- : unit = () - : unit = () h : heuristic = - : unit = () APPLY CRITERIA (Marked dependency pairs) TRS termination of: [1] minus(x,0) -> x [2] minus(s(x),s(y)) -> minus(x,y) [3] quot(0,s(y)) -> 0 [4] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) [5] plus(0,y) -> y [6] plus(s(x),y) -> s(plus(x,y)) [7] minus(minus(x,y),z) -> minus(x,plus(y,z)) [8] app(nil,k) -> k [9] app(l,nil) -> l [10] app(cons(x,l),k) -> cons(x,app(l,k)) [11] sum(cons(x,nil)) -> cons(x,nil) [12] sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) [13] sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) [14] plus(0,y) -> y [15] plus(s(x),y) -> s(plus(x,y)) Sub problem: guided: DP termination of: END GUIDED APPLY CRITERIA (Graph splitting) Found 6 components: { --> } { --> --> --> --> } { --> } { --> } { --> } { --> --> --> --> } APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { minus(minus(x,y),z) >= minus(x,plus(y,z)) ; minus(s(x),s(y)) >= minus(x,y) ; minus(x,0) >= x ; quot(0,s(y)) >= 0 ; quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ; plus(0,y) >= y ; plus(s(x),y) >= s(plus(x,y)) ; app(nil,k) >= k ; app(cons(x,l),k) >= cons(x,app(l,k)) ; app(l,nil) >= l ; sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ; sum(cons(x,nil)) >= cons(x,nil) ; sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ; Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y)) ; } + Disjunctions:{ { Marked_quot(s(x),s(y)) > Marked_quot(minus(x,y),s(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 Sat solver result read === STOPING TIMER real === === STOPING TIMER virtual === constraint: minus(minus(x,y),z) >= minus(x,plus(y,z)) constraint: minus(s(x),s(y)) >= minus(x,y) constraint: minus(x,0) >= x constraint: quot(0,s(y)) >= 0 constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) constraint: plus(0,y) >= y constraint: plus(s(x),y) >= s(plus(x,y)) constraint: app(nil,k) >= k constraint: app(cons(x,l),k) >= cons(x,app(l,k)) constraint: app(l,nil) >= l constraint: sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) constraint: sum(cons(x,nil)) >= cons(x,nil) constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) constraint: Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y)) APPLY CRITERIA (Subterm criterion) ST: Marked_minus -> 1 APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { minus(minus(x,y),z) >= minus(x,plus(y,z)) ; minus(s(x),s(y)) >= minus(x,y) ; minus(x,0) >= x ; quot(0,s(y)) >= 0 ; quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ; plus(0,y) >= y ; plus(s(x),y) >= s(plus(x,y)) ; app(nil,k) >= k ; app(cons(x,l),k) >= cons(x,app(l,k)) ; app(l,nil) >= l ; sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ; sum(cons(x,nil)) >= cons(x,nil) ; sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ; Marked_sum(app(l,cons(x,cons(y,k)))) >= Marked_sum(app(l, sum(cons(x,cons(y,k))))) ; } + Disjunctions:{ { Marked_sum(app(l,cons(x,cons(y,k)))) > Marked_sum(app(l, sum(cons(x,cons(y,k))))) ; } } === 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: minus(minus(x,y),z) >= minus(x,plus(y,z)) constraint: minus(s(x),s(y)) >= minus(x,y) constraint: minus(x,0) >= x constraint: quot(0,s(y)) >= 0 constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) constraint: plus(0,y) >= y constraint: plus(s(x),y) >= s(plus(x,y)) constraint: app(nil,k) >= k constraint: app(cons(x,l),k) >= cons(x,app(l,k)) constraint: app(l,nil) >= l constraint: sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) constraint: sum(cons(x,nil)) >= cons(x,nil) constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) constraint: Marked_sum(app(l,cons(x,cons(y,k)))) >= Marked_sum(app(l, sum(cons( x, cons(y,k))))) APPLY CRITERIA (Subterm criterion) ST: Marked_app -> 1 APPLY CRITERIA (Subterm criterion) APPLY CRITERIA (Choosing graph) Trying to solve the following constraints: { minus(minus(x,y),z) >= minus(x,plus(y,z)) ; minus(s(x),s(y)) >= minus(x,y) ; minus(x,0) >= x ; quot(0,s(y)) >= 0 ; quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ; plus(0,y) >= y ; plus(s(x),y) >= s(plus(x,y)) ; app(nil,k) >= k ; app(cons(x,l),k) >= cons(x,app(l,k)) ; app(l,nil) >= l ; sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ; sum(cons(x,nil)) >= cons(x,nil) ; sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ; Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(plus(x,y),l)) ; } + Disjunctions:{ { Marked_sum(cons(x,cons(y,l))) > Marked_sum(cons(plus(x,y),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: minus(minus(x,y),z) >= minus(x,plus(y,z)) constraint: minus(s(x),s(y)) >= minus(x,y) constraint: minus(x,0) >= x constraint: quot(0,s(y)) >= 0 constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) constraint: plus(0,y) >= y constraint: plus(s(x),y) >= s(plus(x,y)) constraint: app(nil,k) >= k constraint: app(cons(x,l),k) >= cons(x,app(l,k)) constraint: app(l,nil) >= l constraint: sum(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) constraint: sum(cons(x,nil)) >= cons(x,nil) constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) constraint: Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(plus(x,y),l)) APPLY CRITERIA (Subterm criterion) ST: Marked_plus -> 1 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: APPLY CRITERIA (Graph splitting) Found 0 components: SOLVED { TRS termination of: [1] minus(x,0) -> x [2] minus(s(x),s(y)) -> minus(x,y) [3] quot(0,s(y)) -> 0 [4] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) [5] plus(0,y) -> y [6] plus(s(x),y) -> s(plus(x,y)) [7] minus(minus(x,y),z) -> minus(x,plus(y,z)) [8] app(nil,k) -> k [9] app(l,nil) -> l [10] app(cons(x,l),k) -> cons(x,app(l,k)) [11] sum(cons(x,nil)) -> cons(x,nil) [12] sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l)) [13] sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k))))) [14] plus(0,y) -> y [15] plus(s(x),y) -> s(plus(x,y)) , CRITERION: MDP [ { DP termination of: , CRITERION: SG [ { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ minus ] (X0,X1) = 1*X0 + 0; [ sum ] (X0) = 2 + 0; [ plus ] (X0,X1) = 2*X0 + 1*X1 + 0; [ s ] (X0) = 2 + 1*X0 + 0; [ nil ] () = 0; [ 0 ] () = 0; [ app ] (X0,X1) = 2*X0 + 1*X1 + 0; [ Marked_quot ] (X0,X1) = 2*X0 + 0; [ quot ] (X0,X1) = 1*X0 + 0; [ cons ] (X0,X1) = 2 + 0; ]} { DP termination of: , CRITERION: ST [ { DP termination of: , CRITERION: SG [ ]} ]} { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ minus ] (X0,X1) = 2*X0 + 0; [ sum ] (X0) = 2 + 0; [ plus ] (X0,X1) = 2*X1 + 0; [ s ] (X0) = 1*X0 + 0; [ Marked_sum ] (X0) = 2*X0 + 0; [ nil ] () = 0; [ 0 ] () = 0; [ app ] (X0,X1) = 2*X0 + 2*X1 + 0; [ quot ] (X0,X1) = 0; [ cons ] (X0,X1) = 2 + 1*X1 + 0; ]} { DP termination of: , CRITERION: ST [ { DP termination of: , CRITERION: SG [ ]} ]} { DP termination of: , CRITERION: ORD [ Solution found: polynomial interpretation = [ minus ] (X0,X1) = 1*X0 + 2*X1 + 0; [ sum ] (X0) = 2 + 0; [ plus ] (X0,X1) = 1*X1 + 0; [ s ] (X0) = 1*X0 + 0; [ Marked_sum ] (X0) = 1*X0 + 0; [ nil ] () = 0; [ 0 ] () = 0; [ app ] (X0,X1) = 2*X0 + 2*X1 + 0; [ quot ] (X0,X1) = 0; [ cons ] (X0,X1) = 2 + 1*X1 + 0; ]} { DP termination of: , CRITERION: ST [ { DP termination of: , CRITERION: SG [ ]} ]} ]} ]} Cime worked for 0.352342 seconds (real time) Cime Exit Status: 0