/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern subset(g,a) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) PrologToDTProblemTransformerProof [SOUND, 0 ms] (66) TRIPLES (67) TriplesToPiDPProof [SOUND, 0 ms] (68) PiDP (69) DependencyGraphProof [EQUIVALENT, 0 ms] (70) AND (71) PiDP (72) UsableRulesProof [EQUIVALENT, 0 ms] (73) PiDP (74) PiDPToQDPProof [SOUND, 0 ms] (75) QDP (76) NonTerminationLoopProof [COMPLETE, 0 ms] (77) NO (78) PiDP (79) PiDPToQDPProof [SOUND, 0 ms] (80) QDP (81) QDPSizeChangeProof [EQUIVALENT, 0 ms] (82) YES (83) PrologToIRSwTTransformerProof [SOUND, 0 ms] (84) AND (85) IRSwT (86) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (87) IRSwT (88) IntTRSCompressionProof [EQUIVALENT, 33 ms] (89) IRSwT (90) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (91) IRSwT (92) IRSwTTerminationDigraphProof [EQUIVALENT, 1 ms] (93) IRSwT (94) FilterProof [EQUIVALENT, 0 ms] (95) IntTRS (96) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] (97) NO (98) IRSwT (99) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (100) IRSwT ---------------------------------------- (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": 21, "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": { "22": { "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": [] } }, "170": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "171": { "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": "(member T43 T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [], "exprvars": [] } }, "151": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "173": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "155": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "166": { "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": [] } }, "167": { "goal": [{ "clause": 2, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T15 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "168": { "goal": [{ "clause": 3, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "169": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "148": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "129": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "60": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "53": { "goal": [{ "clause": 0, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 21, "to": 22, "label": "CASE" }, { "from": 22, "to": 53, "label": "PARALLEL" }, { "from": 22, "to": 60, "label": "PARALLEL" }, { "from": 53, "to": 126, "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7" }, { "from": 53, "to": 129, "label": "EVAL-BACKTRACK" }, { "from": 60, "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": 60, "to": 151, "label": "EVAL-BACKTRACK" }, { "from": 126, "to": 148, "label": "SUCCESS" }, { "from": 150, "to": 155, "label": "SPLIT 1" }, { "from": 150, "to": 157, "label": "SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:T17 -> T22" }, { "from": 155, "to": 166, "label": "CASE" }, { "from": 157, "to": 21, "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T22" }, { "from": 166, "to": 167, "label": "PARALLEL" }, { "from": 166, "to": 168, "label": "PARALLEL" }, { "from": 167, "to": 169, "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT14 -> T35,\nX34 -> T35,\nX35 -> T36,\nT17 -> .(T35, T36)" }, { "from": 167, "to": 170, "label": "EVAL-BACKTRACK" }, { "from": 168, "to": 172, "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": 168, "to": 173, "label": "EVAL-BACKTRACK" }, { "from": 169, "to": 171, "label": "SUCCESS" }, { "from": 172, "to": 155, "label": "INSTANCE with matching:\nT14 -> T43\nT17 -> T46" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_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: F21_IN(.(T14, T15)) -> U1^1(f150_in(T14, T15), .(T14, T15)) F21_IN(.(T14, T15)) -> F150_IN(T14, T15) F155_IN(T43) -> U2^1(f155_in(T43), T43) F155_IN(T43) -> F155_IN(T43) F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) F150_IN(T14, T15) -> F155_IN(T14) U3^1(f155_out1, T14, T15) -> U4^1(f21_in(T15), T14, T15) U3^1(f155_out1, T14, T15) -> F21_IN(T15) The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_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: F155_IN(T43) -> F155_IN(T43) The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_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: F155_IN(T43) -> F155_IN(T43) The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_out1 The set Q consists of the following terms: f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f155_in(x0) U2(f155_out1, x0) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_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: F155_IN(T43) -> F155_IN(T43) R is empty. The set Q consists of the following terms: f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f155_in(x0) U2(f155_out1, x0) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_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]. f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f155_in(x0) U2(f155_out1, x0) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_out1, x0, x1) ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: F155_IN(T43) -> F155_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 = F155_IN(T43) evaluates to t =F155_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 F155_IN(T43) to F155_IN(T43). ---------------------------------------- (55) NO ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: F21_IN(.(T14, T15)) -> F150_IN(T14, T15) F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) U3^1(f155_out1, T14, T15) -> F21_IN(T15) The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_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: F21_IN(.(T14, T15)) -> F150_IN(T14, T15) F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) U3^1(f155_out1, T14, T15) -> F21_IN(T15) The TRS R consists of the following rules: f21_in([]) -> f21_out1 f21_in(.(T14, T15)) -> U1(f150_in(T14, T15), .(T14, T15)) U1(f150_out1, .(T14, T15)) -> f21_out1 f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 f150_in(T14, T15) -> U3(f155_in(T14), T14, T15) U3(f155_out1, T14, T15) -> U4(f21_in(T15), T14, T15) U4(f21_out1, T14, T15) -> f150_out1 The set Q consists of the following terms: f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f155_in(x0) U2(f155_out1, x0) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_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: F21_IN(.(T14, T15)) -> F150_IN(T14, T15) F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) U3^1(f155_out1, T14, T15) -> F21_IN(T15) The TRS R consists of the following rules: f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 The set Q consists of the following terms: f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f155_in(x0) U2(f155_out1, x0) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_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]. f21_in([]) f21_in(.(x0, x1)) U1(f150_out1, .(x0, x1)) f150_in(x0, x1) U3(f155_out1, x0, x1) U4(f21_out1, x0, x1) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: F21_IN(.(T14, T15)) -> F150_IN(T14, T15) F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) U3^1(f155_out1, T14, T15) -> F21_IN(T15) The TRS R consists of the following rules: f155_in(T35) -> f155_out1 f155_in(T43) -> U2(f155_in(T43), T43) U2(f155_out1, T43) -> f155_out1 The set Q consists of the following terms: f155_in(x0) U2(f155_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: *F150_IN(T14, T15) -> U3^1(f155_in(T14), T14, T15) The graph contains the following edges 1 >= 2, 2 >= 3 *U3^1(f155_out1, T14, T15) -> F21_IN(T15) The graph contains the following edges 3 >= 1 *F21_IN(.(T14, T15)) -> F150_IN(T14, T15) The graph contains the following edges 1 > 1, 1 > 2 ---------------------------------------- (64) YES ---------------------------------------- (65) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 2, "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": { "193": { "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": [] } }, "type": "Nodes", "194": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "195": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "174": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "196": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T9 (. T42 T43))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T9"], "free": [], "exprvars": [] } }, "197": { "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": [] } }, "198": { "goal": [{ "clause": 2, "scope": 3, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "199": { "goal": [{ "clause": 3, "scope": 3, "term": "(member T33 T36)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T33"], "free": [], "exprvars": [] } }, "138": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset ([]) T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "181": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "182": { "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": [] } }, "183": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "184": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (member T8 T11) (subset T9 T11))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T8", "T9" ], "free": [], "exprvars": [] } }, "185": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T9 (. T20 T22))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T9", "T20" ], "free": [], "exprvars": [] } }, "186": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "121": { "goal": [ { "clause": -1, "scope": -1, "term": "(true)" }, { "clause": 1, "scope": 1, "term": "(subset ([]) T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "123": { "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": [] } }, "200": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "146": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "201": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "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": [] } }, "202": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "203": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T64 T67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T64"], "free": [], "exprvars": [] } }, "204": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 121, "label": "EVAL with clause\nsubset([], X5).\nand substitutionT1 -> [],\nT2 -> T4,\nX5 -> T4" }, { "from": 4, "to": 123, "label": "EVAL-BACKTRACK" }, { "from": 121, "to": 138, "label": "SUCCESS" }, { "from": 123, "to": 174, "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": 123, "to": 181, "label": "EVAL-BACKTRACK" }, { "from": 138, "to": 146, "label": "BACKTRACK\nfor clause: subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys))because of non-unification" }, { "from": 174, "to": 182, "label": "CASE" }, { "from": 182, "to": 183, "label": "PARALLEL" }, { "from": 182, "to": 184, "label": "PARALLEL" }, { "from": 183, "to": 185, "label": "EVAL with clause\nmember(X23, .(X23, X24)).\nand substitutionT8 -> T20,\nX23 -> T20,\nX24 -> T22,\nT11 -> .(T20, T22),\nT21 -> T22" }, { "from": 183, "to": 186, "label": "EVAL-BACKTRACK" }, { "from": 184, "to": 193, "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": 184, "to": 194, "label": "EVAL-BACKTRACK" }, { "from": 185, "to": 2, "label": "INSTANCE with matching:\nT1 -> T9\nT2 -> .(T20, T22)" }, { "from": 193, "to": 195, "label": "SPLIT 1" }, { "from": 193, "to": 196, "label": "SPLIT 2\nnew knowledge:\nT33 is ground\nreplacements:T37 -> T42,\nT36 -> T43" }, { "from": 195, "to": 197, "label": "CASE" }, { "from": 196, "to": 2, "label": "INSTANCE with matching:\nT1 -> T9\nT2 -> .(T42, T43)" }, { "from": 197, "to": 198, "label": "PARALLEL" }, { "from": 197, "to": 199, "label": "PARALLEL" }, { "from": 198, "to": 200, "label": "EVAL with clause\nmember(X52, .(X52, X53)).\nand substitutionT33 -> T56,\nX52 -> T56,\nX53 -> T57,\nT36 -> .(T56, T57)" }, { "from": 198, "to": 201, "label": "EVAL-BACKTRACK" }, { "from": 199, "to": 203, "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": 199, "to": 204, "label": "EVAL-BACKTRACK" }, { "from": 200, "to": 202, "label": "SUCCESS" }, { "from": 203, "to": 195, "label": "INSTANCE with matching:\nT33 -> T64\nT36 -> T67" } ], "type": "Graph" } } ---------------------------------------- (66) 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) ---------------------------------------- (67) 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 ---------------------------------------- (68) 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 ---------------------------------------- (69) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (70) Complex Obligation (AND) ---------------------------------------- (71) 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 ---------------------------------------- (72) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (73) 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 ---------------------------------------- (74) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (75) 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. ---------------------------------------- (76) 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 = MEMBERB_IN_AA evaluates to t =MEMBERB_IN_AA 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 MEMBERB_IN_AA to MEMBERB_IN_AA. ---------------------------------------- (77) NO ---------------------------------------- (78) 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 ---------------------------------------- (79) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETA_IN_GA(.(X2)) -> U4_GA(X2, membercB_in_aa) U4_GA(X2, membercB_out_aa) -> SUBSETA_IN_GA(X2) SUBSETA_IN_GA(.(X2)) -> SUBSETA_IN_GA(X2) The TRS R consists of the following rules: membercB_in_aa -> membercB_out_aa membercB_in_aa -> U10_aa(membercB_in_aa) U10_aa(membercB_out_aa) -> membercB_out_aa The set Q consists of the following terms: membercB_in_aa U10_aa(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (81) 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: *U4_GA(X2, membercB_out_aa) -> SUBSETA_IN_GA(X2) The graph contains the following edges 1 >= 1 *SUBSETA_IN_GA(.(X2)) -> SUBSETA_IN_GA(X2) The graph contains the following edges 1 > 1 *SUBSETA_IN_GA(.(X2)) -> U4_GA(X2, membercB_in_aa) The graph contains the following edges 1 > 1 ---------------------------------------- (82) YES ---------------------------------------- (83) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 19, "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": { "23": { "goal": [{ "clause": 0, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "160": { "goal": [{ "clause": 3, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "type": "Nodes", "161": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "162": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T14 T17) (subset T15 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T14", "T15" ], "free": [], "exprvars": [] } }, "163": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "131": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "153": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "164": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T43 T46)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T43"], "free": [], "exprvars": [] } }, "154": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "165": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "156": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T15 T22)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T15"], "free": [], "exprvars": [] } }, "158": { "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": [] } }, "159": { "goal": [{ "clause": 2, "scope": 2, "term": "(member T14 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "127": { "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": [] } }, "20": { "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": [] } } }, "edges": [ { "from": 19, "to": 20, "label": "CASE" }, { "from": 20, "to": 23, "label": "PARALLEL" }, { "from": 20, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 127, "label": "EVAL with clause\nsubset([], X8).\nand substitutionT1 -> [],\nT2 -> T7,\nX8 -> T7" }, { "from": 23, "to": 131, "label": "EVAL-BACKTRACK" }, { "from": 24, "to": 152, "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": 24, "to": 153, "label": "EVAL-BACKTRACK" }, { "from": 127, "to": 149, "label": "SUCCESS" }, { "from": 152, "to": 154, "label": "SPLIT 1" }, { "from": 152, "to": 156, "label": "SPLIT 2\nnew knowledge:\nT14 is ground\nreplacements:T17 -> T22" }, { "from": 154, "to": 158, "label": "CASE" }, { "from": 156, "to": 19, "label": "INSTANCE with matching:\nT1 -> T15\nT2 -> T22" }, { "from": 158, "to": 159, "label": "PARALLEL" }, { "from": 158, "to": 160, "label": "PARALLEL" }, { "from": 159, "to": 161, "label": "EVAL with clause\nmember(X34, .(X34, X35)).\nand substitutionT14 -> T35,\nX34 -> T35,\nX35 -> T36,\nT17 -> .(T35, T36)" }, { "from": 159, "to": 162, "label": "EVAL-BACKTRACK" }, { "from": 160, "to": 164, "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": 160, "to": 165, "label": "EVAL-BACKTRACK" }, { "from": 161, "to": 163, "label": "SUCCESS" }, { "from": 164, "to": 154, "label": "INSTANCE with matching:\nT14 -> T43\nT17 -> T46" } ], "type": "Graph" } } ---------------------------------------- (84) Complex Obligation (AND) ---------------------------------------- (85) Obligation: Rules: f154_in(T14) -> f158_in(T14) :|: TRUE f158_out(x) -> f154_out(x) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f160_in(x1) -> f165_in :|: TRUE f165_out -> f160_out(x2) :|: TRUE f164_out(x3) -> f160_out(x3) :|: TRUE f154_out(x4) -> f164_out(x4) :|: TRUE f164_in(x5) -> f154_in(x5) :|: TRUE f160_out(x6) -> f158_out(x6) :|: TRUE f158_in(x7) -> f159_in(x7) :|: TRUE f158_in(x8) -> f160_in(x8) :|: TRUE f159_out(x9) -> f158_out(x9) :|: TRUE f19_in(T1) -> f20_in(T1) :|: TRUE f20_out(x10) -> f19_out(x10) :|: TRUE f24_out(x11) -> f20_out(x11) :|: TRUE f23_out(x12) -> f20_out(x12) :|: TRUE f20_in(x13) -> f23_in(x13) :|: TRUE f20_in(x14) -> f24_in(x14) :|: TRUE f24_in(.(x15, x16)) -> f152_in(x15, x16) :|: TRUE f153_out -> f24_out(x17) :|: TRUE f24_in(x18) -> f153_in :|: TRUE f152_out(x19, x20) -> f24_out(.(x19, x20)) :|: TRUE f152_in(x21, x22) -> f154_in(x21) :|: TRUE f154_out(x23) -> f156_in(x24) :|: TRUE f156_out(x25) -> f152_out(x26, x25) :|: TRUE Start term: f19_in(T1) ---------------------------------------- (86) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f154_in(T14) -> f158_in(T14) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f164_in(x5) -> f154_in(x5) :|: TRUE f158_in(x8) -> f160_in(x8) :|: TRUE ---------------------------------------- (87) Obligation: Rules: f154_in(T14) -> f158_in(T14) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f164_in(x5) -> f154_in(x5) :|: TRUE f158_in(x8) -> f160_in(x8) :|: TRUE ---------------------------------------- (88) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (89) Obligation: Rules: f160_in(T43:0) -> f160_in(T43:0) :|: TRUE ---------------------------------------- (90) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (91) Obligation: Rules: f160_in(T43:0) -> f160_in(T43:0) :|: TRUE ---------------------------------------- (92) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f160_in(T43:0) -> f160_in(T43:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (93) Obligation: Termination digraph: Nodes: (1) f160_in(T43:0) -> f160_in(T43:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (94) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f160_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (95) Obligation: Rules: f160_in(T43:0) -> f160_in(T43:0) :|: TRUE ---------------------------------------- (96) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, T43:0) -> f(1, T43:0) :|: 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)) ---------------------------------------- (97) NO ---------------------------------------- (98) Obligation: Rules: f154_in(T14) -> f158_in(T14) :|: TRUE f158_out(x) -> f154_out(x) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f160_in(x1) -> f165_in :|: TRUE f165_out -> f160_out(x2) :|: TRUE f164_out(x3) -> f160_out(x3) :|: TRUE f24_in(.(x4, x5)) -> f152_in(x4, x5) :|: TRUE f153_out -> f24_out(T1) :|: TRUE f24_in(x6) -> f153_in :|: TRUE f152_out(x7, x8) -> f24_out(.(x7, x8)) :|: TRUE f152_in(x9, x10) -> f154_in(x9) :|: TRUE f154_out(x11) -> f156_in(x12) :|: TRUE f156_out(x13) -> f152_out(x14, x13) :|: TRUE f160_out(x15) -> f158_out(x15) :|: TRUE f158_in(x16) -> f159_in(x16) :|: TRUE f158_in(x17) -> f160_in(x17) :|: TRUE f159_out(x18) -> f158_out(x18) :|: TRUE f161_in -> f161_out :|: TRUE f162_out -> f159_out(x19) :|: TRUE f161_out -> f159_out(T35) :|: TRUE f159_in(x20) -> f161_in :|: TRUE f159_in(x21) -> f162_in :|: TRUE f19_in(x22) -> f20_in(x22) :|: TRUE f20_out(x23) -> f19_out(x23) :|: TRUE f154_out(x24) -> f164_out(x24) :|: TRUE f164_in(x25) -> f154_in(x25) :|: TRUE f24_out(x26) -> f20_out(x26) :|: TRUE f23_out(x27) -> f20_out(x27) :|: TRUE f20_in(x28) -> f23_in(x28) :|: TRUE f20_in(x29) -> f24_in(x29) :|: TRUE f156_in(T15) -> f19_in(T15) :|: TRUE f19_out(x30) -> f156_out(x30) :|: TRUE Start term: f19_in(T1) ---------------------------------------- (99) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f154_in(T14) -> f158_in(T14) :|: TRUE f158_out(x) -> f154_out(x) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f164_out(x3) -> f160_out(x3) :|: TRUE f24_in(.(x4, x5)) -> f152_in(x4, x5) :|: TRUE f152_in(x9, x10) -> f154_in(x9) :|: TRUE f154_out(x11) -> f156_in(x12) :|: TRUE f160_out(x15) -> f158_out(x15) :|: TRUE f158_in(x16) -> f159_in(x16) :|: TRUE f158_in(x17) -> f160_in(x17) :|: TRUE f159_out(x18) -> f158_out(x18) :|: TRUE f161_in -> f161_out :|: TRUE f161_out -> f159_out(T35) :|: TRUE f159_in(x20) -> f161_in :|: TRUE f19_in(x22) -> f20_in(x22) :|: TRUE f154_out(x24) -> f164_out(x24) :|: TRUE f164_in(x25) -> f154_in(x25) :|: TRUE f20_in(x29) -> f24_in(x29) :|: TRUE f156_in(T15) -> f19_in(T15) :|: TRUE ---------------------------------------- (100) Obligation: Rules: f154_in(T14) -> f158_in(T14) :|: TRUE f158_out(x) -> f154_out(x) :|: TRUE f160_in(T43) -> f164_in(T43) :|: TRUE f164_out(x3) -> f160_out(x3) :|: TRUE f24_in(.(x4, x5)) -> f152_in(x4, x5) :|: TRUE f152_in(x9, x10) -> f154_in(x9) :|: TRUE f154_out(x11) -> f156_in(x12) :|: TRUE f160_out(x15) -> f158_out(x15) :|: TRUE f158_in(x16) -> f159_in(x16) :|: TRUE f158_in(x17) -> f160_in(x17) :|: TRUE f159_out(x18) -> f158_out(x18) :|: TRUE f161_in -> f161_out :|: TRUE f161_out -> f159_out(T35) :|: TRUE f159_in(x20) -> f161_in :|: TRUE f19_in(x22) -> f20_in(x22) :|: TRUE f154_out(x24) -> f164_out(x24) :|: TRUE f164_in(x25) -> f154_in(x25) :|: TRUE f20_in(x29) -> f24_in(x29) :|: TRUE f156_in(T15) -> f19_in(T15) :|: TRUE