- : unit = () h : heuristic = - : unit = () APPLY CRITERIA (Marked dependency pairs) TRS termination of: [1] +(x,0) -> x [2] +(0,x) -> x [3] +(s(x),s(y)) -> s(s(+(x,y))) [4] +(+(x,y),z) -> +(x,+(y,z)) [5] *(x,0) -> 0 [6] *(0,x) -> 0 [7] *(s(x),s(y)) -> s(+( *(x,y),+(x,y))) [8] *( *(x,y),z) -> *(x, *(y,z)) [9] app(nil,l) -> l [10] app(cons(x,l1),l2) -> cons(x,app(l1,l2)) [11] sum(nil) -> 0 [12] sum(cons(x,l)) -> +(x,sum(l)) [13] sum(app(l1,l2)) -> +(sum(l1),sum(l2)) [14] prod(nil) -> s(0) [15] prod(cons(x,l)) -> *(x,prod(l)) [16] prod(app(l1,l2)) -> *(prod(l1),prod(l2)) Sub problem: guided: DP termination of: END GUIDED APPLY CRITERIA (Graph splitting) Found 5 components: { --> } { --> --> --> --> --> --> --> --> --> } { --> --> --> --> --> --> --> --> --> } { --> --> --> --> --> --> --> --> --> } { --> --> --> --> --> --> --> --> --> } APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { +(+(x,y),z) >= +(x,+(y,z)) ; +(0,x) >= x ; +(s(x),s(y)) >= s(s(+(x,y))) ; +(x,0) >= x ; *(0,x) >= 0 ; *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) ; *( *(x,y),z) >= *(x, *(y,z)) ; *(x,0) >= 0 ; app(nil,l) >= l ; app(cons(x,l1),l2) >= cons(x,app(l1,l2)) ; sum(app(l1,l2)) >= +(sum(l1),sum(l2)) ; sum(nil) >= 0 ; sum(cons(x,l)) >= +(x,sum(l)) ; prod(app(l1,l2)) >= *(prod(l1),prod(l2)) ; prod(nil) >= s(0) ; prod(cons(x,l)) >= *(x,prod(l)) ; Marked_app(cons(x,l1),l2) >= Marked_app(l1,l2) ; } + Disjunctions:{ { Marked_app(cons(x,l1),l2) > Marked_app(l1,l2) ; } } === 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: +(+(x,y),z) >= +(x,+(y,z)) constraint: +(0,x) >= x constraint: +(s(x),s(y)) >= s(s(+(x,y))) constraint: +(x,0) >= x constraint: *(0,x) >= 0 constraint: *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) constraint: *( *(x,y),z) >= *(x, *(y,z)) constraint: *(x,0) >= 0 constraint: app(nil,l) >= l constraint: app(cons(x,l1),l2) >= cons(x,app(l1,l2)) constraint: sum(app(l1,l2)) >= +(sum(l1),sum(l2)) constraint: sum(nil) >= 0 constraint: sum(cons(x,l)) >= +(x,sum(l)) constraint: prod(app(l1,l2)) >= *(prod(l1),prod(l2)) constraint: prod(nil) >= s(0) constraint: prod(cons(x,l)) >= *(x,prod(l)) constraint: Marked_app(cons(x,l1),l2) >= Marked_app(l1,l2) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { +(+(x,y),z) >= +(x,+(y,z)) ; +(0,x) >= x ; +(s(x),s(y)) >= s(s(+(x,y))) ; +(x,0) >= x ; *(0,x) >= 0 ; *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) ; *( *(x,y),z) >= *(x, *(y,z)) ; *(x,0) >= 0 ; app(nil,l) >= l ; app(cons(x,l1),l2) >= cons(x,app(l1,l2)) ; sum(app(l1,l2)) >= +(sum(l1),sum(l2)) ; sum(nil) >= 0 ; sum(cons(x,l)) >= +(x,sum(l)) ; prod(app(l1,l2)) >= *(prod(l1),prod(l2)) ; prod(nil) >= s(0) ; prod(cons(x,l)) >= *(x,prod(l)) ; Marked_sum(app(l1,l2)) >= Marked_sum(l2) ; Marked_sum(app(l1,l2)) >= Marked_sum(l1) ; Marked_sum(cons(x,l)) >= Marked_sum(l) ; } + Disjunctions:{ { Marked_sum(app(l1,l2)) > Marked_sum(l2) ; } { Marked_sum(app(l1,l2)) > Marked_sum(l1) ; } { Marked_sum(cons(x,l)) > Marked_sum(l) ; } } === 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: +(+(x,y),z) >= +(x,+(y,z)) constraint: +(0,x) >= x constraint: +(s(x),s(y)) >= s(s(+(x,y))) constraint: +(x,0) >= x constraint: *(0,x) >= 0 constraint: *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) constraint: *( *(x,y),z) >= *(x, *(y,z)) constraint: *(x,0) >= 0 constraint: app(nil,l) >= l constraint: app(cons(x,l1),l2) >= cons(x,app(l1,l2)) constraint: sum(app(l1,l2)) >= +(sum(l1),sum(l2)) constraint: sum(nil) >= 0 constraint: sum(cons(x,l)) >= +(x,sum(l)) constraint: prod(app(l1,l2)) >= *(prod(l1),prod(l2)) constraint: prod(nil) >= s(0) constraint: prod(cons(x,l)) >= *(x,prod(l)) constraint: Marked_sum(app(l1,l2)) >= Marked_sum(l2) constraint: Marked_sum(app(l1,l2)) >= Marked_sum(l1) constraint: Marked_sum(cons(x,l)) >= Marked_sum(l) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { +(+(x,y),z) >= +(x,+(y,z)) ; +(0,x) >= x ; +(s(x),s(y)) >= s(s(+(x,y))) ; +(x,0) >= x ; *(0,x) >= 0 ; *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) ; *( *(x,y),z) >= *(x, *(y,z)) ; *(x,0) >= 0 ; app(nil,l) >= l ; app(cons(x,l1),l2) >= cons(x,app(l1,l2)) ; sum(app(l1,l2)) >= +(sum(l1),sum(l2)) ; sum(nil) >= 0 ; sum(cons(x,l)) >= +(x,sum(l)) ; prod(app(l1,l2)) >= *(prod(l1),prod(l2)) ; prod(nil) >= s(0) ; prod(cons(x,l)) >= *(x,prod(l)) ; Marked_prod(app(l1,l2)) >= Marked_prod(l2) ; Marked_prod(app(l1,l2)) >= Marked_prod(l1) ; Marked_prod(cons(x,l)) >= Marked_prod(l) ; } + Disjunctions:{ { Marked_prod(app(l1,l2)) > Marked_prod(l2) ; } { Marked_prod(app(l1,l2)) > Marked_prod(l1) ; } { Marked_prod(cons(x,l)) > Marked_prod(l) ; } } === 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: +(+(x,y),z) >= +(x,+(y,z)) constraint: +(0,x) >= x constraint: +(s(x),s(y)) >= s(s(+(x,y))) constraint: +(x,0) >= x constraint: *(0,x) >= 0 constraint: *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) constraint: *( *(x,y),z) >= *(x, *(y,z)) constraint: *(x,0) >= 0 constraint: app(nil,l) >= l constraint: app(cons(x,l1),l2) >= cons(x,app(l1,l2)) constraint: sum(app(l1,l2)) >= +(sum(l1),sum(l2)) constraint: sum(nil) >= 0 constraint: sum(cons(x,l)) >= +(x,sum(l)) constraint: prod(app(l1,l2)) >= *(prod(l1),prod(l2)) constraint: prod(nil) >= s(0) constraint: prod(cons(x,l)) >= *(x,prod(l)) constraint: Marked_prod(app(l1,l2)) >= Marked_prod(l2) constraint: Marked_prod(app(l1,l2)) >= Marked_prod(l1) constraint: Marked_prod(cons(x,l)) >= Marked_prod(l) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { +(+(x,y),z) >= +(x,+(y,z)) ; +(0,x) >= x ; +(s(x),s(y)) >= s(s(+(x,y))) ; +(x,0) >= x ; *(0,x) >= 0 ; *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) ; *( *(x,y),z) >= *(x, *(y,z)) ; *(x,0) >= 0 ; app(nil,l) >= l ; app(cons(x,l1),l2) >= cons(x,app(l1,l2)) ; sum(app(l1,l2)) >= +(sum(l1),sum(l2)) ; sum(nil) >= 0 ; sum(cons(x,l)) >= +(x,sum(l)) ; prod(app(l1,l2)) >= *(prod(l1),prod(l2)) ; prod(nil) >= s(0) ; prod(cons(x,l)) >= *(x,prod(l)) ; Marked_*(s(x),s(y)) >= Marked_*(x,y) ; Marked_*( *(x,y),z) >= Marked_*(y,z) ; Marked_*( *(x,y),z) >= Marked_*(x, *(y,z)) ; } + Disjunctions:{ { Marked_*(s(x),s(y)) > Marked_*(x,y) ; } { Marked_*( *(x,y),z) > Marked_*(y,z) ; } { Marked_*( *(x,y),z) > Marked_*(x, *(y,z)) ; } } === 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 === === STOPING TIMER virtual === constraint: +(+(x,y),z) >= +(x,+(y,z)) constraint: +(0,x) >= x constraint: +(s(x),s(y)) >= s(s(+(x,y))) constraint: +(x,0) >= x constraint: *(0,x) >= 0 constraint: *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) constraint: *( *(x,y),z) >= *(x, *(y,z)) constraint: *(x,0) >= 0 constraint: app(nil,l) >= l constraint: app(cons(x,l1),l2) >= cons(x,app(l1,l2)) constraint: sum(app(l1,l2)) >= +(sum(l1),sum(l2)) constraint: sum(nil) >= 0 constraint: sum(cons(x,l)) >= +(x,sum(l)) constraint: prod(app(l1,l2)) >= *(prod(l1),prod(l2)) constraint: prod(nil) >= s(0) constraint: prod(cons(x,l)) >= *(x,prod(l)) constraint: Marked_*(s(x),s(y)) >= Marked_*(x,y) constraint: Marked_*( *(x,y),z) >= Marked_*(y,z) constraint: Marked_*( *(x,y),z) >= Marked_*(x, *(y,z)) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { +(+(x,y),z) >= +(x,+(y,z)) ; +(0,x) >= x ; +(s(x),s(y)) >= s(s(+(x,y))) ; +(x,0) >= x ; *(0,x) >= 0 ; *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) ; *( *(x,y),z) >= *(x, *(y,z)) ; *(x,0) >= 0 ; app(nil,l) >= l ; app(cons(x,l1),l2) >= cons(x,app(l1,l2)) ; sum(app(l1,l2)) >= +(sum(l1),sum(l2)) ; sum(nil) >= 0 ; sum(cons(x,l)) >= +(x,sum(l)) ; prod(app(l1,l2)) >= *(prod(l1),prod(l2)) ; prod(nil) >= s(0) ; prod(cons(x,l)) >= *(x,prod(l)) ; Marked_+(+(x,y),z) >= Marked_+(y,z) ; Marked_+(+(x,y),z) >= Marked_+(x,+(y,z)) ; Marked_+(s(x),s(y)) >= Marked_+(x,y) ; } + Disjunctions:{ { Marked_+(+(x,y),z) > Marked_+(y,z) ; } { Marked_+(+(x,y),z) > Marked_+(x,+(y,z)) ; } { Marked_+(s(x),s(y)) > Marked_+(x,y) ; } } === 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 === === STOPING TIMER virtual === constraint: +(+(x,y),z) >= +(x,+(y,z)) constraint: +(0,x) >= x constraint: +(s(x),s(y)) >= s(s(+(x,y))) constraint: +(x,0) >= x constraint: *(0,x) >= 0 constraint: *(s(x),s(y)) >= s(+( *(x,y),+(x,y))) constraint: *( *(x,y),z) >= *(x, *(y,z)) constraint: *(x,0) >= 0 constraint: app(nil,l) >= l constraint: app(cons(x,l1),l2) >= cons(x,app(l1,l2)) constraint: sum(app(l1,l2)) >= +(sum(l1),sum(l2)) constraint: sum(nil) >= 0 constraint: sum(cons(x,l)) >= +(x,sum(l)) constraint: prod(app(l1,l2)) >= *(prod(l1),prod(l2)) constraint: prod(nil) >= s(0) constraint: prod(cons(x,l)) >= *(x,prod(l)) constraint: Marked_+(+(x,y),z) >= Marked_+(y,z) constraint: Marked_+(+(x,y),z) >= Marked_+(x,+(y,z)) constraint: Marked_+(s(x),s(y)) >= Marked_+(x,y) APPLY CRITERIA (Graph splitting) Found 0 components: APPLY CRITERIA (Graph splitting) Found 0 components: APPLY CRITERIA (Graph splitting) Found 0 components: APPLY CRITERIA (Graph splitting) Found 0 components: APPLY CRITERIA (Graph splitting) Found 0 components: SOLVED { TRS termination of: [1] +(x,0) -> x [2] +(0,x) -> x [3] +(s(x),s(y)) -> s(s(+(x,y))) [4] +(+(x,y),z) -> +(x,+(y,z)) [5] *(x,0) -> 0 [6] *(0,x) -> 0 [7] *(s(x),s(y)) -> s(+( *(x,y),+(x,y))) [8] *( *(x,y),z) -> *(x, *(y,z)) [9] app(nil,l) -> l [10] app(cons(x,l1),l2) -> cons(x,app(l1,l2)) [11] sum(nil) -> 0 [12] sum(cons(x,l)) -> +(x,sum(l)) [13] sum(app(l1,l2)) -> +(sum(l1),sum(l2)) [14] prod(nil) -> s(0) [15] prod(cons(x,l)) -> *(x,prod(l)) [16] prod(app(l1,l2)) -> *(prod(l1),prod(l2)) , CRITERION: MDP [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ + ] (X0,X1) = 2*X0 + 1*X1 + 0; [ prod ] (X0) = 2 + 0; [ app ] (X0,X1) = 2*X0 + 2*X1 + 0; [ s ] (X0) = 0; [ cons ] (X0,X1) = 2 + 3*X0 + 1*X1 + 0; [ 0 ] () = 0; [ nil ] () = 0; [ Marked_app ] (X0,X1) = 3*X0 + 0; [ * ] (X0,X1) = 0; [ sum ] (X0) = 3*X0 + 0; ]} { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ + ] (X0,X1) = 1*X0 + 1*X1 + 0; [ prod ] (X0) = 2 + 2*X0 + 0; [ app ] (X0,X1) = 2 + 3*X0 + 1*X1 + 0; [ s ] (X0) = 0; [ cons ] (X0,X1) = 2 + 1*X0 + 1*X1 + 0; [ 0 ] () = 0; [ nil ] () = 2 + 0; [ * ] (X0,X1) = 0; [ Marked_sum ] (X0) = 3*X0 + 0; [ sum ] (X0) = 3*X0 + 0; ]} { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ + ] (X0,X1) = 1*X0 + 1*X1 + 0; [ prod ] (X0) = 2 + 2*X0 + 0; [ app ] (X0,X1) = 2 + 3*X0 + 1*X1 + 0; [ s ] (X0) = 0; [ cons ] (X0,X1) = 2 + 1*X0 + 1*X1 + 0; [ 0 ] () = 0; [ Marked_prod ] (X0) = 3*X0 + 0; [ nil ] () = 2 + 0; [ * ] (X0,X1) = 0; [ sum ] (X0) = 3*X0 + 0; ]} { DP termination of: , CRITERION: ORD [ Solution found: RPO with AFS = AFS: and precedence: prec (All symbols are Lex.): { + > s ; + < * ; + < sum ; + < prod ; + < Marked_* ; 0 < sum ; 0 < prod ; s < + ; s < * ; s < sum ; s < prod ; s < Marked_* ; * > + ; * > s ; * < prod ; * < Marked_* ; app > cons ; cons < app ; sum > + ; sum > 0 ; sum > s ; prod > + ; prod > 0 ; prod > s ; prod > * ; Marked_* > + ; Marked_* > s ; Marked_* > * ; } ]} { DP termination of: , CRITERION: ORD [ Solution found: RPO with AFS = AFS: and precedence: prec (All symbols are Lex.): { + > s ; + < * ; + < sum ; + < prod ; + < Marked_+ ; 0 < sum ; 0 < prod ; s < + ; s < * ; s < sum ; s < prod ; s < Marked_+ ; * > + ; * > s ; * < prod ; app > cons ; cons < app ; sum > + ; sum > 0 ; sum > s ; prod > + ; prod > 0 ; prod > s ; prod > * ; Marked_+ > + ; Marked_+ > s ; } ]} ]} ]} Cime worked for 0.349029 seconds (real time) Cime Exit Status: 0