/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern subset(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) PrologToPiTRSProof [SOUND, 0 ms] (20) PiTRS (21) DependencyPairsProof [EQUIVALENT, 0 ms] (22) PiDP (23) DependencyGraphProof [EQUIVALENT, 0 ms] (24) AND (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDPToQDPProof [SOUND, 0 ms] (29) QDP (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] (31) YES (32) PiDP (33) UsableRulesProof [EQUIVALENT, 0 ms] (34) PiDP (35) PiDPToQDPProof [SOUND, 0 ms] (36) QDP (37) TransformationProof [SOUND, 0 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) DependencyPairsProof [EQUIVALENT, 0 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 1 ms] (46) AND (47) QDP (48) UsableRulesProof [EQUIVALENT, 0 ms] (49) QDP (50) QDPSizeChangeProof [EQUIVALENT, 0 ms] (51) YES (52) QDP (53) NonTerminationLoopProof [COMPLETE, 3 ms] (54) NO (55) PrologToDTProblemTransformerProof [SOUND, 0 ms] (56) TRIPLES (57) TriplesToPiDPProof [SOUND, 0 ms] (58) PiDP (59) DependencyGraphProof [EQUIVALENT, 0 ms] (60) AND (61) PiDP (62) UsableRulesProof [EQUIVALENT, 0 ms] (63) PiDP (64) PiDPToQDPProof [SOUND, 0 ms] (65) QDP (66) QDPSizeChangeProof [EQUIVALENT, 0 ms] (67) YES (68) PiDP (69) PiDPToQDPProof [SOUND, 0 ms] (70) QDP (71) PrologToIRSwTTransformerProof [SOUND, 0 ms] (72) AND (73) IRSwT (74) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (75) IRSwT (76) IntTRSCompressionProof [EQUIVALENT, 0 ms] (77) IRSwT (78) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (79) IRSwT (80) IRSwTTerminationDigraphProof [EQUIVALENT, 7 ms] (81) IRSwT (82) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 10 ms] (83) IRSwT (84) TempFilterProof [SOUND, 3 ms] (85) IRSwT (86) IRSwTToQDPProof [SOUND, 0 ms] (87) QDP (88) QDPSizeChangeProof [EQUIVALENT, 0 ms] (89) YES (90) IRSwT (91) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (92) IRSwT ---------------------------------------- (0) Obligation: Clauses: member(X, .(Y, Xs)) :- member(X, Xs). member(X, .(X, Xs)). subset(.(X, Xs), Ys) :- ','(member(X, Ys), subset(Xs, Ys)). subset([], Ys). Query: subset(a,g) ---------------------------------------- (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: (f,b) member_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) 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_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) ---------------------------------------- (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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys)) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4) = U3_AG(x1, x3, 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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys)) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4) = U3_AG(x1, x3, 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_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x3, x4) subset_out_ag(x1, x2) = subset_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, 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: U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) The argument filtering Pi contains the following mapping: member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x2, x3, x4) member_out_ag(x1, x2) = member_out_ag(x1, x2) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, 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: U2_AG(Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Ys) SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys)) The TRS R consists of the following rules: member_in_ag(.(Y, Xs)) -> U1_ag(Y, Xs, member_in_ag(Xs)) member_in_ag(.(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) The set Q consists of the following terms: member_in_ag(x0) U1_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) 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: (f,b) member_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (20) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) ---------------------------------------- (21) 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_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys)) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x4) U3_AG(x1, x2, x3, x4) = U3_AG(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) SUBSET_IN_AG(.(X, Xs), Ys) -> MEMBER_IN_AG(X, Ys) MEMBER_IN_AG(X, .(Y, Xs)) -> U1_AG(X, Y, Xs, member_in_ag(X, Xs)) MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_AG(X, Xs, Ys, subset_in_ag(Xs, Ys)) U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x4) U3_AG(x1, x2, x3, x4) = U3_AG(x1, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (24) Complex Obligation (AND) ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (27) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(X, .(Y, Xs)) -> MEMBER_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER_IN_AG(x1, x2) = MEMBER_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) 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: *MEMBER_IN_AG(.(Y, Xs)) -> MEMBER_IN_AG(Xs) The graph contains the following edges 1 > 1 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: subset_in_ag(.(X, Xs), Ys) -> U2_ag(X, Xs, Ys, member_in_ag(X, Ys)) member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) U2_ag(X, Xs, Ys, member_out_ag(X, Ys)) -> U3_ag(X, Xs, Ys, subset_in_ag(Xs, Ys)) subset_in_ag([], Ys) -> subset_out_ag([], Ys) U3_ag(X, Xs, Ys, subset_out_ag(Xs, Ys)) -> subset_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset_in_ag(x1, x2) = subset_in_ag(x2) U2_ag(x1, x2, x3, x4) = U2_ag(x3, x4) member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) U3_ag(x1, x2, x3, x4) = U3_ag(x1, x4) subset_out_ag(x1, x2) = subset_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AG(X, Xs, Ys, member_out_ag(X, Ys)) -> SUBSET_IN_AG(Xs, Ys) SUBSET_IN_AG(.(X, Xs), Ys) -> U2_AG(X, Xs, Ys, member_in_ag(X, Ys)) The TRS R consists of the following rules: member_in_ag(X, .(Y, Xs)) -> U1_ag(X, Y, Xs, member_in_ag(X, Xs)) member_in_ag(X, .(X, Xs)) -> member_out_ag(X, .(X, Xs)) U1_ag(X, Y, Xs, member_out_ag(X, Xs)) -> member_out_ag(X, .(Y, Xs)) The argument filtering Pi contains the following mapping: member_in_ag(x1, x2) = member_in_ag(x2) .(x1, x2) = .(x1, x2) U1_ag(x1, x2, x3, x4) = U1_ag(x4) member_out_ag(x1, x2) = member_out_ag(x1) SUBSET_IN_AG(x1, x2) = SUBSET_IN_AG(x2) U2_AG(x1, x2, x3, x4) = U2_AG(x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys) SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys)) The TRS R consists of the following rules: member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs)) member_in_ag(.(X, Xs)) -> member_out_ag(X) U1_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U1_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (SOUND) By narrowing [LPAR04] the rule SUBSET_IN_AG(Ys) -> U2_AG(Ys, member_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]: (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1))),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1)))) (SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0)),SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys) SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1))) SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0)) The TRS R consists of the following rules: member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs)) member_in_ag(.(X, Xs)) -> member_out_ag(X) U1_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U1_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U2_AG(Ys, member_out_ag(X)) -> SUBSET_IN_AG(Ys) we obtained the following new rules [LPAR04]: (U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)),U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1))) (U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)),U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), U1_ag(member_in_ag(x1))) SUBSET_IN_AG(.(x0, x1)) -> U2_AG(.(x0, x1), member_out_ag(x0)) U2_AG(.(z0, z1), member_out_ag(x1)) -> SUBSET_IN_AG(.(z0, z1)) U2_AG(.(z0, z1), member_out_ag(z0)) -> SUBSET_IN_AG(.(z0, z1)) The TRS R consists of the following rules: member_in_ag(.(Y, Xs)) -> U1_ag(member_in_ag(Xs)) member_in_ag(.(X, Xs)) -> member_out_ag(X) U1_ag(member_out_ag(X)) -> member_out_ag(X) The set Q consists of the following terms: member_in_ag(x0) U1_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 131, "program": { "directives": [], "clauses": [ [ "(member X (. Y Xs))", "(member X Xs)" ], [ "(member X (. X Xs))", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(subset ([]) Ys)", null ] ] }, "graph": { "nodes": { "160": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "150": { "goal": [ { "clause": 0, "scope": 2, "term": "(member T18 T17)" }, { "clause": 1, "scope": 2, "term": "(member T18 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T18 T17) (subset T19 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "131": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "133": { "goal": [ { "clause": 2, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 3, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "144": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": 0, "scope": 2, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": 1, "scope": 2, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "148": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "159": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T42 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [], "exprvars": [] } }, "138": { "goal": [{ "clause": 2, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "149": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T23 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "204": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "139": { "goal": [{ "clause": 3, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "205": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "206": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "207": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "208": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "209": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 131, "to": 133, "label": "CASE" }, { "from": 133, "to": 138, "label": "PARALLEL" }, { "from": 133, "to": 139, "label": "PARALLEL" }, { "from": 138, "to": 141, "label": "EVAL with clause\nsubset(.(X13, X14), X15) :- ','(member(X13, X15), subset(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 138, "to": 144, "label": "EVAL-BACKTRACK" }, { "from": 139, "to": 207, "label": "EVAL with clause\nsubset([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57" }, { "from": 139, "to": 208, "label": "EVAL-BACKTRACK" }, { "from": 141, "to": 148, "label": "SPLIT 1" }, { "from": 141, "to": 149, "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23" }, { "from": 148, "to": 150, "label": "CASE" }, { "from": 149, "to": 131, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17" }, { "from": 150, "to": 157, "label": "PARALLEL" }, { "from": 150, "to": 158, "label": "PARALLEL" }, { "from": 157, "to": 159, "label": "EVAL with clause\nmember(X34, .(X35, X36)) :- member(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42" }, { "from": 157, "to": 160, "label": "EVAL-BACKTRACK" }, { "from": 158, "to": 204, "label": "EVAL with clause\nmember(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)" }, { "from": 158, "to": 205, "label": "EVAL-BACKTRACK" }, { "from": 159, "to": 148, "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41" }, { "from": 204, "to": 206, "label": "SUCCESS" }, { "from": 207, "to": 209, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f131_in(T17) -> U1(f141_in(T17), T17) U1(f141_out1(T18, T19), T17) -> f131_out1(.(T18, T19)) f131_in(T57) -> f131_out1([]) f148_in(.(T40, T41)) -> U2(f148_in(T41), .(T40, T41)) U2(f148_out1(T42), .(T40, T41)) -> f148_out1(T42) f148_in(.(T50, T51)) -> f148_out1(T50) f141_in(T17) -> U3(f148_in(T17), T17) U3(f148_out1(T18), T17) -> U4(f131_in(T17), T17, T18) U4(f131_out1(T23), T17, T18) -> f141_out1(T18, T23) 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: F131_IN(T17) -> U1^1(f141_in(T17), T17) F131_IN(T17) -> F141_IN(T17) F148_IN(.(T40, T41)) -> U2^1(f148_in(T41), .(T40, T41)) F148_IN(.(T40, T41)) -> F148_IN(T41) F141_IN(T17) -> U3^1(f148_in(T17), T17) F141_IN(T17) -> F148_IN(T17) U3^1(f148_out1(T18), T17) -> U4^1(f131_in(T17), T17, T18) U3^1(f148_out1(T18), T17) -> F131_IN(T17) The TRS R consists of the following rules: f131_in(T17) -> U1(f141_in(T17), T17) U1(f141_out1(T18, T19), T17) -> f131_out1(.(T18, T19)) f131_in(T57) -> f131_out1([]) f148_in(.(T40, T41)) -> U2(f148_in(T41), .(T40, T41)) U2(f148_out1(T42), .(T40, T41)) -> f148_out1(T42) f148_in(.(T50, T51)) -> f148_out1(T50) f141_in(T17) -> U3(f148_in(T17), T17) U3(f148_out1(T18), T17) -> U4(f131_in(T17), T17, T18) U4(f131_out1(T23), T17, T18) -> f141_out1(T18, T23) 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: F148_IN(.(T40, T41)) -> F148_IN(T41) The TRS R consists of the following rules: f131_in(T17) -> U1(f141_in(T17), T17) U1(f141_out1(T18, T19), T17) -> f131_out1(.(T18, T19)) f131_in(T57) -> f131_out1([]) f148_in(.(T40, T41)) -> U2(f148_in(T41), .(T40, T41)) U2(f148_out1(T42), .(T40, T41)) -> f148_out1(T42) f148_in(.(T50, T51)) -> f148_out1(T50) f141_in(T17) -> U3(f148_in(T17), T17) U3(f148_out1(T18), T17) -> U4(f131_in(T17), T17, T18) U4(f131_out1(T23), T17, T18) -> f141_out1(T18, T23) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: F148_IN(.(T40, T41)) -> F148_IN(T41) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) 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: *F148_IN(.(T40, T41)) -> F148_IN(T41) The graph contains the following edges 1 > 1 ---------------------------------------- (51) YES ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F131_IN(T17) -> F141_IN(T17) F141_IN(T17) -> U3^1(f148_in(T17), T17) U3^1(f148_out1(T18), T17) -> F131_IN(T17) The TRS R consists of the following rules: f131_in(T17) -> U1(f141_in(T17), T17) U1(f141_out1(T18, T19), T17) -> f131_out1(.(T18, T19)) f131_in(T57) -> f131_out1([]) f148_in(.(T40, T41)) -> U2(f148_in(T41), .(T40, T41)) U2(f148_out1(T42), .(T40, T41)) -> f148_out1(T42) f148_in(.(T50, T51)) -> f148_out1(T50) f141_in(T17) -> U3(f148_in(T17), T17) U3(f148_out1(T18), T17) -> U4(f131_in(T17), T17, T18) U4(f131_out1(T23), T17, T18) -> f141_out1(T18, T23) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = F141_IN(.(T50, T51)) evaluates to t =F141_IN(.(T50, T51)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F141_IN(.(T50, T51)) -> U3^1(f148_in(.(T50, T51)), .(T50, T51)) with rule F141_IN(T17) -> U3^1(f148_in(T17), T17) at position [] and matcher [T17 / .(T50, T51)] U3^1(f148_in(.(T50, T51)), .(T50, T51)) -> U3^1(f148_out1(T50), .(T50, T51)) with rule f148_in(.(T50', T51')) -> f148_out1(T50') at position [0] and matcher [T50' / T50, T51' / T51] U3^1(f148_out1(T50), .(T50, T51)) -> F131_IN(.(T50, T51)) with rule U3^1(f148_out1(T18), T17') -> F131_IN(T17') at position [] and matcher [T18 / T50, T17' / .(T50, T51)] F131_IN(.(T50, T51)) -> F141_IN(.(T50, T51)) with rule F131_IN(T17) -> F141_IN(T17) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (54) NO ---------------------------------------- (55) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 134, "program": { "directives": [], "clauses": [ [ "(member X (. Y Xs))", "(member X Xs)" ], [ "(member X (. X Xs))", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(subset ([]) Ys)", null ] ] }, "graph": { "nodes": { "190": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (member T9 T8) (subset T10 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "191": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(subset T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "192": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T74 (. T72 T73))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T73" ], "free": [], "exprvars": [] } }, "171": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (member T9 T8) (subset T10 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "193": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "172": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (member T9 T8) (subset T10 T8))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(subset T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "194": { "goal": [{ "clause": 3, "scope": 1, "term": "(subset T1 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "195": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "196": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "175": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T26 T25) (subset T27 (. T24 T25)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [], "exprvars": [] } }, "197": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "176": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "198": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "177": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "199": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "134": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "178": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T31 (. T24 T25))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [], "exprvars": [] } }, "135": { "goal": [ { "clause": 2, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 3, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "179": { "goal": [ { "clause": 0, "scope": 3, "term": "(member T26 T25)" }, { "clause": 1, "scope": 3, "term": "(member T26 T25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "180": { "goal": [{ "clause": 0, "scope": 3, "term": "(member T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "181": { "goal": [{ "clause": 1, "scope": 3, "term": "(member T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "182": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T50 T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T49"], "free": [], "exprvars": [] } }, "183": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "142": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (member T9 T8) (subset T10 T8))" }, { "clause": 3, "scope": 1, "term": "(subset T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "187": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "188": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "145": { "goal": [{ "clause": 3, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [[ "(subset T1 T2)", "(subset (. X4 X5) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X4", "X5", "X6" ], "exprvars": [] } }, "189": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "200": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "168": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (member T9 T8) (subset T10 T8))" }, { "clause": 1, "scope": 2, "term": "(',' (member T9 T8) (subset T10 T8))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 3, "scope": 1, "term": "(subset T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 134, "to": 135, "label": "CASE" }, { "from": 135, "to": 142, "label": "EVAL with clause\nsubset(.(X4, X5), X6) :- ','(member(X4, X6), subset(X5, X6)).\nand substitutionX4 -> T9,\nX5 -> T10,\nT1 -> .(T9, T10),\nT2 -> T8,\nX6 -> T8,\nT6 -> T9,\nT7 -> T10" }, { "from": 135, "to": 145, "label": "EVAL-BACKTRACK" }, { "from": 142, "to": 168, "label": "CASE" }, { "from": 145, "to": 198, "label": "EVAL with clause\nsubset([], X72).\nand substitutionT1 -> [],\nT2 -> T83,\nX72 -> T83" }, { "from": 145, "to": 199, "label": "EVAL-BACKTRACK" }, { "from": 168, "to": 171, "label": "PARALLEL" }, { "from": 168, "to": 172, "label": "PARALLEL" }, { "from": 171, "to": 175, "label": "EVAL with clause\nmember(X19, .(X20, X21)) :- member(X19, X21).\nand substitutionT9 -> T26,\nX19 -> T26,\nX20 -> T24,\nX21 -> T25,\nT8 -> .(T24, T25),\nT23 -> T26,\nT10 -> T27" }, { "from": 171, "to": 176, "label": "EVAL-BACKTRACK" }, { "from": 172, "to": 190, "label": "PARALLEL" }, { "from": 172, "to": 191, "label": "PARALLEL" }, { "from": 175, "to": 177, "label": "SPLIT 1" }, { "from": 175, "to": 178, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nT25 is ground\nreplacements:T27 -> T31" }, { "from": 177, "to": 179, "label": "CASE" }, { "from": 178, "to": 134, "label": "INSTANCE with matching:\nT1 -> T31\nT2 -> .(T24, T25)" }, { "from": 179, "to": 180, "label": "PARALLEL" }, { "from": 179, "to": 181, "label": "PARALLEL" }, { "from": 180, "to": 182, "label": "EVAL with clause\nmember(X40, .(X41, X42)) :- member(X40, X42).\nand substitutionT26 -> T50,\nX40 -> T50,\nX41 -> T48,\nX42 -> T49,\nT25 -> .(T48, T49),\nT47 -> T50" }, { "from": 180, "to": 183, "label": "EVAL-BACKTRACK" }, { "from": 181, "to": 187, "label": "EVAL with clause\nmember(X50, .(X50, X51)).\nand substitutionT26 -> T58,\nX50 -> T58,\nX51 -> T59,\nT25 -> .(T58, T59)" }, { "from": 181, "to": 188, "label": "EVAL-BACKTRACK" }, { "from": 182, "to": 177, "label": "INSTANCE with matching:\nT26 -> T50\nT25 -> T49" }, { "from": 187, "to": 189, "label": "SUCCESS" }, { "from": 190, "to": 192, "label": "EVAL with clause\nmember(X63, .(X63, X64)).\nand substitutionT9 -> T72,\nX63 -> T72,\nX64 -> T73,\nT8 -> .(T72, T73),\nT10 -> T74" }, { "from": 190, "to": 193, "label": "EVAL-BACKTRACK" }, { "from": 191, "to": 194, "label": "FAILURE" }, { "from": 192, "to": 134, "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> .(T72, T73)" }, { "from": 194, "to": 195, "label": "EVAL with clause\nsubset([], X70).\nand substitutionT1 -> [],\nT8 -> T81,\nX70 -> T81" }, { "from": 194, "to": 196, "label": "EVAL-BACKTRACK" }, { "from": 195, "to": 197, "label": "SUCCESS" }, { "from": 198, "to": 200, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (56) Obligation: Triples: memberB(X1, .(X2, X3)) :- memberB(X1, X3). subsetA(.(X1, X2), .(X3, X4)) :- memberB(X1, X4). subsetA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetA(X2, .(X3, X4))). subsetA(.(X1, X2), .(X1, X3)) :- subsetA(X2, .(X1, X3)). Clauses: subsetcA(.(X1, X2), .(X3, X4)) :- ','(membercB(X1, X4), subsetcA(X2, .(X3, X4))). subsetcA(.(X1, X2), .(X1, X3)) :- subsetcA(X2, .(X1, X3)). subsetcA([], X1). subsetcA([], X1). membercB(X1, .(X2, X3)) :- membercB(X1, X3). membercB(X1, .(X1, X2)). Afs: subsetA(x1, x2) = subsetA(x2) ---------------------------------------- (57) 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: (f,b) memberB_in_2: (f,b) membercB_in_2: (f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, memberB_in_ag(X1, X4)) SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AG(X1, X4) MEMBERB_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberB_in_ag(X1, X3)) MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subsetA_in_ag(X2, .(X3, X4))) U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4)) SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3))) SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subsetA_in_ag(x1, x2) = subsetA_in_ag(x2) .(x1, x2) = .(x1, x2) memberB_in_ag(x1, x2) = memberB_in_ag(x2) membercB_in_ag(x1, x2) = membercB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) membercB_out_ag(x1, x2) = membercB_out_ag(x1, x2) SUBSETA_IN_AG(x1, x2) = SUBSETA_IN_AG(x2) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x4, x5) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x4, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x4, x5) U5_AG(x1, x2, x3, x4) = U5_AG(x1, x3, x4) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (58) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, memberB_in_ag(X1, X4)) SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBERB_IN_AG(X1, X4) MEMBERB_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, memberB_in_ag(X1, X3)) MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subsetA_in_ag(X2, .(X3, X4))) U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4)) SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subsetA_in_ag(X2, .(X1, X3))) SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subsetA_in_ag(x1, x2) = subsetA_in_ag(x2) .(x1, x2) = .(x1, x2) memberB_in_ag(x1, x2) = memberB_in_ag(x2) membercB_in_ag(x1, x2) = membercB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) membercB_out_ag(x1, x2) = membercB_out_ag(x1, x2) SUBSETA_IN_AG(x1, x2) = SUBSETA_IN_AG(x2) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x4, x5) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) U1_AG(x1, x2, x3, x4) = U1_AG(x2, x3, x4) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x4, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x3, x4, x5) U5_AG(x1, x2, x3, x4) = U5_AG(x1, x3, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (59) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (60) Complex Obligation (AND) ---------------------------------------- (61) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) The TRS R consists of the following rules: membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercB_in_ag(x1, x2) = membercB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) membercB_out_ag(x1, x2) = membercB_out_ag(x1, x2) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (62) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (63) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(X1, .(X2, X3)) -> MEMBERB_IN_AG(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBERB_IN_AG(x1, x2) = MEMBERB_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (64) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBERB_IN_AG(.(X2, X3)) -> MEMBERB_IN_AG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (66) 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: *MEMBERB_IN_AG(.(X2, X3)) -> MEMBERB_IN_AG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (67) YES ---------------------------------------- (68) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSETA_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, membercB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(X2, .(X3, X4)) SUBSETA_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSETA_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: membercB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, membercB_in_ag(X1, X3)) membercB_in_ag(X1, .(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) membercB_in_ag(x1, x2) = membercB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) membercB_out_ag(x1, x2) = membercB_out_ag(x1, x2) SUBSETA_IN_AG(x1, x2) = SUBSETA_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (69) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSETA_IN_AG(.(X3, X4)) -> U3_AG(X3, X4, membercB_in_ag(X4)) U3_AG(X3, X4, membercB_out_ag(X1, X4)) -> SUBSETA_IN_AG(.(X3, X4)) SUBSETA_IN_AG(.(X1, X3)) -> SUBSETA_IN_AG(.(X1, X3)) The TRS R consists of the following rules: membercB_in_ag(.(X2, X3)) -> U10_ag(X2, X3, membercB_in_ag(X3)) membercB_in_ag(.(X1, X2)) -> membercB_out_ag(X1, .(X1, X2)) U10_ag(X2, X3, membercB_out_ag(X1, X3)) -> membercB_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: membercB_in_ag(x0) U10_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (71) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 130, "program": { "directives": [], "clauses": [ [ "(member X (. Y Xs))", "(member X Xs)" ], [ "(member X (. X Xs))", null ], [ "(subset (. X Xs) Ys)", "(',' (member X Ys) (subset Xs Ys))" ], [ "(subset ([]) Ys)", null ] ] }, "graph": { "nodes": { "170": { "goal": [{ "clause": 1, "scope": 2, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "type": "Nodes", "140": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member T18 T17) (subset T19 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "173": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T42 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [], "exprvars": [] } }, "184": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "130": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "174": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "185": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "186": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "132": { "goal": [ { "clause": 2, "scope": 1, "term": "(subset T1 T2)" }, { "clause": 3, "scope": 1, "term": "(subset T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "143": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "167": { "goal": [ { "clause": 0, "scope": 2, "term": "(member T18 T17)" }, { "clause": 1, "scope": 2, "term": "(member T18 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "146": { "goal": [{ "clause": -1, "scope": -1, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "201": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "136": { "goal": [{ "clause": 2, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "147": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset T23 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "169": { "goal": [{ "clause": 0, "scope": 2, "term": "(member T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "202": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "137": { "goal": [{ "clause": 3, "scope": 1, "term": "(subset T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "203": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 130, "to": 132, "label": "CASE" }, { "from": 132, "to": 136, "label": "PARALLEL" }, { "from": 132, "to": 137, "label": "PARALLEL" }, { "from": 136, "to": 140, "label": "EVAL with clause\nsubset(.(X13, X14), X15) :- ','(member(X13, X15), subset(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 136, "to": 143, "label": "EVAL-BACKTRACK" }, { "from": 137, "to": 201, "label": "EVAL with clause\nsubset([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57" }, { "from": 137, "to": 202, "label": "EVAL-BACKTRACK" }, { "from": 140, "to": 146, "label": "SPLIT 1" }, { "from": 140, "to": 147, "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23" }, { "from": 146, "to": 167, "label": "CASE" }, { "from": 147, "to": 130, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17" }, { "from": 167, "to": 169, "label": "PARALLEL" }, { "from": 167, "to": 170, "label": "PARALLEL" }, { "from": 169, "to": 173, "label": "EVAL with clause\nmember(X34, .(X35, X36)) :- member(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42" }, { "from": 169, "to": 174, "label": "EVAL-BACKTRACK" }, { "from": 170, "to": 184, "label": "EVAL with clause\nmember(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)" }, { "from": 170, "to": 185, "label": "EVAL-BACKTRACK" }, { "from": 173, "to": 146, "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41" }, { "from": 184, "to": 186, "label": "SUCCESS" }, { "from": 201, "to": 203, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (72) Complex Obligation (AND) ---------------------------------------- (73) Obligation: Rules: f167_in(T17) -> f170_in(T17) :|: TRUE f170_out(x) -> f167_out(x) :|: TRUE f167_in(x1) -> f169_in(x1) :|: TRUE f169_out(x2) -> f167_out(x2) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_out(x3) -> f169_out(.(x4, x3)) :|: TRUE f169_in(x5) -> f174_in :|: TRUE f174_out -> f169_out(x6) :|: TRUE f146_out(x7) -> f173_out(x7) :|: TRUE f173_in(x8) -> f146_in(x8) :|: TRUE f146_in(x9) -> f167_in(x9) :|: TRUE f167_out(x10) -> f146_out(x10) :|: TRUE f130_in(T2) -> f132_in(T2) :|: TRUE f132_out(x11) -> f130_out(x11) :|: TRUE f136_out(x12) -> f132_out(x12) :|: TRUE f137_out(x13) -> f132_out(x13) :|: TRUE f132_in(x14) -> f136_in(x14) :|: TRUE f132_in(x15) -> f137_in(x15) :|: TRUE f136_in(x16) -> f140_in(x16) :|: TRUE f136_in(x17) -> f143_in :|: TRUE f140_out(x18) -> f136_out(x18) :|: TRUE f143_out -> f136_out(x19) :|: TRUE f147_out(x20) -> f140_out(x20) :|: TRUE f146_out(x21) -> f147_in(x21) :|: TRUE f140_in(x22) -> f146_in(x22) :|: TRUE Start term: f130_in(T2) ---------------------------------------- (74) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f167_in(x1) -> f169_in(x1) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_in(x8) -> f146_in(x8) :|: TRUE f146_in(x9) -> f167_in(x9) :|: TRUE ---------------------------------------- (75) Obligation: Rules: f167_in(x1) -> f169_in(x1) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_in(x8) -> f146_in(x8) :|: TRUE f146_in(x9) -> f167_in(x9) :|: TRUE ---------------------------------------- (76) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (77) Obligation: Rules: f167_in(.(T40:0, T41:0)) -> f167_in(T41:0) :|: TRUE ---------------------------------------- (78) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (79) Obligation: Rules: f167_in(.(T40:0, T41:0)) -> f167_in(T41:0) :|: TRUE ---------------------------------------- (80) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f167_in(.(T40:0, T41:0)) -> f167_in(T41:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (81) Obligation: Termination digraph: Nodes: (1) f167_in(.(T40:0, T41:0)) -> f167_in(T41:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (82) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: .(x1, x2) -> .(x2) ---------------------------------------- (83) Obligation: Rules: f167_in(.(T41:0)) -> f167_in(T41:0) :|: TRUE ---------------------------------------- (84) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f167_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (85) Obligation: Rules: f167_in(.(T41:0)) -> f167_in(T41:0) ---------------------------------------- (86) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: f167_in(.(T41:0)) -> f167_in(T41:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (88) 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: *f167_in(.(T41:0)) -> f167_in(T41:0) The graph contains the following edges 1 > 1 ---------------------------------------- (89) YES ---------------------------------------- (90) Obligation: Rules: f136_out(T2) -> f132_out(T2) :|: TRUE f137_out(x) -> f132_out(x) :|: TRUE f132_in(x1) -> f136_in(x1) :|: TRUE f132_in(x2) -> f137_in(x2) :|: TRUE f147_in(T17) -> f130_in(T17) :|: TRUE f130_out(x3) -> f147_out(x3) :|: TRUE f167_in(x4) -> f170_in(x4) :|: TRUE f170_out(x5) -> f167_out(x5) :|: TRUE f167_in(x6) -> f169_in(x6) :|: TRUE f169_out(x7) -> f167_out(x7) :|: TRUE f185_out -> f170_out(x8) :|: TRUE f170_in(.(T50, T51)) -> f184_in :|: TRUE f170_in(x9) -> f185_in :|: TRUE f184_out -> f170_out(.(x10, x11)) :|: TRUE f147_out(x12) -> f140_out(x12) :|: TRUE f146_out(x13) -> f147_in(x13) :|: TRUE f140_in(x14) -> f146_in(x14) :|: TRUE f136_in(x15) -> f140_in(x15) :|: TRUE f136_in(x16) -> f143_in :|: TRUE f140_out(x17) -> f136_out(x17) :|: TRUE f143_out -> f136_out(x18) :|: TRUE f146_in(x19) -> f167_in(x19) :|: TRUE f167_out(x20) -> f146_out(x20) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_out(x21) -> f169_out(.(x22, x21)) :|: TRUE f169_in(x23) -> f174_in :|: TRUE f174_out -> f169_out(x24) :|: TRUE f184_in -> f184_out :|: TRUE f146_out(x25) -> f173_out(x25) :|: TRUE f173_in(x26) -> f146_in(x26) :|: TRUE f130_in(x27) -> f132_in(x27) :|: TRUE f132_out(x28) -> f130_out(x28) :|: TRUE Start term: f130_in(T2) ---------------------------------------- (91) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f132_in(x1) -> f136_in(x1) :|: TRUE f147_in(T17) -> f130_in(T17) :|: TRUE f167_in(x4) -> f170_in(x4) :|: TRUE f170_out(x5) -> f167_out(x5) :|: TRUE f167_in(x6) -> f169_in(x6) :|: TRUE f169_out(x7) -> f167_out(x7) :|: TRUE f170_in(.(T50, T51)) -> f184_in :|: TRUE f184_out -> f170_out(.(x10, x11)) :|: TRUE f146_out(x13) -> f147_in(x13) :|: TRUE f140_in(x14) -> f146_in(x14) :|: TRUE f136_in(x15) -> f140_in(x15) :|: TRUE f146_in(x19) -> f167_in(x19) :|: TRUE f167_out(x20) -> f146_out(x20) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_out(x21) -> f169_out(.(x22, x21)) :|: TRUE f184_in -> f184_out :|: TRUE f146_out(x25) -> f173_out(x25) :|: TRUE f173_in(x26) -> f146_in(x26) :|: TRUE f130_in(x27) -> f132_in(x27) :|: TRUE ---------------------------------------- (92) Obligation: Rules: f132_in(x1) -> f136_in(x1) :|: TRUE f147_in(T17) -> f130_in(T17) :|: TRUE f167_in(x4) -> f170_in(x4) :|: TRUE f170_out(x5) -> f167_out(x5) :|: TRUE f167_in(x6) -> f169_in(x6) :|: TRUE f169_out(x7) -> f167_out(x7) :|: TRUE f170_in(.(T50, T51)) -> f184_in :|: TRUE f184_out -> f170_out(.(x10, x11)) :|: TRUE f146_out(x13) -> f147_in(x13) :|: TRUE f140_in(x14) -> f146_in(x14) :|: TRUE f136_in(x15) -> f140_in(x15) :|: TRUE f146_in(x19) -> f167_in(x19) :|: TRUE f167_out(x20) -> f146_out(x20) :|: TRUE f169_in(.(T40, T41)) -> f173_in(T41) :|: TRUE f173_out(x21) -> f169_out(.(x22, x21)) :|: TRUE f184_in -> f184_out :|: TRUE f146_out(x25) -> f173_out(x25) :|: TRUE f173_in(x26) -> f146_in(x26) :|: TRUE f130_in(x27) -> f132_in(x27) :|: TRUE