/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern subset(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) NonTerminationLoopProof [COMPLETE, 0 ms] (13) NO (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PrologToPiTRSProof [SOUND, 0 ms] (22) PiTRS (23) DependencyPairsProof [EQUIVALENT, 0 ms] (24) PiDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) AND (27) PiDP (28) UsableRulesProof [EQUIVALENT, 0 ms] (29) PiDP (30) PiDPToQDPProof [SOUND, 0 ms] (31) QDP (32) NonTerminationLoopProof [COMPLETE, 0 ms] (33) NO (34) PiDP (35) UsableRulesProof [EQUIVALENT, 0 ms] (36) PiDP (37) PiDPToQDPProof [SOUND, 0 ms] (38) QDP (39) QDPSizeChangeProof [EQUIVALENT, 0 ms] (40) YES (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) DependencyPairsProof [EQUIVALENT, 0 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 0 ms] (46) AND (47) QDP (48) MNOCProof [EQUIVALENT, 0 ms] (49) QDP (50) UsableRulesProof [EQUIVALENT, 0 ms] (51) QDP (52) QReductionProof [EQUIVALENT, 0 ms] (53) QDP (54) NonTerminationLoopProof [COMPLETE, 0 ms] (55) NO (56) QDP (57) MNOCProof [EQUIVALENT, 0 ms] (58) QDP (59) UsableRulesProof [EQUIVALENT, 0 ms] (60) QDP (61) QReductionProof [EQUIVALENT, 0 ms] (62) QDP (63) QDPSizeChangeProof [EQUIVALENT, 0 ms] (64) YES (65) PrologToIRSwTTransformerProof [SOUND, 0 ms] (66) AND (67) IRSwT (68) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 2 ms] (69) IRSwT (70) IntTRSCompressionProof [EQUIVALENT, 24 ms] (71) IRSwT (72) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (73) IRSwT (74) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] (75) IRSwT (76) FilterProof [EQUIVALENT, 0 ms] (77) IntTRS (78) IntTRSNonPeriodicNontermProof [COMPLETE, 6 ms] (79) NO (80) IRSwT (81) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (82) IRSwT (83) IntTRSCompressionProof [EQUIVALENT, 8 ms] (84) IRSwT (85) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (86) IRSwT (87) PrologToDTProblemTransformerProof [SOUND, 0 ms] (88) TRIPLES (89) TriplesToPiDPProof [SOUND, 5 ms] (90) PiDP (91) DependencyGraphProof [EQUIVALENT, 0 ms] (92) AND (93) PiDP (94) UsableRulesProof [EQUIVALENT, 0 ms] (95) PiDP (96) PiDPToQDPProof [SOUND, 0 ms] (97) QDP (98) PiDP ---------------------------------------- (0) Obligation: Clauses: subset([], X1). subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)). member(X, .(X, X2)). member(X, .(X3, Xs)) :- member(X, Xs). Query: subset(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: subset_in_2: (b,f) member_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) ---------------------------------------- (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: SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) SUBSET_IN_GA(.(X, Xs), Ys) -> MEMBER_IN_GA(X, Ys) MEMBER_IN_GA(X, .(X3, Xs)) -> U3_GA(X, X3, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys)) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) SUBSET_IN_GA(.(X, Xs), Ys) -> MEMBER_IN_GA(X, Ys) MEMBER_IN_GA(X, .(X3, Xs)) -> U3_GA(X, X3, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys)) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x1, x4) U2_GA(x1, x2, x3, x4) = U2_GA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) 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: MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) 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: MEMBER_IN_GA(X) -> MEMBER_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = MEMBER_IN_GA(X) evaluates to t =MEMBER_IN_GA(X) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from MEMBER_IN_GA(X) to MEMBER_IN_GA(X). ---------------------------------------- (13) NO ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga(x1) .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) The TRS R consists of the following rules: member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga(x1) U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, member_out_ga(X)) -> SUBSET_IN_GA(Xs) SUBSET_IN_GA(.(X, Xs)) -> U1_GA(X, Xs, member_in_ga(X)) The TRS R consists of the following rules: member_in_ga(X) -> member_out_ga(X) member_in_ga(X) -> U3_ga(X, member_in_ga(X)) U3_ga(X, member_out_ga(X)) -> member_out_ga(X) The set Q consists of the following terms: member_in_ga(x0) U3_ga(x0, x1) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: *SUBSET_IN_GA(.(X, Xs)) -> U1_GA(X, Xs, member_in_ga(X)) The graph contains the following edges 1 > 1, 1 > 2 *U1_GA(X, Xs, member_out_ga(X)) -> SUBSET_IN_GA(Xs) The graph contains the following edges 2 >= 1 ---------------------------------------- (20) YES ---------------------------------------- (21) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: subset_in_2: (b,f) member_in_2: (b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (22) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) ---------------------------------------- (23) 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: SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) SUBSET_IN_GA(.(X, Xs), Ys) -> MEMBER_IN_GA(X, Ys) MEMBER_IN_GA(X, .(X3, Xs)) -> U3_GA(X, X3, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys)) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U2_GA(x1, x2, x3, x4) = U2_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) SUBSET_IN_GA(.(X, Xs), Ys) -> MEMBER_IN_GA(X, Ys) MEMBER_IN_GA(X, .(X3, Xs)) -> U3_GA(X, X3, Xs, member_in_ga(X, Xs)) MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_GA(X, Xs, Ys, subset_in_ga(Xs, Ys)) U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) U3_GA(x1, x2, x3, x4) = U3_GA(x4) U2_GA(x1, x2, x3, x4) = U2_GA(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (26) Complex Obligation (AND) ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (29) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X, .(X3, Xs)) -> MEMBER_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_GA(x1, x2) = MEMBER_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (30) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_GA(X) -> MEMBER_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (32) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = MEMBER_IN_GA(X) evaluates to t =MEMBER_IN_GA(X) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from MEMBER_IN_GA(X) to MEMBER_IN_GA(X). ---------------------------------------- (33) NO ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) The TRS R consists of the following rules: subset_in_ga([], X1) -> subset_out_ga([], X1) subset_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, member_in_ga(X, Ys)) member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) U1_ga(X, Xs, Ys, member_out_ga(X, Ys)) -> U2_ga(X, Xs, Ys, subset_in_ga(Xs, Ys)) U2_ga(X, Xs, Ys, subset_out_ga(Xs, Ys)) -> subset_out_ga(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ga(x1, x2) = subset_in_ga(x1) [] = [] subset_out_ga(x1, x2) = subset_out_ga .(x1, x2) = .(x1, x2) U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) U2_ga(x1, x2, x3, x4) = U2_ga(x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Xs, Ys, member_out_ga(X, Ys)) -> SUBSET_IN_GA(Xs, Ys) SUBSET_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, member_in_ga(X, Ys)) The TRS R consists of the following rules: member_in_ga(X, .(X, X2)) -> member_out_ga(X, .(X, X2)) member_in_ga(X, .(X3, Xs)) -> U3_ga(X, X3, Xs, member_in_ga(X, Xs)) U3_ga(X, X3, Xs, member_out_ga(X, Xs)) -> member_out_ga(X, .(X3, Xs)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member_in_ga(x1, x2) = member_in_ga(x1) member_out_ga(x1, x2) = member_out_ga U3_ga(x1, x2, x3, x4) = U3_ga(x4) SUBSET_IN_GA(x1, x2) = SUBSET_IN_GA(x1) U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GA(Xs, member_out_ga) -> SUBSET_IN_GA(Xs) SUBSET_IN_GA(.(X, Xs)) -> U1_GA(Xs, member_in_ga(X)) The TRS R consists of the following rules: member_in_ga(X) -> member_out_ga member_in_ga(X) -> U3_ga(member_in_ga(X)) U3_ga(member_out_ga) -> member_out_ga The set Q consists of the following terms: member_in_ga(x0) U3_ga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) 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: *SUBSET_IN_GA(.(X, Xs)) -> U1_GA(Xs, member_in_ga(X)) The graph contains the following edges 1 > 1 *U1_GA(Xs, member_out_ga) -> SUBSET_IN_GA(Xs) The graph contains the following edges 1 >= 1 ---------------------------------------- (40) YES ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 11, "program": { "directives": [], "clauses": [ [ "(subset ([]) X1)", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(member X (. X X2))", null ], [ "(member X (. X3 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "12": { "goal": [ { "clause": 0, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 1, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": 0, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "25": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "170": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "163": { "goal": [ { "clause": 2, "scope": 2, "term": "(member T14 T17)" }, { "clause": 3, "scope": 2, "term": "(member T14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "164": { "goal": [{ "clause": 2, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "165": { "goal": [{ "clause": 3, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "166": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "167": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "168": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "147": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T14 T17) (subset T15 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T14", "T15" ], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T15 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "169": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T43 T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [], "exprvars": [] } }, "148": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 11, "to": 12, "label": "CASE" }, { "from": 12, "to": 13, "label": "PARALLEL" }, { "from": 12, "to": 14, "label": "PARALLEL" }, { "from": 13, "to": 23, "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7" }, { "from": 13, "to": 25, "label": "EVAL-BACKTRACK" }, { "from": 14, "to": 147, "label": "EVAL with clause\nsubset(.(X15, X16), X17) :- ','(member(X15, X17), subset(X16, X17)).\nand substitutionX15 -> T14,\nX16 -> T15,\nT1 -> .(T14, T15),\nT2 -> T17,\nX17 -> T17,\nT16 -> T17" }, { "from": 14, "to": 148, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 146, "label": "SUCCESS" }, { "from": 147, "to": 157, "label": "SPLIT 1" }, { "from": 147, "to": 158, "label": "SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:T17 -> T22" }, { "from": 157, "to": 163, "label": "CASE" }, { "from": 158, "to": 11, "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T22" }, { "from": 163, "to": 164, "label": "PARALLEL" }, { "from": 163, "to": 165, "label": "PARALLEL" }, { "from": 164, "to": 166, "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT14 -> T35,\nX34 -> T35,\nX35 -> T36,\nT17 -> .(T35, T36)" }, { "from": 164, "to": 167, "label": "EVAL-BACKTRACK" }, { "from": 165, "to": 169, "label": "EVAL with clause\nmember(X42, .(X43, X44)) :- member(X42, X44).\nand substitutionT14 -> T43,\nX42 -> T43,\nX43 -> T44,\nX44 -> T46,\nT17 -> .(T44, T46),\nT45 -> T46" }, { "from": 165, "to": 170, "label": "EVAL-BACKTRACK" }, { "from": 166, "to": 168, "label": "SUCCESS" }, { "from": 169, "to": 157, "label": "INSTANCE with matching:\nT14 -> T43\nT17 -> T46" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 Q is empty. ---------------------------------------- (43) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: F11_IN(.(T14, T15)) -> U1^1(f147_in(T14, T15), .(T14, T15)) F11_IN(.(T14, T15)) -> F147_IN(T14, T15) F157_IN(T43) -> U2^1(f157_in(T43), T43) F157_IN(T43) -> F157_IN(T43) F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) F147_IN(T14, T15) -> F157_IN(T14) U3^1(f157_out1, T14, T15) -> U4^1(f11_in(T15), T14, T15) U3^1(f157_out1, T14, T15) -> F11_IN(T15) The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (46) Complex Obligation (AND) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: F157_IN(T43) -> F157_IN(T43) The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: F157_IN(T43) -> F157_IN(T43) The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 The set Q consists of the following terms: f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f157_in(x0) U2(f157_out1, x0) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: F157_IN(T43) -> F157_IN(T43) R is empty. The set Q consists of the following terms: f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f157_in(x0) U2(f157_out1, x0) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f157_in(x0) U2(f157_out1, x0) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: F157_IN(T43) -> F157_IN(T43) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by semiunifying a rule from P directly. s = F157_IN(T43) evaluates to t =F157_IN(T43) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence The DP semiunifies directly so there is only one rewrite step from F157_IN(T43) to F157_IN(T43). ---------------------------------------- (55) NO ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F11_IN(.(T14, T15)) -> F147_IN(T14, T15) F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) U3^1(f157_out1, T14, T15) -> F11_IN(T15) The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) MNOCProof (EQUIVALENT) We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: F11_IN(.(T14, T15)) -> F147_IN(T14, T15) F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) U3^1(f157_out1, T14, T15) -> F11_IN(T15) The TRS R consists of the following rules: f11_in([]) -> f11_out1 f11_in(.(T14, T15)) -> U1(f147_in(T14, T15), .(T14, T15)) U1(f147_out1, .(T14, T15)) -> f11_out1 f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 f147_in(T14, T15) -> U3(f157_in(T14), T14, T15) U3(f157_out1, T14, T15) -> U4(f11_in(T15), T14, T15) U4(f11_out1, T14, T15) -> f147_out1 The set Q consists of the following terms: f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f157_in(x0) U2(f157_out1, x0) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: F11_IN(.(T14, T15)) -> F147_IN(T14, T15) F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) U3^1(f157_out1, T14, T15) -> F11_IN(T15) The TRS R consists of the following rules: f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 The set Q consists of the following terms: f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f157_in(x0) U2(f157_out1, x0) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f11_in([]) f11_in(.(x0, x1)) U1(f147_out1, .(x0, x1)) f147_in(x0, x1) U3(f157_out1, x0, x1) U4(f11_out1, x0, x1) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: F11_IN(.(T14, T15)) -> F147_IN(T14, T15) F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) U3^1(f157_out1, T14, T15) -> F11_IN(T15) The TRS R consists of the following rules: f157_in(T35) -> f157_out1 f157_in(T43) -> U2(f157_in(T43), T43) U2(f157_out1, T43) -> f157_out1 The set Q consists of the following terms: f157_in(x0) U2(f157_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) 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: *F147_IN(T14, T15) -> U3^1(f157_in(T14), T14, T15) The graph contains the following edges 1 >= 2, 2 >= 3 *U3^1(f157_out1, T14, T15) -> F11_IN(T15) The graph contains the following edges 3 >= 1 *F11_IN(.(T14, T15)) -> F147_IN(T14, T15) The graph contains the following edges 1 > 1, 1 > 2 ---------------------------------------- (64) YES ---------------------------------------- (65) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 27, "program": { "directives": [], "clauses": [ [ "(subset ([]) X1)", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(member X (. X X2))", null ], [ "(member X (. X3 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "45": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "27": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "38": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "28": { "goal": [ { "clause": 0, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 1, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "29": { "goal": [{ "clause": 0, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "181": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "171": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "182": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T14 T17) (subset T15 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T14", "T15" ], "free": [], "exprvars": [] } }, "172": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T15 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "151": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "173": { "goal": [ { "clause": 2, "scope": 2, "term": "(member T14 T17)" }, { "clause": 3, "scope": 2, "term": "(member T14 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "184": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T43 T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [], "exprvars": [] } }, "186": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "176": { "goal": [{ "clause": 2, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "177": { "goal": [{ "clause": 3, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "179": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "149": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "30": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 27, "to": 28, "label": "CASE" }, { "from": 28, "to": 29, "label": "PARALLEL" }, { "from": 28, "to": 30, "label": "PARALLEL" }, { "from": 29, "to": 38, "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7" }, { "from": 29, "to": 45, "label": "EVAL-BACKTRACK" }, { "from": 30, "to": 150, "label": "EVAL with clause\nsubset(.(X15, X16), X17) :- ','(member(X15, X17), subset(X16, X17)).\nand substitutionX15 -> T14,\nX16 -> T15,\nT1 -> .(T14, T15),\nT2 -> T17,\nX17 -> T17,\nT16 -> T17" }, { "from": 30, "to": 151, "label": "EVAL-BACKTRACK" }, { "from": 38, "to": 149, "label": "SUCCESS" }, { "from": 150, "to": 171, "label": "SPLIT 1" }, { "from": 150, "to": 172, "label": "SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:T17 -> T22" }, { "from": 171, "to": 173, "label": "CASE" }, { "from": 172, "to": 27, "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T22" }, { "from": 173, "to": 176, "label": "PARALLEL" }, { "from": 173, "to": 177, "label": "PARALLEL" }, { "from": 176, "to": 179, "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT14 -> T35,\nX34 -> T35,\nX35 -> T36,\nT17 -> .(T35, T36)" }, { "from": 176, "to": 181, "label": "EVAL-BACKTRACK" }, { "from": 177, "to": 184, "label": "EVAL with clause\nmember(X42, .(X43, X44)) :- member(X42, X44).\nand substitutionT14 -> T43,\nX42 -> T43,\nX43 -> T44,\nX44 -> T46,\nT17 -> .(T44, T46),\nT45 -> T46" }, { "from": 177, "to": 186, "label": "EVAL-BACKTRACK" }, { "from": 179, "to": 182, "label": "SUCCESS" }, { "from": 184, "to": 171, "label": "INSTANCE with matching:\nT14 -> T43\nT17 -> T46" } ], "type": "Graph" } } ---------------------------------------- (66) Complex Obligation (AND) ---------------------------------------- (67) Obligation: Rules: f171_out(T43) -> f184_out(T43) :|: TRUE f184_in(x) -> f171_in(x) :|: TRUE f177_out(T14) -> f173_out(T14) :|: TRUE f173_in(x1) -> f177_in(x1) :|: TRUE f176_out(x2) -> f173_out(x2) :|: TRUE f173_in(x3) -> f176_in(x3) :|: TRUE f186_out -> f177_out(x4) :|: TRUE f177_in(x5) -> f186_in :|: TRUE f184_out(x6) -> f177_out(x6) :|: TRUE f177_in(x7) -> f184_in(x7) :|: TRUE f173_out(x8) -> f171_out(x8) :|: TRUE f171_in(x9) -> f173_in(x9) :|: TRUE f28_out(T1) -> f27_out(T1) :|: TRUE f27_in(x10) -> f28_in(x10) :|: TRUE f28_in(x11) -> f29_in(x11) :|: TRUE f30_out(x12) -> f28_out(x12) :|: TRUE f29_out(x13) -> f28_out(x13) :|: TRUE f28_in(x14) -> f30_in(x14) :|: TRUE f150_out(x15, x16) -> f30_out(.(x15, x16)) :|: TRUE f151_out -> f30_out(x17) :|: TRUE f30_in(.(x18, x19)) -> f150_in(x18, x19) :|: TRUE f30_in(x20) -> f151_in :|: TRUE f171_out(x21) -> f172_in(x22) :|: TRUE f150_in(x23, x24) -> f171_in(x23) :|: TRUE f172_out(x25) -> f150_out(x26, x25) :|: TRUE Start term: f27_in(T1) ---------------------------------------- (68) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f184_in(x) -> f171_in(x) :|: TRUE f173_in(x1) -> f177_in(x1) :|: TRUE f177_in(x7) -> f184_in(x7) :|: TRUE f171_in(x9) -> f173_in(x9) :|: TRUE ---------------------------------------- (69) Obligation: Rules: f184_in(x) -> f171_in(x) :|: TRUE f173_in(x1) -> f177_in(x1) :|: TRUE f177_in(x7) -> f184_in(x7) :|: TRUE f171_in(x9) -> f173_in(x9) :|: TRUE ---------------------------------------- (70) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (71) Obligation: Rules: f173_in(x1:0) -> f173_in(x1:0) :|: TRUE ---------------------------------------- (72) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (73) Obligation: Rules: f173_in(x1:0) -> f173_in(x1:0) :|: TRUE ---------------------------------------- (74) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f173_in(x1:0) -> f173_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (75) Obligation: Termination digraph: Nodes: (1) f173_in(x1:0) -> f173_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (76) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f173_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (77) Obligation: Rules: f173_in(x1:0) -> f173_in(x1:0) :|: TRUE ---------------------------------------- (78) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x1:0) -> f(1, x1:0) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: (((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: ((run2_0 = ((1 * 1)) and run2_1 = ((run1_1 * 1))) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (79) NO ---------------------------------------- (80) Obligation: Rules: f150_out(T14, T15) -> f30_out(.(T14, T15)) :|: TRUE f151_out -> f30_out(T1) :|: TRUE f30_in(.(x, x1)) -> f150_in(x, x1) :|: TRUE f30_in(x2) -> f151_in :|: TRUE f171_out(x3) -> f172_in(x4) :|: TRUE f150_in(x5, x6) -> f171_in(x5) :|: TRUE f172_out(x7) -> f150_out(x8, x7) :|: TRUE f28_in(x9) -> f29_in(x9) :|: TRUE f30_out(x10) -> f28_out(x10) :|: TRUE f29_out(x11) -> f28_out(x11) :|: TRUE f28_in(x12) -> f30_in(x12) :|: TRUE f177_out(x13) -> f173_out(x13) :|: TRUE f173_in(x14) -> f177_in(x14) :|: TRUE f176_out(x15) -> f173_out(x15) :|: TRUE f173_in(x16) -> f176_in(x16) :|: TRUE f28_out(x17) -> f27_out(x17) :|: TRUE f27_in(x18) -> f28_in(x18) :|: TRUE f171_out(T43) -> f184_out(T43) :|: TRUE f184_in(x19) -> f171_in(x19) :|: TRUE f172_in(x20) -> f27_in(x20) :|: TRUE f27_out(x21) -> f172_out(x21) :|: TRUE f186_out -> f177_out(x22) :|: TRUE f177_in(x23) -> f186_in :|: TRUE f184_out(x24) -> f177_out(x24) :|: TRUE f177_in(x25) -> f184_in(x25) :|: TRUE f179_in -> f179_out :|: TRUE f179_out -> f176_out(T35) :|: TRUE f176_in(x26) -> f179_in :|: TRUE f176_in(x27) -> f181_in :|: TRUE f181_out -> f176_out(x28) :|: TRUE f173_out(x29) -> f171_out(x29) :|: TRUE f171_in(x30) -> f173_in(x30) :|: TRUE Start term: f27_in(T1) ---------------------------------------- (81) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f30_in(.(x, x1)) -> f150_in(x, x1) :|: TRUE f171_out(x3) -> f172_in(x4) :|: TRUE f150_in(x5, x6) -> f171_in(x5) :|: TRUE f28_in(x12) -> f30_in(x12) :|: TRUE f177_out(x13) -> f173_out(x13) :|: TRUE f173_in(x14) -> f177_in(x14) :|: TRUE f176_out(x15) -> f173_out(x15) :|: TRUE f173_in(x16) -> f176_in(x16) :|: TRUE f27_in(x18) -> f28_in(x18) :|: TRUE f171_out(T43) -> f184_out(T43) :|: TRUE f184_in(x19) -> f171_in(x19) :|: TRUE f172_in(x20) -> f27_in(x20) :|: TRUE f184_out(x24) -> f177_out(x24) :|: TRUE f177_in(x25) -> f184_in(x25) :|: TRUE f179_in -> f179_out :|: TRUE f179_out -> f176_out(T35) :|: TRUE f176_in(x26) -> f179_in :|: TRUE f173_out(x29) -> f171_out(x29) :|: TRUE f171_in(x30) -> f173_in(x30) :|: TRUE ---------------------------------------- (82) Obligation: Rules: f30_in(.(x, x1)) -> f150_in(x, x1) :|: TRUE f171_out(x3) -> f172_in(x4) :|: TRUE f150_in(x5, x6) -> f171_in(x5) :|: TRUE f28_in(x12) -> f30_in(x12) :|: TRUE f177_out(x13) -> f173_out(x13) :|: TRUE f173_in(x14) -> f177_in(x14) :|: TRUE f176_out(x15) -> f173_out(x15) :|: TRUE f173_in(x16) -> f176_in(x16) :|: TRUE f27_in(x18) -> f28_in(x18) :|: TRUE f171_out(T43) -> f184_out(T43) :|: TRUE f184_in(x19) -> f171_in(x19) :|: TRUE f172_in(x20) -> f27_in(x20) :|: TRUE f184_out(x24) -> f177_out(x24) :|: TRUE f177_in(x25) -> f184_in(x25) :|: TRUE f179_in -> f179_out :|: TRUE f179_out -> f176_out(T35) :|: TRUE f176_in(x26) -> f179_in :|: TRUE f173_out(x29) -> f171_out(x29) :|: TRUE f171_in(x30) -> f173_in(x30) :|: TRUE ---------------------------------------- (83) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (84) Obligation: Rules: f171_out(T43:0) -> f171_out(T43:0) :|: TRUE f173_in(x14:0) -> f173_in(x14:0) :|: TRUE f171_out(x3:0) -> f173_in(x:0) :|: TRUE f173_in(x16:0) -> f171_out(T35:0) :|: TRUE ---------------------------------------- (85) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (86) Obligation: Rules: f171_out(T43:0) -> f171_out(T43:0) :|: TRUE f173_in(x14:0) -> f173_in(x14:0) :|: TRUE f171_out(x3:0) -> f173_in(x:0) :|: TRUE f173_in(x16:0) -> f171_out(T35:0) :|: TRUE ---------------------------------------- (87) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 9, "program": { "directives": [], "clauses": [ [ "(subset ([]) X1)", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(member X (. X X2))", null ], [ "(member X (. X3 Xs))", "(member X Xs)" ] ] }, "graph": { "nodes": { "24": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(subset ([]) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [[ "(subset T1 T2)", "(subset ([]) X5)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": ["X5"], "exprvars": [] } }, "190": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "191": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T64 T67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T64"], "free": [], "exprvars": [] } }, "192": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "152": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset ([]) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "174": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T33 T36) (subset T9 (. T37 T36)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T33" ], "free": [], "exprvars": [] } }, "153": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "175": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "155": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "156": { "goal": [ { "clause": 2, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" }, { "clause": 3, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "178": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "159": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "10": { "goal": [ { "clause": 0, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 1, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "180": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T9 (. T42 T43))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [], "exprvars": [] } }, "160": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "161": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T9 (. T20 T22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T20" ], "free": [], "exprvars": [] } }, "183": { "goal": [ { "clause": 2, "scope": 3, "term": "(member T33 T36)" }, { "clause": 3, "scope": 3, "term": "(member T33 T36)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "162": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "185": { "goal": [{ "clause": 2, "scope": 3, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "187": { "goal": [{ "clause": 3, "scope": 3, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "188": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "189": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 9, "to": 10, "label": "CASE" }, { "from": 10, "to": 24, "label": "EVAL with clause\nsubset([], X5).\nand substitutionT1 -> [],\nT2 -> T4,\nX5 -> T4" }, { "from": 10, "to": 26, "label": "EVAL-BACKTRACK" }, { "from": 24, "to": 152, "label": "SUCCESS" }, { "from": 26, "to": 154, "label": "EVAL with clause\nsubset(.(X12, X13), X14) :- ','(member(X12, X14), subset(X13, X14)).\nand substitutionX12 -> T8,\nX13 -> T9,\nT1 -> .(T8, T9),\nT2 -> T11,\nX14 -> T11,\nT10 -> T11" }, { "from": 26, "to": 155, "label": "EVAL-BACKTRACK" }, { "from": 152, "to": 153, "label": "BACKTRACK\nfor clause: subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys))because of non-unification" }, { "from": 154, "to": 156, "label": "CASE" }, { "from": 156, "to": 159, "label": "PARALLEL" }, { "from": 156, "to": 160, "label": "PARALLEL" }, { "from": 159, "to": 161, "label": "EVAL with clause\nmember(X23, .(X23, X24)).\nand substitutionT8 -> T20,\nX23 -> T20,\nX24 -> T22,\nT11 -> .(T20, T22),\nT21 -> T22" }, { "from": 159, "to": 162, "label": "EVAL-BACKTRACK" }, { "from": 160, "to": 174, "label": "EVAL with clause\nmember(X33, .(X34, X35)) :- member(X33, X35).\nand substitutionT8 -> T33,\nX33 -> T33,\nX34 -> T37,\nX35 -> T36,\nT11 -> .(T37, T36),\nT35 -> T36,\nT34 -> T37" }, { "from": 160, "to": 175, "label": "EVAL-BACKTRACK" }, { "from": 161, "to": 9, "label": "INSTANCE with matching:\nT1 -> T9\nT2 -> .(T20, T22)" }, { "from": 174, "to": 178, "label": "SPLIT 1" }, { "from": 174, "to": 180, "label": "SPLIT 2\nnew knowledge:\nT33 is ground\nreplacements:T37 -> T42,\nT36 -> T43" }, { "from": 178, "to": 183, "label": "CASE" }, { "from": 180, "to": 9, "label": "INSTANCE with matching:\nT1 -> T9\nT2 -> .(T42, T43)" }, { "from": 183, "to": 185, "label": "PARALLEL" }, { "from": 183, "to": 187, "label": "PARALLEL" }, { "from": 185, "to": 188, "label": "EVAL with clause\nmember(X52, .(X52, X53)).\nand substitutionT33 -> T56,\nX52 -> T56,\nX53 -> T57,\nT36 -> .(T56, T57)" }, { "from": 185, "to": 189, "label": "EVAL-BACKTRACK" }, { "from": 187, "to": 191, "label": "EVAL with clause\nmember(X60, .(X61, X62)) :- member(X60, X62).\nand substitutionT33 -> T64,\nX60 -> T64,\nX61 -> T65,\nX62 -> T67,\nT36 -> .(T65, T67),\nT66 -> T67" }, { "from": 187, "to": 192, "label": "EVAL-BACKTRACK" }, { "from": 188, "to": 190, "label": "SUCCESS" }, { "from": 191, "to": 178, "label": "INSTANCE with matching:\nT33 -> T64\nT36 -> T67" } ], "type": "Graph" } } ---------------------------------------- (88) Obligation: Triples: memberB(X1, .(X2, X3)) :- memberB(X1, X3). subsetA(.(X1, X2), .(X1, X3)) :- subsetA(X2, .(X1, X3)). subsetA(.(X1, X2), .(X3, X4)) :- memberB(X1, X4). subsetA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetA(X2, .(X3, X4))). Clauses: subsetcA([], X1). subsetcA(.(X1, X2), .(X1, X3)) :- subsetcA(X2, .(X1, X3)). subsetcA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetcA(X2, .(X3, X4))). membercB(X1, .(X1, X2)). membercB(X1, .(X2, X3)) :- membercB(X1, X3). Afs: subsetA(x1, x2) = subsetA(x1) ---------------------------------------- (89) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: subsetA_in_2: (b,f) memberB_in_2: (f,f) membercB_in_2: (f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_GA(.(X1, X2), .(X1, X3)) -> U2_GA(X1, X2, X3, subsetA_in_ga(X2, .(X1, X3))) SUBSETA_IN_GA(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_GA(X2, .(X1, X3)) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> U3_GA(X1, X2, X3, X4, memberB_in_aa(X1, X4)) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AA(X1, X4) MEMBERB_IN_AA(X1, .(X2, X3)) -> U1_AA(X1, X2, X3, memberB_in_aa(X1, X3)) MEMBERB_IN_AA(X1, .(X2, X3)) -> MEMBERB_IN_AA(X1, X3) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> U4_GA(X1, X2, X3, X4, membercB_in_aa(X1, X4)) U4_GA(X1, X2, X3, X4, membercB_out_aa(X1, X4)) -> U5_GA(X1, X2, X3, X4, subsetA_in_ga(X2, .(X3, X4))) U4_GA(X1, X2, X3, X4, membercB_out_aa(X1, X4)) -> SUBSETA_IN_GA(X2, .(X3, X4)) The TRS R consists of the following rules: membercB_in_aa(X1, .(X1, X2)) -> membercB_out_aa(X1, .(X1, X2)) membercB_in_aa(X1, .(X2, X3)) -> U10_aa(X1, X2, X3, membercB_in_aa(X1, X3)) U10_aa(X1, X2, X3, membercB_out_aa(X1, X3)) -> membercB_out_aa(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subsetA_in_ga(x1, x2) = subsetA_in_ga(x1) .(x1, x2) = .(x2) memberB_in_aa(x1, x2) = memberB_in_aa membercB_in_aa(x1, x2) = membercB_in_aa membercB_out_aa(x1, x2) = membercB_out_aa U10_aa(x1, x2, x3, x4) = U10_aa(x4) SUBSETA_IN_GA(x1, x2) = SUBSETA_IN_GA(x1) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x2, x5) MEMBERB_IN_AA(x1, x2) = MEMBERB_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x2, x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x2, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (90) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_GA(.(X1, X2), .(X1, X3)) -> U2_GA(X1, X2, X3, subsetA_in_ga(X2, .(X1, X3))) SUBSETA_IN_GA(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_GA(X2, .(X1, X3)) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> U3_GA(X1, X2, X3, X4, memberB_in_aa(X1, X4)) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AA(X1, X4) MEMBERB_IN_AA(X1, .(X2, X3)) -> U1_AA(X1, X2, X3, memberB_in_aa(X1, X3)) MEMBERB_IN_AA(X1, .(X2, X3)) -> MEMBERB_IN_AA(X1, X3) SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> U4_GA(X1, X2, X3, X4, membercB_in_aa(X1, X4)) U4_GA(X1, X2, X3, X4, membercB_out_aa(X1, X4)) -> U5_GA(X1, X2, X3, X4, subsetA_in_ga(X2, .(X3, X4))) U4_GA(X1, X2, X3, X4, membercB_out_aa(X1, X4)) -> SUBSETA_IN_GA(X2, .(X3, X4)) The TRS R consists of the following rules: membercB_in_aa(X1, .(X1, X2)) -> membercB_out_aa(X1, .(X1, X2)) membercB_in_aa(X1, .(X2, X3)) -> U10_aa(X1, X2, X3, membercB_in_aa(X1, X3)) U10_aa(X1, X2, X3, membercB_out_aa(X1, X3)) -> membercB_out_aa(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subsetA_in_ga(x1, x2) = subsetA_in_ga(x1) .(x1, x2) = .(x2) memberB_in_aa(x1, x2) = memberB_in_aa membercB_in_aa(x1, x2) = membercB_in_aa membercB_out_aa(x1, x2) = membercB_out_aa U10_aa(x1, x2, x3, x4) = U10_aa(x4) SUBSETA_IN_GA(x1, x2) = SUBSETA_IN_GA(x1) U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x2, x5) MEMBERB_IN_AA(x1, x2) = MEMBERB_IN_AA U1_AA(x1, x2, x3, x4) = U1_AA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x2, x5) U5_GA(x1, x2, x3, x4, x5) = U5_GA(x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (91) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (92) Complex Obligation (AND) ---------------------------------------- (93) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AA(X1, .(X2, X3)) -> MEMBERB_IN_AA(X1, X3) The TRS R consists of the following rules: membercB_in_aa(X1, .(X1, X2)) -> membercB_out_aa(X1, .(X1, X2)) membercB_in_aa(X1, .(X2, X3)) -> U10_aa(X1, X2, X3, membercB_in_aa(X1, X3)) U10_aa(X1, X2, X3, membercB_out_aa(X1, X3)) -> membercB_out_aa(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) membercB_in_aa(x1, x2) = membercB_in_aa membercB_out_aa(x1, x2) = membercB_out_aa U10_aa(x1, x2, x3, x4) = U10_aa(x4) MEMBERB_IN_AA(x1, x2) = MEMBERB_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (94) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (95) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AA(X1, .(X2, X3)) -> MEMBERB_IN_AA(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) MEMBERB_IN_AA(x1, x2) = MEMBERB_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (96) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERB_IN_AA -> MEMBERB_IN_AA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (98) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_GA(.(X1, X2), .(X3, X4)) -> U4_GA(X1, X2, X3, X4, membercB_in_aa(X1, X4)) U4_GA(X1, X2, X3, X4, membercB_out_aa(X1, X4)) -> SUBSETA_IN_GA(X2, .(X3, X4)) SUBSETA_IN_GA(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_GA(X2, .(X1, X3)) The TRS R consists of the following rules: membercB_in_aa(X1, .(X1, X2)) -> membercB_out_aa(X1, .(X1, X2)) membercB_in_aa(X1, .(X2, X3)) -> U10_aa(X1, X2, X3, membercB_in_aa(X1, X3)) U10_aa(X1, X2, X3, membercB_out_aa(X1, X3)) -> membercB_out_aa(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) membercB_in_aa(x1, x2) = membercB_in_aa membercB_out_aa(x1, x2) = membercB_out_aa U10_aa(x1, x2, x3, x4) = U10_aa(x4) SUBSETA_IN_GA(x1, x2) = SUBSETA_IN_GA(x1) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x2, x5) We have to consider all (P,R,Pi)-chains