5.04/2.12 YES 5.12/2.14 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.12/2.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.12/2.14 5.12/2.14 5.12/2.14 Left Termination of the query pattern 5.12/2.14 5.12/2.14 ordered(g) 5.12/2.14 5.12/2.14 w.r.t. the given Prolog program could successfully be proven: 5.12/2.14 5.12/2.14 (0) Prolog 5.12/2.14 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.12/2.14 (2) PiTRS 5.12/2.14 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.12/2.14 (4) PiDP 5.12/2.14 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.12/2.14 (6) AND 5.12/2.14 (7) PiDP 5.12/2.14 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.12/2.14 (9) PiDP 5.12/2.14 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.12/2.14 (11) QDP 5.12/2.14 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.12/2.14 (13) YES 5.12/2.14 (14) PiDP 5.12/2.14 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.12/2.14 (16) PiDP 5.12/2.14 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.12/2.14 (18) QDP 5.12/2.14 (19) MRRProof [EQUIVALENT, 0 ms] 5.12/2.14 (20) QDP 5.12/2.14 (21) PisEmptyProof [EQUIVALENT, 0 ms] 5.12/2.14 (22) YES 5.12/2.14 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (0) 5.12/2.14 Obligation: 5.12/2.14 Clauses: 5.12/2.14 5.12/2.14 ordered([]). 5.12/2.14 ordered(.(X1, [])). 5.12/2.14 ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))). 5.12/2.14 less(0, s(X2)). 5.12/2.14 less(s(X), s(Y)) :- less(X, Y). 5.12/2.14 5.12/2.14 5.12/2.14 Query: ordered(g) 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (1) PrologToPiTRSProof (SOUND) 5.12/2.14 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.12/2.14 5.12/2.14 ordered_in_1: (b) 5.12/2.14 5.12/2.14 less_in_2: (b,b) 5.12/2.14 5.12/2.14 Transforming Prolog into the following Term Rewriting System: 5.12/2.14 5.12/2.14 Pi-finite rewrite system: 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 5.12/2.14 5.12/2.14 5.12/2.14 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.12/2.14 5.12/2.14 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (2) 5.12/2.14 Obligation: 5.12/2.14 Pi-finite rewrite system: 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (3) DependencyPairsProof (EQUIVALENT) 5.12/2.14 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y)) 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> U3_GG(X, Y, less_in_gg(X, Y)) 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_G(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (4) 5.12/2.14 Obligation: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> LESS_IN_GG(X, s(Y)) 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> U3_GG(X, Y, less_in_gg(X, Y)) 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_G(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (5) DependencyGraphProof (EQUIVALENT) 5.12/2.14 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (6) 5.12/2.14 Complex Obligation (AND) 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (7) 5.12/2.14 Obligation: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (8) UsableRulesProof (EQUIVALENT) 5.12/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (9) 5.12/2.14 Obligation: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 5.12/2.14 R is empty. 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (10) PiDPToQDPProof (EQUIVALENT) 5.12/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (11) 5.12/2.14 Obligation: 5.12/2.14 Q DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 5.12/2.14 R is empty. 5.12/2.14 Q is empty. 5.12/2.14 We have to consider all (P,Q,R)-chains. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (12) QDPSizeChangeProof (EQUIVALENT) 5.12/2.14 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. 5.12/2.14 5.12/2.14 From the DPs we obtained the following set of size-change graphs: 5.12/2.14 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.12/2.14 The graph contains the following edges 1 > 1, 2 > 2 5.12/2.14 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (13) 5.12/2.14 YES 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (14) 5.12/2.14 Obligation: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 ordered_in_g([]) -> ordered_out_g([]) 5.12/2.14 ordered_in_g(.(X1, [])) -> ordered_out_g(.(X1, [])) 5.12/2.14 ordered_in_g(.(X, .(Y, Xs))) -> U1_g(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 U1_g(X, Y, Xs, less_out_gg(X, s(Y))) -> U2_g(X, Y, Xs, ordered_in_g(.(Y, Xs))) 5.12/2.14 U2_g(X, Y, Xs, ordered_out_g(.(Y, Xs))) -> ordered_out_g(.(X, .(Y, Xs))) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (15) UsableRulesProof (EQUIVALENT) 5.12/2.14 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (16) 5.12/2.14 Obligation: 5.12/2.14 Pi DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 5.12/2.14 Pi is empty. 5.12/2.14 We have to consider all (P,R,Pi)-chains 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (17) PiDPToQDPProof (EQUIVALENT) 5.12/2.14 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (18) 5.12/2.14 Obligation: 5.12/2.14 Q DP problem: 5.12/2.14 The TRS P consists of the following rules: 5.12/2.14 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 5.12/2.14 The set Q consists of the following terms: 5.12/2.14 5.12/2.14 less_in_gg(x0, x1) 5.12/2.14 U3_gg(x0, x1, x2) 5.12/2.14 5.12/2.14 We have to consider all (P,Q,R)-chains. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (19) MRRProof (EQUIVALENT) 5.12/2.14 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 5.12/2.14 5.12/2.14 Strictly oriented dependency pairs: 5.12/2.14 5.12/2.14 U1_G(X, Y, Xs, less_out_gg(X, s(Y))) -> ORDERED_IN_G(.(Y, Xs)) 5.12/2.14 ORDERED_IN_G(.(X, .(Y, Xs))) -> U1_G(X, Y, Xs, less_in_gg(X, s(Y))) 5.12/2.14 5.12/2.14 5.12/2.14 Used ordering: Polynomial interpretation [POLO]: 5.12/2.14 5.12/2.14 POL(.(x_1, x_2)) = 2*x_1 + x_2 5.12/2.14 POL(0) = 2 5.12/2.14 POL(ORDERED_IN_G(x_1)) = 1 + 2*x_1 5.12/2.14 POL(U1_G(x_1, x_2, x_3, x_4)) = 2*x_1 + 2*x_2 + 2*x_3 + x_4 5.12/2.14 POL(U3_gg(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3 5.12/2.14 POL(less_in_gg(x_1, x_2)) = 2*x_1 + x_2 5.12/2.14 POL(less_out_gg(x_1, x_2)) = 2 + x_1 + x_2 5.12/2.14 POL(s(x_1)) = 2*x_1 5.12/2.14 5.12/2.14 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (20) 5.12/2.14 Obligation: 5.12/2.14 Q DP problem: 5.12/2.14 P is empty. 5.12/2.14 The TRS R consists of the following rules: 5.12/2.14 5.12/2.14 less_in_gg(0, s(X2)) -> less_out_gg(0, s(X2)) 5.12/2.14 less_in_gg(s(X), s(Y)) -> U3_gg(X, Y, less_in_gg(X, Y)) 5.12/2.14 U3_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.12/2.14 5.12/2.14 The set Q consists of the following terms: 5.12/2.14 5.12/2.14 less_in_gg(x0, x1) 5.12/2.14 U3_gg(x0, x1, x2) 5.12/2.14 5.12/2.14 We have to consider all (P,Q,R)-chains. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (21) PisEmptyProof (EQUIVALENT) 5.12/2.14 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.12/2.14 ---------------------------------------- 5.12/2.14 5.12/2.14 (22) 5.12/2.14 YES 5.12/2.17 EOF