/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern in_order(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) PiDPToQDPProof [SOUND, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: Clauses: in_order(void, []). in_order(tree(X, Left, Right), Xs) :- ','(in_order(Left, Ls), ','(in_order(Right, Rs), app(Ls, .(X, Rs), Xs))). app([], X, X). app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs). Query: in_order(g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: in_order_in_2: (b,f) app_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_GA(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_GA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> APP_IN_GGA(Ls, .(X, Rs), Xs) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x3, x5) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x5, x6) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_GA(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_GA(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) U2_GA(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> APP_IN_GGA(Ls, .(X, Rs), Xs) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U4_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x3, x5) U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x5, x6) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4, x5) = U4_GGA(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APP_IN_GGA(x1, x2, x3) = APP_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> IN_ORDER_IN_GA(Right, Rs) IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> U1_GA(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) IN_ORDER_IN_GA(tree(X, Left, Right), Xs) -> IN_ORDER_IN_GA(Left, Ls) The TRS R consists of the following rules: in_order_in_ga(void, []) -> in_order_out_ga(void, []) in_order_in_ga(tree(X, Left, Right), Xs) -> U1_ga(X, Left, Right, Xs, in_order_in_ga(Left, Ls)) U1_ga(X, Left, Right, Xs, in_order_out_ga(Left, Ls)) -> U2_ga(X, Left, Right, Xs, Ls, in_order_in_ga(Right, Rs)) U2_ga(X, Left, Right, Xs, Ls, in_order_out_ga(Right, Rs)) -> U3_ga(X, Left, Right, Xs, app_in_gga(Ls, .(X, Rs), Xs)) app_in_gga([], X, X) -> app_out_gga([], X, X) app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U4_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs)) U4_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs)) U3_ga(X, Left, Right, Xs, app_out_gga(Ls, .(X, Rs), Xs)) -> in_order_out_ga(tree(X, Left, Right), Xs) The argument filtering Pi contains the following mapping: in_order_in_ga(x1, x2) = in_order_in_ga(x1) void = void in_order_out_ga(x1, x2) = in_order_out_ga(x2) tree(x1, x2, x3) = tree(x1, x2, x3) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x3, x5) U2_ga(x1, x2, x3, x4, x5, x6) = U2_ga(x1, x5, x6) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5) app_in_gga(x1, x2, x3) = app_in_gga(x1, x2) [] = [] app_out_gga(x1, x2, x3) = app_out_gga(x3) .(x1, x2) = .(x1, x2) U4_gga(x1, x2, x3, x4, x5) = U4_gga(x1, x5) IN_ORDER_IN_GA(x1, x2) = IN_ORDER_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(X, Right, in_order_out_ga(Ls)) -> IN_ORDER_IN_GA(Right) IN_ORDER_IN_GA(tree(X, Left, Right)) -> U1_GA(X, Right, in_order_in_ga(Left)) IN_ORDER_IN_GA(tree(X, Left, Right)) -> IN_ORDER_IN_GA(Left) The TRS R consists of the following rules: in_order_in_ga(void) -> in_order_out_ga([]) in_order_in_ga(tree(X, Left, Right)) -> U1_ga(X, Right, in_order_in_ga(Left)) U1_ga(X, Right, in_order_out_ga(Ls)) -> U2_ga(X, Ls, in_order_in_ga(Right)) U2_ga(X, Ls, in_order_out_ga(Rs)) -> U3_ga(app_in_gga(Ls, .(X, Rs))) app_in_gga([], X) -> app_out_gga(X) app_in_gga(.(X, Xs), Ys) -> U4_gga(X, app_in_gga(Xs, Ys)) U4_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs)) U3_ga(app_out_gga(Xs)) -> in_order_out_ga(Xs) The set Q consists of the following terms: in_order_in_ga(x0) U1_ga(x0, x1, x2) U2_ga(x0, x1, x2) app_in_gga(x0, x1) U4_gga(x0, x1) U3_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *IN_ORDER_IN_GA(tree(X, Left, Right)) -> U1_GA(X, Right, in_order_in_ga(Left)) The graph contains the following edges 1 > 1, 1 > 2 *IN_ORDER_IN_GA(tree(X, Left, Right)) -> IN_ORDER_IN_GA(Left) The graph contains the following edges 1 > 1 *U1_GA(X, Right, in_order_out_ga(Ls)) -> IN_ORDER_IN_GA(Right) The graph contains the following edges 2 >= 1 ---------------------------------------- (18) YES