros >= cons(0,zeros) constraint: a__zeros >= zeros constraint: a__nats >= a__adx(a__zeros) constraint: a__nats >= nats constraint: a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) constraint: a__incr(X) >= incr(X) constraint: mark(cons(X1,X2)) >= cons(X1,X2) constraint: mark(0) >= 0 constraint: mark(zeros) >= a__zeros constraint: mark(s(X)) >= s(X) constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(adx(X)) >= a__adx(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(hd(X)) >= a__hd(mark(X)) constraint: mark(tl(X)) >= a__tl(mark(X)) constraint: a__hd(cons(X,Y)) >= mark(X) constraint: a__hd(X) >= hd(X) constraint: a__tl(cons(X,Y)) >= mark(Y) constraint: a__tl(X) >= tl(X) constraint: Marked_a__tl(cons(X,Y)) >= Marked_mark(Y) constraint: Marked_a__hd(cons(X,Y)) >= Marked_mark(X) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(adx(X)) >= Marked_mark(X) constraint: Marked_mark(hd(X)) >= Marked_a__hd(mark(X)) constraint: Marked_mark(hd(X)) >= Marked_mark(X) constraint: Marked_mark(tl(X)) >= Marked_a__tl(mark(X)) constraint: Marked_mark(tl(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) ; a__adx(X) >= adx(X) ; a__zeros >= cons(0,zeros) ; a__zeros >= zeros ; a__nats >= a__adx(a__zeros) ; a__nats >= nats ; a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) ; a__incr(X) >= incr(X) ; mark(cons(X1,X2)) >= cons(X1,X2) ; mark(0) >= 0 ; mark(zeros) >= a__zeros ; mark(s(X)) >= s(X) ; mark(incr(X)) >= a__incr(mark(X)) ; mark(adx(X)) >= a__adx(mark(X)) ; mark(nats) >= a__nats ; mark(hd(X)) >= a__hd(mark(X)) ; mark(tl(X)) >= a__tl(mark(X)) ; a__hd(cons(X,Y)) >= mark(X) ; a__hd(X) >= hd(X) ; a__tl(cons(X,Y)) >= mark(Y) ; a__tl(X) >= tl(X) ; Marked_a__hd(cons(X,Y)) >= Marked_mark(X) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(adx(X)) >= Marked_mark(X) ; Marked_mark(hd(X)) >= Marked_a__hd(mark(X)) ; Marked_mark(hd(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_a__hd(cons(X,Y)) > Marked_mark(X) ; } { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(adx(X)) > Marked_mark(X) ; } { Marked_mark(hd(X)) > Marked_a__hd(mark(X)) ; } { Marked_mark(hd(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__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) constraint: a__adx(X) >= adx(X) constraint: a__zeros >= cons(0,zeros) constraint: a__zeros >= zeros constraint: a__nats >= a__adx(a__zeros) constraint: a__nats >= nats constraint: a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) constraint: a__incr(X) >= incr(X) constraint: mark(cons(X1,X2)) >= cons(X1,X2) constraint: mark(0) >= 0 constraint: mark(zeros) >= a__zeros constraint: mark(s(X)) >= s(X) constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(adx(X)) >= a__adx(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(hd(X)) >= a__hd(mark(X)) constraint: mark(tl(X)) >= a__tl(mark(X)) constraint: a__hd(cons(X,Y)) >= mark(X) constraint: a__hd(X) >= hd(X) constraint: a__tl(cons(X,Y)) >= mark(Y) constraint: a__tl(X) >= tl(X) constraint: Marked_a__hd(cons(X,Y)) >= Marked_mark(X) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(adx(X)) >= Marked_mark(X) constraint: Marked_mark(hd(X)) >= Marked_a__hd(mark(X)) constraint: Marked_mark(hd(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> --> --> --> } APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) ; a__adx(X) >= adx(X) ; a__zeros >= cons(0,zeros) ; a__zeros >= zeros ; a__nats >= a__adx(a__zeros) ; a__nats >= nats ; a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) ; a__incr(X) >= incr(X) ; mark(cons(X1,X2)) >= cons(X1,X2) ; mark(0) >= 0 ; mark(zeros) >= a__zeros ; mark(s(X)) >= s(X) ; mark(incr(X)) >= a__incr(mark(X)) ; mark(adx(X)) >= a__adx(mark(X)) ; mark(nats) >= a__nats ; mark(hd(X)) >= a__hd(mark(X)) ; mark(tl(X)) >= a__tl(mark(X)) ; a__hd(cons(X,Y)) >= mark(X) ; a__hd(X) >= hd(X) ; a__tl(cons(X,Y)) >= mark(Y) ; a__tl(X) >= tl(X) ; Marked_mark(incr(X)) >= Marked_mark(X) ; Marked_mark(adx(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_mark(incr(X)) > Marked_mark(X) ; } { Marked_mark(adx(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__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) constraint: a__adx(X) >= adx(X) constraint: a__zeros >= cons(0,zeros) constraint: a__zeros >= zeros constraint: a__nats >= a__adx(a__zeros) constraint: a__nats >= nats constraint: a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) constraint: a__incr(X) >= incr(X) constraint: mark(cons(X1,X2)) >= cons(X1,X2) constraint: mark(0) >= 0 constraint: mark(zeros) >= a__zeros constraint: mark(s(X)) >= s(X) constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(adx(X)) >= a__adx(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(hd(X)) >= a__hd(mark(X)) constraint: mark(tl(X)) >= a__tl(mark(X)) constraint: a__hd(cons(X,Y)) >= mark(X) constraint: a__hd(X) >= hd(X) constraint: a__tl(cons(X,Y)) >= mark(Y) constraint: a__tl(X) >= tl(X) constraint: Marked_mark(incr(X)) >= Marked_mark(X) constraint: Marked_mark(adx(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 1 components: { --> } APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { a__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) ; a__adx(X) >= adx(X) ; a__zeros >= cons(0,zeros) ; a__zeros >= zeros ; a__nats >= a__adx(a__zeros) ; a__nats >= nats ; a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) ; a__incr(X) >= incr(X) ; mark(cons(X1,X2)) >= cons(X1,X2) ; mark(0) >= 0 ; mark(zeros) >= a__zeros ; mark(s(X)) >= s(X) ; mark(incr(X)) >= a__incr(mark(X)) ; mark(adx(X)) >= a__adx(mark(X)) ; mark(nats) >= a__nats ; mark(hd(X)) >= a__hd(mark(X)) ; mark(tl(X)) >= a__tl(mark(X)) ; a__hd(cons(X,Y)) >= mark(X) ; a__hd(X) >= hd(X) ; a__tl(cons(X,Y)) >= mark(Y) ; a__tl(X) >= tl(X) ; Marked_mark(incr(X)) >= Marked_mark(X) ; } + Disjunctions:{ { Marked_mark(incr(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__adx(cons(X,Y)) >= a__incr(cons(X,adx(Y))) constraint: a__adx(X) >= adx(X) constraint: a__zeros >= cons(0,zeros) constraint: a__zeros >= zeros constraint: a__nats >= a__adx(a__zeros) constraint: a__nats >= nats constraint: a__incr(cons(X,Y)) >= cons(s(X),incr(Y)) constraint: a__incr(X) >= incr(X) constraint: mark(cons(X1,X2)) >= cons(X1,X2) constraint: mark(0) >= 0 constraint: mark(zeros) >= a__zeros constraint: mark(s(X)) >= s(X) constraint: mark(incr(X)) >= a__incr(mark(X)) constraint: mark(adx(X)) >= a__adx(mark(X)) constraint: mark(nats) >= a__nats constraint: mark(hd(X)) >= a__hd(mark(X)) constraint: mark(tl(X)) >= a__tl(mark(X)) constraint: a__hd(cons(X,Y)) >= mark(X) constraint: a__hd(X) >= hd(X) constraint: a__tl(cons(X,Y)) >= mark(Y) constraint: a__tl(X) >= tl(X) constraint: Marked_mark(incr(X)) >= Marked_mark(X) APPLY CRITERIA (Graph splitting) Found 0 components: SOLVED { TRS termination of: [1] a__nats -> a__adx(a__zeros) [2] a__zeros -> cons(0,zeros) [3] a__incr(cons(X,Y)) -> cons(s(X),incr(Y)) [4] a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y))) [5] a__hd(cons(X,Y)) -> mark(X) [6] a__tl(cons(X,Y)) -> mark(Y) [7] mark(nats) -> a__nats [8] mark(adx(X)) -> a__adx(mark(X)) [9] mark(zeros) -> a__zeros [10] mark(incr(X)) -> a__incr(mark(X)) [11] mark(hd(X)) -> a__hd(mark(X)) [12] mark(tl(X)) -> a__tl(mark(X)) [13] mark(cons(X1,X2)) -> cons(X1,X2) [14] mark(0) -> 0 [15] mark(s(X)) -> s(X) [16] a__nats -> nats [17] a__adx(X) -> adx(X) [18] a__zeros -> zeros [19] a__incr(X) -> incr(X) [20] a__hd(X) -> hd(X) [21] a__tl(X) -> tl(X) , CRITERION: MDP [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: CG using polynomial interpretation = [ a__adx ] (X0) = 2*X0; [ Marked_a__tl ] (X0) = 3*X0 + 2; [ a__incr ] (X0) = 1*X0; [ 0 ] () = 0; [ a__tl ] (X0) = 2*X0 + 2; [ a__nats ] () = 0; [ mark ] (X0) = 2*X0; [ s ] (X0) = 0; [ Marked_mark ] (X0) = 3*X0 + 2; [ hd ] (X0) = 2*X0; [ a__zeros ] () = 0; [ Marked_a__hd ] (X0) = 2*X0 + 2; [ adx ] (X0) = 2*X0; [ zeros ] () = 0; [ nats ] () = 0; [ cons ] (X0,X1) = 1*X1 + 2*X0; [ a__hd ] (X0) = 2*X0; [ incr ] (X0) = 1*X0; [ tl ] (X0) = 2*X0 + 2; removing [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: CG using polynomial interpretation = [ a__adx ] (X0) = 1*X0; [ a__incr ] (X0) = 1*X0; [ 0 ] () = 0; [ a__tl ] (X0) = 1*X0; [ a__nats ] () = 0; [ mark ] (X0) = 1*X0; [ s ] (X0) = 1*X0; [ Marked_mark ] (X0) = 1*X0 + 1; [ hd ] (X0) = 1*X0 + 3; [ a__zeros ] () = 0; [ Marked_a__hd ] (X0) = 1*X0 + 3; [ adx ] (X0) = 1*X0; [ zeros ] () = 0; [ nats ] () = 0; [ cons ] (X0,X1) = 1*X1 + 1*X0; [ a__hd ] (X0) = 1*X0 + 3; [ incr ] (X0) = 1*X0; [ tl ] (X0) = 1*X0; removing [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: CG using polynomial interpretation = [ a__adx ] (X0) = 1*X0 + 3; [ a__incr ] (X0) = 1*X0; [ 0 ] () = 0; [ a__tl ] (X0) = 3*X0 + 2; [ a__nats ] () = 3; [ mark ] (X0) = 3*X0 + 1; [ s ] (X0) = 0; [ Marked_mark ] (X0) = 3*X0; [ hd ] (X0) = 3*X0 + 2; [ a__zeros ] () = 0; [ adx ] (X0) = 1*X0 + 1; [ zeros ] () = 0; [ nats ] () = 3; [ cons ] (X0,X1) = 3*X1 + 3*X0; [ a__hd ] (X0) = 3*X0 + 2; [ incr ] (X0) = 1*X0; [ tl ] (X0) = 3*X0 + 2; removing [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ a__adx ] (X0) = 2 + 1*X0 + 0; [ a__incr ] (X0) = 1 + 1*X0 + 0; [ 0 ] () = 0; [ a__tl ] (X0) = 2*X0 + 0; [ a__nats ] () = 2 + 0; [ mark ] (X0) = 2*X0 + 0; [ s ] (X0) = 0; [ Marked_mark ] (X0) = 3*X0 + 0; [ hd ] (X0) = 1 + 2*X0 + 0; [ a__zeros ] () = 0; [ adx ] (X0) = 1 + 1*X0 + 0; [ zeros ] () = 0; [ nats ] () = 2 + 0; [ cons ] (X0,X1) = 2*X0 + 1*X1 + 0; [ a__hd ] (X0) = 1 + 2*X0 + 0; [ incr ] (X0) = 1 + 1*X0 + 0; [ tl ] (X0) = 2*X0 + 0; ]} ]} ]} ]} ]} ]} ]} ]} ]} Cime worked for 0.226143 seconds (real time) Cime Exit Status: 0