/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern subset1(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms]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, 0 ms] (46) AND (47) QDP (48) UsableRulesProof [EQUIVALENT, 0 ms] (49) QDP (50) QDPSizeChangeProof [EQUIVALENT, 0 ms] (51) YES (52) QDP (53) NonTerminationLoopProof [COMPLETE, 0 ms] (54) NO (55) PrologToIRSwTTransformerProof [SOUND, 0 ms] (56) AND (57) IRSwT (58) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (59) IRSwT (60) IntTRSCompressionProof [EQUIVALENT, 37 ms] (61) IRSwT (62) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (63) IRSwT (64) IRSwTTerminationDigraphProof [EQUIVALENT, 6 ms] (65) IRSwT (66) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (67) IRSwT (68) TempFilterProof [SOUND, 2 ms] (69) IRSwT (70) IRSwTToQDPProof [SOUND, 0 ms] (71) QDP (72) QDPSizeChangeProof [EQUIVALENT, 0 ms] (73) YES (74) IRSwT (75) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (76) IRSwT (77) IntTRSCompressionProof [EQUIVALENT, 9 ms] (78) IRSwT (79) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (80) IRSwT (81) PrologToDTProblemTransformerProof [SOUND, 0 ms] (82) TRIPLES (83) TriplesToPiDPProof [SOUND, 0 ms] (84) PiDP (85) DependencyGraphProof [EQUIVALENT, 0 ms] (86) AND (87) PiDP (88) UsableRulesProof [EQUIVALENT, 0 ms] (89) PiDP (90) PiDPToQDPProof [SOUND, 0 ms] (91) QDP (92) QDPSizeChangeProof [EQUIVALENT, 0 ms] (93) YES (94) PiDP (95) PiDPToQDPProof [SOUND, 0 ms] (96) QDP ---------------------------------------- (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). member1(X, .(Y, Xs)) :- member1(X, Xs). member1(X, .(X, Xs)). subset1(.(X, Xs), Ys) :- ','(member1(X, Ys), subset1(Xs, Ys)). subset1([], Ys). Query: subset1(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: subset1_in_2: (f,b) member1_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_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: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_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: SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) SUBSET1_IN_AG(.(X, Xs), Ys) -> MEMBER1_IN_AG(X, Ys) MEMBER1_IN_AG(X, .(Y, Xs)) -> U4_AG(X, Y, Xs, member1_in_ag(X, Xs)) MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_AG(X, Xs, Ys, subset1_in_ag(Xs, Ys)) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1, x2) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x3, x4) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4) U6_AG(x1, x2, x3, x4) = U6_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: SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) SUBSET1_IN_AG(.(X, Xs), Ys) -> MEMBER1_IN_AG(X, Ys) MEMBER1_IN_AG(X, .(Y, Xs)) -> U4_AG(X, Y, Xs, member1_in_ag(X, Xs)) MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_AG(X, Xs, Ys, subset1_in_ag(Xs, Ys)) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1, x2) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x3, x4) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4) U6_AG(x1, x2, x3, x4) = U6_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: MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1, x2) MEMBER1_IN_AG(x1, x2) = MEMBER1_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: MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1_IN_AG(x1, x2) = MEMBER1_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: MEMBER1_IN_AG(.(Y, Xs)) -> MEMBER1_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: *MEMBER1_IN_AG(.(Y, Xs)) -> MEMBER1_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: U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x3, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1, x2) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_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: U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) The TRS R consists of the following rules: member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) The argument filtering Pi contains the following mapping: member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4) member1_out_ag(x1, x2) = member1_out_ag(x1, x2) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_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: U5_AG(Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Ys) SUBSET1_IN_AG(Ys) -> U5_AG(Ys, member1_in_ag(Ys)) The TRS R consists of the following rules: member1_in_ag(.(Y, Xs)) -> U4_ag(Y, Xs, member1_in_ag(Xs)) member1_in_ag(.(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) The set Q consists of the following terms: member1_in_ag(x0) U4_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: subset1_in_2: (f,b) member1_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_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: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_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: SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) SUBSET1_IN_AG(.(X, Xs), Ys) -> MEMBER1_IN_AG(X, Ys) MEMBER1_IN_AG(X, .(Y, Xs)) -> U4_AG(X, Y, Xs, member1_in_ag(X, Xs)) MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_AG(X, Xs, Ys, subset1_in_ag(Xs, Ys)) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x3, x4) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) U6_AG(x1, x2, x3, x4) = U6_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: SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) SUBSET1_IN_AG(.(X, Xs), Ys) -> MEMBER1_IN_AG(X, Ys) MEMBER1_IN_AG(X, .(Y, Xs)) -> U4_AG(X, Y, Xs, member1_in_ag(X, Xs)) MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_AG(X, Xs, Ys, subset1_in_ag(Xs, Ys)) U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x3, x4) MEMBER1_IN_AG(x1, x2) = MEMBER1_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) U6_AG(x1, x2, x3, x4) = U6_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: MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1) MEMBER1_IN_AG(x1, x2) = MEMBER1_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: MEMBER1_IN_AG(X, .(Y, Xs)) -> MEMBER1_IN_AG(X, Xs) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1_IN_AG(x1, x2) = MEMBER1_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: MEMBER1_IN_AG(.(Y, Xs)) -> MEMBER1_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: *MEMBER1_IN_AG(.(Y, Xs)) -> MEMBER1_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: U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) The TRS R consists of the following rules: subset1_in_ag(.(X, Xs), Ys) -> U5_ag(X, Xs, Ys, member1_in_ag(X, Ys)) member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) U5_ag(X, Xs, Ys, member1_out_ag(X, Ys)) -> U6_ag(X, Xs, Ys, subset1_in_ag(Xs, Ys)) subset1_in_ag([], Ys) -> subset1_out_ag([], Ys) U6_ag(X, Xs, Ys, subset1_out_ag(Xs, Ys)) -> subset1_out_ag(.(X, Xs), Ys) The argument filtering Pi contains the following mapping: subset1_in_ag(x1, x2) = subset1_in_ag(x2) U5_ag(x1, x2, x3, x4) = U5_ag(x3, x4) member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) subset1_out_ag(x1, x2) = subset1_out_ag(x1) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_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: U5_AG(X, Xs, Ys, member1_out_ag(X, Ys)) -> SUBSET1_IN_AG(Xs, Ys) SUBSET1_IN_AG(.(X, Xs), Ys) -> U5_AG(X, Xs, Ys, member1_in_ag(X, Ys)) The TRS R consists of the following rules: member1_in_ag(X, .(Y, Xs)) -> U4_ag(X, Y, Xs, member1_in_ag(X, Xs)) member1_in_ag(X, .(X, Xs)) -> member1_out_ag(X, .(X, Xs)) U4_ag(X, Y, Xs, member1_out_ag(X, Xs)) -> member1_out_ag(X, .(Y, Xs)) The argument filtering Pi contains the following mapping: member1_in_ag(x1, x2) = member1_in_ag(x2) .(x1, x2) = .(x1, x2) U4_ag(x1, x2, x3, x4) = U4_ag(x4) member1_out_ag(x1, x2) = member1_out_ag(x1) SUBSET1_IN_AG(x1, x2) = SUBSET1_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_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: U5_AG(Ys, member1_out_ag(X)) -> SUBSET1_IN_AG(Ys) SUBSET1_IN_AG(Ys) -> U5_AG(Ys, member1_in_ag(Ys)) The TRS R consists of the following rules: member1_in_ag(.(Y, Xs)) -> U4_ag(member1_in_ag(Xs)) member1_in_ag(.(X, Xs)) -> member1_out_ag(X) U4_ag(member1_out_ag(X)) -> member1_out_ag(X) The set Q consists of the following terms: member1_in_ag(x0) U4_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (SOUND) By narrowing [LPAR04] the rule SUBSET1_IN_AG(Ys) -> U5_AG(Ys, member1_in_ag(Ys)) at position [1] we obtained the following new rules [LPAR04]: (SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), U4_ag(member1_in_ag(x1))),SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), U4_ag(member1_in_ag(x1)))) (SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), member1_out_ag(x0)),SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), member1_out_ag(x0))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: U5_AG(Ys, member1_out_ag(X)) -> SUBSET1_IN_AG(Ys) SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), U4_ag(member1_in_ag(x1))) SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), member1_out_ag(x0)) The TRS R consists of the following rules: member1_in_ag(.(Y, Xs)) -> U4_ag(member1_in_ag(Xs)) member1_in_ag(.(X, Xs)) -> member1_out_ag(X) U4_ag(member1_out_ag(X)) -> member1_out_ag(X) The set Q consists of the following terms: member1_in_ag(x0) U4_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule U5_AG(Ys, member1_out_ag(X)) -> SUBSET1_IN_AG(Ys) we obtained the following new rules [LPAR04]: (U5_AG(.(z0, z1), member1_out_ag(x1)) -> SUBSET1_IN_AG(.(z0, z1)),U5_AG(.(z0, z1), member1_out_ag(x1)) -> SUBSET1_IN_AG(.(z0, z1))) (U5_AG(.(z0, z1), member1_out_ag(z0)) -> SUBSET1_IN_AG(.(z0, z1)),U5_AG(.(z0, z1), member1_out_ag(z0)) -> SUBSET1_IN_AG(.(z0, z1))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), U4_ag(member1_in_ag(x1))) SUBSET1_IN_AG(.(x0, x1)) -> U5_AG(.(x0, x1), member1_out_ag(x0)) U5_AG(.(z0, z1), member1_out_ag(x1)) -> SUBSET1_IN_AG(.(z0, z1)) U5_AG(.(z0, z1), member1_out_ag(z0)) -> SUBSET1_IN_AG(.(z0, z1)) The TRS R consists of the following rules: member1_in_ag(.(Y, Xs)) -> U4_ag(member1_in_ag(Xs)) member1_in_ag(.(X, Xs)) -> member1_out_ag(X) U4_ag(member1_out_ag(X)) -> member1_out_ag(X) The set Q consists of the following terms: member1_in_ag(x0) U4_ag(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 5, "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 ], [ "(member1 X (. Y Xs))", "(member1 X Xs)" ], [ "(member1 X (. X Xs))", null ], [ "(subset1 (. X Xs) Ys)", "(',' (member1 X Ys) (subset1 Xs Ys))" ], [ "(subset1 ([]) Ys)", null ] ] }, "graph": { "nodes": { "24": { "goal": [{ "clause": 6, "scope": 1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "290": { "goal": [ { "clause": 4, "scope": 2, "term": "(member1 T18 T17)" }, { "clause": 5, "scope": 2, "term": "(member1 T18 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "291": { "goal": [{ "clause": 4, "scope": 2, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "292": { "goal": [{ "clause": 5, "scope": 2, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "type": "Nodes", "293": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T42 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [], "exprvars": [] } }, "195": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 T18 T17) (subset1 T19 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "250": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T23 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "294": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "295": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "296": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "297": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "298": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "299": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "300": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "202": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 6, "scope": 1, "term": "(subset1 T1 T2)" }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "249": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 24, "label": "PARALLEL" }, { "from": 6, "to": 25, "label": "PARALLEL" }, { "from": 24, "to": 195, "label": "EVAL with clause\nsubset1(.(X13, X14), X15) :- ','(member1(X13, X15), subset1(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 24, "to": 202, "label": "EVAL-BACKTRACK" }, { "from": 25, "to": 298, "label": "EVAL with clause\nsubset1([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57" }, { "from": 25, "to": 299, "label": "EVAL-BACKTRACK" }, { "from": 195, "to": 249, "label": "SPLIT 1" }, { "from": 195, "to": 250, "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23" }, { "from": 249, "to": 290, "label": "CASE" }, { "from": 250, "to": 5, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17" }, { "from": 290, "to": 291, "label": "PARALLEL" }, { "from": 290, "to": 292, "label": "PARALLEL" }, { "from": 291, "to": 293, "label": "EVAL with clause\nmember1(X34, .(X35, X36)) :- member1(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42" }, { "from": 291, "to": 294, "label": "EVAL-BACKTRACK" }, { "from": 292, "to": 295, "label": "EVAL with clause\nmember1(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)" }, { "from": 292, "to": 296, "label": "EVAL-BACKTRACK" }, { "from": 293, "to": 249, "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41" }, { "from": 295, "to": 297, "label": "SUCCESS" }, { "from": 298, "to": 300, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f5_in(T17) -> U1(f195_in(T17), T17) U1(f195_out1(T18, T19), T17) -> f5_out1(.(T18, T19)) f5_in(T57) -> f5_out1([]) f249_in(.(T40, T41)) -> U2(f249_in(T41), .(T40, T41)) U2(f249_out1(T42), .(T40, T41)) -> f249_out1(T42) f249_in(.(T50, T51)) -> f249_out1(T50) f195_in(T17) -> U3(f249_in(T17), T17) U3(f249_out1(T18), T17) -> U4(f5_in(T17), T17, T18) U4(f5_out1(T23), T17, T18) -> f195_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: F5_IN(T17) -> U1^1(f195_in(T17), T17) F5_IN(T17) -> F195_IN(T17) F249_IN(.(T40, T41)) -> U2^1(f249_in(T41), .(T40, T41)) F249_IN(.(T40, T41)) -> F249_IN(T41) F195_IN(T17) -> U3^1(f249_in(T17), T17) F195_IN(T17) -> F249_IN(T17) U3^1(f249_out1(T18), T17) -> U4^1(f5_in(T17), T17, T18) U3^1(f249_out1(T18), T17) -> F5_IN(T17) The TRS R consists of the following rules: f5_in(T17) -> U1(f195_in(T17), T17) U1(f195_out1(T18, T19), T17) -> f5_out1(.(T18, T19)) f5_in(T57) -> f5_out1([]) f249_in(.(T40, T41)) -> U2(f249_in(T41), .(T40, T41)) U2(f249_out1(T42), .(T40, T41)) -> f249_out1(T42) f249_in(.(T50, T51)) -> f249_out1(T50) f195_in(T17) -> U3(f249_in(T17), T17) U3(f249_out1(T18), T17) -> U4(f5_in(T17), T17, T18) U4(f5_out1(T23), T17, T18) -> f195_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: F249_IN(.(T40, T41)) -> F249_IN(T41) The TRS R consists of the following rules: f5_in(T17) -> U1(f195_in(T17), T17) U1(f195_out1(T18, T19), T17) -> f5_out1(.(T18, T19)) f5_in(T57) -> f5_out1([]) f249_in(.(T40, T41)) -> U2(f249_in(T41), .(T40, T41)) U2(f249_out1(T42), .(T40, T41)) -> f249_out1(T42) f249_in(.(T50, T51)) -> f249_out1(T50) f195_in(T17) -> U3(f249_in(T17), T17) U3(f249_out1(T18), T17) -> U4(f5_in(T17), T17, T18) U4(f5_out1(T23), T17, T18) -> f195_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: F249_IN(.(T40, T41)) -> F249_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: *F249_IN(.(T40, T41)) -> F249_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: F5_IN(T17) -> F195_IN(T17) F195_IN(T17) -> U3^1(f249_in(T17), T17) U3^1(f249_out1(T18), T17) -> F5_IN(T17) The TRS R consists of the following rules: f5_in(T17) -> U1(f195_in(T17), T17) U1(f195_out1(T18, T19), T17) -> f5_out1(.(T18, T19)) f5_in(T57) -> f5_out1([]) f249_in(.(T40, T41)) -> U2(f249_in(T41), .(T40, T41)) U2(f249_out1(T42), .(T40, T41)) -> f249_out1(T42) f249_in(.(T50, T51)) -> f249_out1(T50) f195_in(T17) -> U3(f249_in(T17), T17) U3(f249_out1(T18), T17) -> U4(f5_in(T17), T17, T18) U4(f5_out1(T23), T17, T18) -> f195_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 = F195_IN(.(T50, T51)) evaluates to t =F195_IN(.(T50, T51)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence F195_IN(.(T50, T51)) -> U3^1(f249_in(.(T50, T51)), .(T50, T51)) with rule F195_IN(T17) -> U3^1(f249_in(T17), T17) at position [] and matcher [T17 / .(T50, T51)] U3^1(f249_in(.(T50, T51)), .(T50, T51)) -> U3^1(f249_out1(T50), .(T50, T51)) with rule f249_in(.(T50', T51')) -> f249_out1(T50') at position [0] and matcher [T50' / T50, T51' / T51] U3^1(f249_out1(T50), .(T50, T51)) -> F5_IN(.(T50, T51)) with rule U3^1(f249_out1(T18), T17') -> F5_IN(T17') at position [] and matcher [T18 / T50, T17' / .(T50, T51)] F5_IN(.(T50, T51)) -> F195_IN(.(T50, T51)) with rule F5_IN(T17) -> F195_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) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "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 ], [ "(member1 X (. Y Xs))", "(member1 X Xs)" ], [ "(member1 X (. X Xs))", null ], [ "(subset1 (. X Xs) Ys)", "(',' (member1 X Ys) (subset1 Xs Ys))" ], [ "(subset1 ([]) Ys)", null ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 6, "scope": 1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "26": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 T18 T17) (subset1 T19 T17))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "27": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "251": { "goal": [ { "clause": 4, "scope": 2, "term": "(member1 T18 T17)" }, { "clause": 5, "scope": 2, "term": "(member1 T18 T17)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "284": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "252": { "goal": [{ "clause": 4, "scope": 2, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "285": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "253": { "goal": [{ "clause": 5, "scope": 2, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "286": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "254": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T42 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [], "exprvars": [] } }, "287": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "255": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "288": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "289": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "4": { "goal": [ { "clause": 6, "scope": 1, "term": "(subset1 T1 T2)" }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "247": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T18 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } }, "248": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T23 T17)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T17"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 22, "label": "PARALLEL" }, { "from": 4, "to": 23, "label": "PARALLEL" }, { "from": 22, "to": 26, "label": "EVAL with clause\nsubset1(.(X13, X14), X15) :- ','(member1(X13, X15), subset1(X14, X15)).\nand substitutionX13 -> T18,\nX14 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX15 -> T17,\nT15 -> T18,\nT16 -> T19" }, { "from": 22, "to": 27, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 287, "label": "EVAL with clause\nsubset1([], X51).\nand substitutionT1 -> [],\nT2 -> T57,\nX51 -> T57" }, { "from": 23, "to": 288, "label": "EVAL-BACKTRACK" }, { "from": 26, "to": 247, "label": "SPLIT 1" }, { "from": 26, "to": 248, "label": "SPLIT 2\nnew knowledge:\nT18 is ground\nT17 is ground\nreplacements:T19 -> T23" }, { "from": 247, "to": 251, "label": "CASE" }, { "from": 248, "to": 2, "label": "INSTANCE with matching:\nT1 -> T23\nT2 -> T17" }, { "from": 251, "to": 252, "label": "PARALLEL" }, { "from": 251, "to": 253, "label": "PARALLEL" }, { "from": 252, "to": 254, "label": "EVAL with clause\nmember1(X34, .(X35, X36)) :- member1(X34, X36).\nand substitutionT18 -> T42,\nX34 -> T42,\nX35 -> T40,\nX36 -> T41,\nT17 -> .(T40, T41),\nT39 -> T42" }, { "from": 252, "to": 255, "label": "EVAL-BACKTRACK" }, { "from": 253, "to": 284, "label": "EVAL with clause\nmember1(X44, .(X44, X45)).\nand substitutionT18 -> T50,\nX44 -> T50,\nX45 -> T51,\nT17 -> .(T50, T51)" }, { "from": 253, "to": 285, "label": "EVAL-BACKTRACK" }, { "from": 254, "to": 247, "label": "INSTANCE with matching:\nT18 -> T42\nT17 -> T41" }, { "from": 284, "to": 286, "label": "SUCCESS" }, { "from": 287, "to": 289, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (56) Complex Obligation (AND) ---------------------------------------- (57) Obligation: Rules: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f255_out -> f252_out(T17) :|: TRUE f254_out(x) -> f252_out(.(x1, x)) :|: TRUE f252_in(x2) -> f255_in :|: TRUE f251_out(x3) -> f247_out(x3) :|: TRUE f247_in(x4) -> f251_in(x4) :|: TRUE f251_in(x5) -> f252_in(x5) :|: TRUE f253_out(x6) -> f251_out(x6) :|: TRUE f252_out(x7) -> f251_out(x7) :|: TRUE f251_in(x8) -> f253_in(x8) :|: TRUE f254_in(x9) -> f247_in(x9) :|: TRUE f247_out(x10) -> f254_out(x10) :|: TRUE f2_in(T2) -> f4_in(T2) :|: TRUE f4_out(x11) -> f2_out(x11) :|: TRUE f4_in(x12) -> f23_in(x12) :|: TRUE f23_out(x13) -> f4_out(x13) :|: TRUE f4_in(x14) -> f22_in(x14) :|: TRUE f22_out(x15) -> f4_out(x15) :|: TRUE f22_in(x16) -> f27_in :|: TRUE f22_in(x17) -> f26_in(x17) :|: TRUE f27_out -> f22_out(x18) :|: TRUE f26_out(x19) -> f22_out(x19) :|: TRUE f248_out(x20) -> f26_out(x20) :|: TRUE f26_in(x21) -> f247_in(x21) :|: TRUE f247_out(x22) -> f248_in(x22) :|: TRUE Start term: f2_in(T2) ---------------------------------------- (58) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f247_in(x4) -> f251_in(x4) :|: TRUE f251_in(x5) -> f252_in(x5) :|: TRUE f254_in(x9) -> f247_in(x9) :|: TRUE ---------------------------------------- (59) Obligation: Rules: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f247_in(x4) -> f251_in(x4) :|: TRUE f251_in(x5) -> f252_in(x5) :|: TRUE f254_in(x9) -> f247_in(x9) :|: TRUE ---------------------------------------- (60) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (61) Obligation: Rules: f247_in(.(T40:0, T41:0)) -> f247_in(T41:0) :|: TRUE ---------------------------------------- (62) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (63) Obligation: Rules: f247_in(.(T40:0, T41:0)) -> f247_in(T41:0) :|: TRUE ---------------------------------------- (64) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f247_in(.(T40:0, T41:0)) -> f247_in(T41:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (65) Obligation: Termination digraph: Nodes: (1) f247_in(.(T40:0, T41:0)) -> f247_in(T41:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (66) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: .(x1, x2) -> .(x2) ---------------------------------------- (67) Obligation: Rules: f247_in(.(T41:0)) -> f247_in(T41:0) :|: TRUE ---------------------------------------- (68) TempFilterProof (SOUND) Used the following sort dictionary for filtering: f247_in(VARIABLE) .(VARIABLE) Removed predefined arithmetic. ---------------------------------------- (69) Obligation: Rules: f247_in(.(T41:0)) -> f247_in(T41:0) ---------------------------------------- (70) IRSwTToQDPProof (SOUND) Removed the integers and created a QDP-Problem. ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: f247_in(.(T41:0)) -> f247_in(T41:0) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (72) 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: *f247_in(.(T41:0)) -> f247_in(T41:0) The graph contains the following edges 1 > 1 ---------------------------------------- (73) YES ---------------------------------------- (74) Obligation: Rules: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f255_out -> f252_out(T17) :|: TRUE f254_out(x) -> f252_out(.(x1, x)) :|: TRUE f252_in(x2) -> f255_in :|: TRUE f4_in(T2) -> f23_in(T2) :|: TRUE f23_out(x3) -> f4_out(x3) :|: TRUE f4_in(x4) -> f22_in(x4) :|: TRUE f22_out(x5) -> f4_out(x5) :|: TRUE f2_out(x6) -> f248_out(x6) :|: TRUE f248_in(x7) -> f2_in(x7) :|: TRUE f2_in(x8) -> f4_in(x8) :|: TRUE f4_out(x9) -> f2_out(x9) :|: TRUE f254_in(x10) -> f247_in(x10) :|: TRUE f247_out(x11) -> f254_out(x11) :|: TRUE f251_out(x12) -> f247_out(x12) :|: TRUE f247_in(x13) -> f251_in(x13) :|: TRUE f284_in -> f284_out :|: TRUE f284_out -> f253_out(.(T50, T51)) :|: TRUE f285_out -> f253_out(x14) :|: TRUE f253_in(x15) -> f285_in :|: TRUE f253_in(.(x16, x17)) -> f284_in :|: TRUE f248_out(x18) -> f26_out(x18) :|: TRUE f26_in(x19) -> f247_in(x19) :|: TRUE f247_out(x20) -> f248_in(x20) :|: TRUE f22_in(x21) -> f27_in :|: TRUE f22_in(x22) -> f26_in(x22) :|: TRUE f27_out -> f22_out(x23) :|: TRUE f26_out(x24) -> f22_out(x24) :|: TRUE f251_in(x25) -> f252_in(x25) :|: TRUE f253_out(x26) -> f251_out(x26) :|: TRUE f252_out(x27) -> f251_out(x27) :|: TRUE f251_in(x28) -> f253_in(x28) :|: TRUE Start term: f2_in(T2) ---------------------------------------- (75) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f254_out(x) -> f252_out(.(x1, x)) :|: TRUE f4_in(x4) -> f22_in(x4) :|: TRUE f248_in(x7) -> f2_in(x7) :|: TRUE f2_in(x8) -> f4_in(x8) :|: TRUE f254_in(x10) -> f247_in(x10) :|: TRUE f247_out(x11) -> f254_out(x11) :|: TRUE f251_out(x12) -> f247_out(x12) :|: TRUE f247_in(x13) -> f251_in(x13) :|: TRUE f284_in -> f284_out :|: TRUE f284_out -> f253_out(.(T50, T51)) :|: TRUE f253_in(.(x16, x17)) -> f284_in :|: TRUE f26_in(x19) -> f247_in(x19) :|: TRUE f247_out(x20) -> f248_in(x20) :|: TRUE f22_in(x22) -> f26_in(x22) :|: TRUE f251_in(x25) -> f252_in(x25) :|: TRUE f253_out(x26) -> f251_out(x26) :|: TRUE f252_out(x27) -> f251_out(x27) :|: TRUE f251_in(x28) -> f253_in(x28) :|: TRUE ---------------------------------------- (76) Obligation: Rules: f252_in(.(T40, T41)) -> f254_in(T41) :|: TRUE f254_out(x) -> f252_out(.(x1, x)) :|: TRUE f4_in(x4) -> f22_in(x4) :|: TRUE f248_in(x7) -> f2_in(x7) :|: TRUE f2_in(x8) -> f4_in(x8) :|: TRUE f254_in(x10) -> f247_in(x10) :|: TRUE f247_out(x11) -> f254_out(x11) :|: TRUE f251_out(x12) -> f247_out(x12) :|: TRUE f247_in(x13) -> f251_in(x13) :|: TRUE f284_in -> f284_out :|: TRUE f284_out -> f253_out(.(T50, T51)) :|: TRUE f253_in(.(x16, x17)) -> f284_in :|: TRUE f26_in(x19) -> f247_in(x19) :|: TRUE f247_out(x20) -> f248_in(x20) :|: TRUE f22_in(x22) -> f26_in(x22) :|: TRUE f251_in(x25) -> f252_in(x25) :|: TRUE f253_out(x26) -> f251_out(x26) :|: TRUE f252_out(x27) -> f251_out(x27) :|: TRUE f251_in(x28) -> f253_in(x28) :|: TRUE ---------------------------------------- (77) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (78) Obligation: Rules: f251_in(.(x16:0, x17:0)) -> f251_out(.(T50:0, T51:0)) :|: TRUE f251_out(x12:0) -> f251_in(x12:0) :|: TRUE f251_out(x) -> f251_out(.(x1, x)) :|: TRUE f251_in(.(T40:0, T41:0)) -> f251_in(T41:0) :|: TRUE ---------------------------------------- (79) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (80) Obligation: Rules: f251_in(.(x16:0, x17:0)) -> f251_out(.(T50:0, T51:0)) :|: TRUE f251_out(x12:0) -> f251_in(x12:0) :|: TRUE f251_out(x) -> f251_out(.(x1, x)) :|: TRUE f251_in(.(T40:0, T41:0)) -> f251_in(T41:0) :|: TRUE ---------------------------------------- (81) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "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 ], [ "(member1 X (. Y Xs))", "(member1 X Xs)" ], [ "(member1 X (. X Xs))", null ], [ "(subset1 (. X Xs) Ys)", "(',' (member1 X Ys) (subset1 Xs Ys))" ], [ "(subset1 ([]) Ys)", null ] ] }, "graph": { "nodes": { "270": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "type": "Nodes", "271": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T31 (. T24 T25))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [], "exprvars": [] } }, "272": { "goal": [ { "clause": 4, "scope": 3, "term": "(member1 T26 T25)" }, { "clause": 5, "scope": 3, "term": "(member1 T26 T25)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "273": { "goal": [{ "clause": 4, "scope": 3, "term": "(member1 T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "274": { "goal": [{ "clause": 5, "scope": 3, "term": "(member1 T26 T25)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T25"], "free": [], "exprvars": [] } }, "275": { "goal": [{ "clause": -1, "scope": -1, "term": "(member1 T50 T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T49"], "free": [], "exprvars": [] } }, "276": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "277": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "278": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "279": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "280": { "goal": [{ "clause": 5, "scope": 2, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "281": { "goal": [ { "clause": -1, "scope": 2, "term": null }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "282": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T74 (. T72 T73))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T73" ], "free": [], "exprvars": [] } }, "283": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "221": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "244": { "goal": [ { "clause": 4, "scope": 2, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }, { "clause": 5, "scope": 2, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "3": { "goal": [ { "clause": 6, "scope": 1, "term": "(subset1 T1 T2)" }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "245": { "goal": [{ "clause": 4, "scope": 2, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "224": { "goal": [{ "clause": 7, "scope": 1, "term": "(subset1 T1 T2)" }], "kb": { "nonunifying": [[ "(subset1 T1 T2)", "(subset1 (. X4 X5) X6)" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X4", "X5", "X6" ], "exprvars": [] } }, "246": { "goal": [ { "clause": 5, "scope": 2, "term": "(',' (member1 T9 T8) (subset1 T10 T8))" }, { "clause": -1, "scope": 2, "term": null }, { "clause": 7, "scope": 1, "term": "(subset1 T1 T8)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "268": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (member1 T26 T25) (subset1 T27 (. T24 T25)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T24", "T25" ], "free": [], "exprvars": [] } }, "301": { "goal": [{ "clause": 7, "scope": 1, "term": "(subset1 T1 T8)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T8"], "free": [], "exprvars": [] } }, "269": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "302": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "303": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "304": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "305": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "306": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "307": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 3, "label": "CASE" }, { "from": 3, "to": 221, "label": "EVAL with clause\nsubset1(.(X4, X5), X6) :- ','(member1(X4, X6), subset1(X5, X6)).\nand substitutionX4 -> T9,\nX5 -> T10,\nT1 -> .(T9, T10),\nT2 -> T8,\nX6 -> T8,\nT6 -> T9,\nT7 -> T10" }, { "from": 3, "to": 224, "label": "EVAL-BACKTRACK" }, { "from": 221, "to": 244, "label": "CASE" }, { "from": 224, "to": 305, "label": "EVAL with clause\nsubset1([], X72).\nand substitutionT1 -> [],\nT2 -> T83,\nX72 -> T83" }, { "from": 224, "to": 306, "label": "EVAL-BACKTRACK" }, { "from": 244, "to": 245, "label": "PARALLEL" }, { "from": 244, "to": 246, "label": "PARALLEL" }, { "from": 245, "to": 268, "label": "EVAL with clause\nmember1(X19, .(X20, X21)) :- member1(X19, X21).\nand substitutionT9 -> T26,\nX19 -> T26,\nX20 -> T24,\nX21 -> T25,\nT8 -> .(T24, T25),\nT23 -> T26,\nT10 -> T27" }, { "from": 245, "to": 269, "label": "EVAL-BACKTRACK" }, { "from": 246, "to": 280, "label": "PARALLEL" }, { "from": 246, "to": 281, "label": "PARALLEL" }, { "from": 268, "to": 270, "label": "SPLIT 1" }, { "from": 268, "to": 271, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nT25 is ground\nreplacements:T27 -> T31" }, { "from": 270, "to": 272, "label": "CASE" }, { "from": 271, "to": 1, "label": "INSTANCE with matching:\nT1 -> T31\nT2 -> .(T24, T25)" }, { "from": 272, "to": 273, "label": "PARALLEL" }, { "from": 272, "to": 274, "label": "PARALLEL" }, { "from": 273, "to": 275, "label": "EVAL with clause\nmember1(X40, .(X41, X42)) :- member1(X40, X42).\nand substitutionT26 -> T50,\nX40 -> T50,\nX41 -> T48,\nX42 -> T49,\nT25 -> .(T48, T49),\nT47 -> T50" }, { "from": 273, "to": 276, "label": "EVAL-BACKTRACK" }, { "from": 274, "to": 277, "label": "EVAL with clause\nmember1(X50, .(X50, X51)).\nand substitutionT26 -> T58,\nX50 -> T58,\nX51 -> T59,\nT25 -> .(T58, T59)" }, { "from": 274, "to": 278, "label": "EVAL-BACKTRACK" }, { "from": 275, "to": 270, "label": "INSTANCE with matching:\nT26 -> T50\nT25 -> T49" }, { "from": 277, "to": 279, "label": "SUCCESS" }, { "from": 280, "to": 282, "label": "EVAL with clause\nmember1(X63, .(X63, X64)).\nand substitutionT9 -> T72,\nX63 -> T72,\nX64 -> T73,\nT8 -> .(T72, T73),\nT10 -> T74" }, { "from": 280, "to": 283, "label": "EVAL-BACKTRACK" }, { "from": 281, "to": 301, "label": "FAILURE" }, { "from": 282, "to": 1, "label": "INSTANCE with matching:\nT1 -> T74\nT2 -> .(T72, T73)" }, { "from": 301, "to": 302, "label": "EVAL with clause\nsubset1([], X70).\nand substitutionT1 -> [],\nT8 -> T81,\nX70 -> T81" }, { "from": 301, "to": 303, "label": "EVAL-BACKTRACK" }, { "from": 302, "to": 304, "label": "SUCCESS" }, { "from": 305, "to": 307, "label": "SUCCESS" } ], "type": "Graph" } } ---------------------------------------- (82) Obligation: Triples: member1B(X1, .(X2, X3)) :- member1B(X1, X3). subset1A(.(X1, X2), .(X3, X4)) :- member1B(X1, X4). subset1A(.(X1, X2), .(X3, X4)) :- ','(member1cB(X1, X4), subset1A(X2, .(X3, X4))). subset1A(.(X1, X2), .(X1, X3)) :- subset1A(X2, .(X1, X3)). Clauses: subset1cA(.(X1, X2), .(X3, X4)) :- ','(member1cB(X1, X4), subset1cA(X2, .(X3, X4))). subset1cA(.(X1, X2), .(X1, X3)) :- subset1cA(X2, .(X1, X3)). subset1cA([], X1). subset1cA([], X1). member1cB(X1, .(X2, X3)) :- member1cB(X1, X3). member1cB(X1, .(X1, X2)). Afs: subset1A(x1, x2) = subset1A(x2) ---------------------------------------- (83) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: subset1A_in_2: (f,b) member1B_in_2: (f,b) member1cB_in_2: (f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, member1B_in_ag(X1, X4)) SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBER1B_IN_AG(X1, X4) MEMBER1B_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, member1B_in_ag(X1, X3)) MEMBER1B_IN_AG(X1, .(X2, X3)) -> MEMBER1B_IN_AG(X1, X3) SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, member1cB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, member1cB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subset1A_in_ag(X2, .(X3, X4))) U3_AG(X1, X2, X3, X4, member1cB_out_ag(X1, X4)) -> SUBSET1A_IN_AG(X2, .(X3, X4)) SUBSET1A_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subset1A_in_ag(X2, .(X1, X3))) SUBSET1A_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSET1A_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: member1cB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, member1cB_in_ag(X1, X3)) member1cB_in_ag(X1, .(X1, X2)) -> member1cB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, member1cB_out_ag(X1, X3)) -> member1cB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subset1A_in_ag(x1, x2) = subset1A_in_ag(x2) .(x1, x2) = .(x1, x2) member1B_in_ag(x1, x2) = member1B_in_ag(x2) member1cB_in_ag(x1, x2) = member1cB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) member1cB_out_ag(x1, x2) = member1cB_out_ag(x1, x2) SUBSET1A_IN_AG(x1, x2) = SUBSET1A_IN_AG(x2) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x4, x5) MEMBER1B_IN_AG(x1, x2) = MEMBER1B_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 ---------------------------------------- (84) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> U2_AG(X1, X2, X3, X4, member1B_in_ag(X1, X4)) SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> MEMBER1B_IN_AG(X1, X4) MEMBER1B_IN_AG(X1, .(X2, X3)) -> U1_AG(X1, X2, X3, member1B_in_ag(X1, X3)) MEMBER1B_IN_AG(X1, .(X2, X3)) -> MEMBER1B_IN_AG(X1, X3) SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, member1cB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, member1cB_out_ag(X1, X4)) -> U4_AG(X1, X2, X3, X4, subset1A_in_ag(X2, .(X3, X4))) U3_AG(X1, X2, X3, X4, member1cB_out_ag(X1, X4)) -> SUBSET1A_IN_AG(X2, .(X3, X4)) SUBSET1A_IN_AG(.(X1, X2), .(X1, X3)) -> U5_AG(X1, X2, X3, subset1A_in_ag(X2, .(X1, X3))) SUBSET1A_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSET1A_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: member1cB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, member1cB_in_ag(X1, X3)) member1cB_in_ag(X1, .(X1, X2)) -> member1cB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, member1cB_out_ag(X1, X3)) -> member1cB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: subset1A_in_ag(x1, x2) = subset1A_in_ag(x2) .(x1, x2) = .(x1, x2) member1B_in_ag(x1, x2) = member1B_in_ag(x2) member1cB_in_ag(x1, x2) = member1cB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) member1cB_out_ag(x1, x2) = member1cB_out_ag(x1, x2) SUBSET1A_IN_AG(x1, x2) = SUBSET1A_IN_AG(x2) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x3, x4, x5) MEMBER1B_IN_AG(x1, x2) = MEMBER1B_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 ---------------------------------------- (85) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (86) Complex Obligation (AND) ---------------------------------------- (87) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1B_IN_AG(X1, .(X2, X3)) -> MEMBER1B_IN_AG(X1, X3) The TRS R consists of the following rules: member1cB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, member1cB_in_ag(X1, X3)) member1cB_in_ag(X1, .(X1, X2)) -> member1cB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, member1cB_out_ag(X1, X3)) -> member1cB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_ag(x1, x2) = member1cB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) member1cB_out_ag(x1, x2) = member1cB_out_ag(x1, x2) MEMBER1B_IN_AG(x1, x2) = MEMBER1B_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (88) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (89) Obligation: Pi DP problem: The TRS P consists of the following rules: MEMBER1B_IN_AG(X1, .(X2, X3)) -> MEMBER1B_IN_AG(X1, X3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) MEMBER1B_IN_AG(x1, x2) = MEMBER1B_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (90) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: MEMBER1B_IN_AG(.(X2, X3)) -> MEMBER1B_IN_AG(X3) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (92) 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: *MEMBER1B_IN_AG(.(X2, X3)) -> MEMBER1B_IN_AG(X3) The graph contains the following edges 1 > 1 ---------------------------------------- (93) YES ---------------------------------------- (94) Obligation: Pi DP problem: The TRS P consists of the following rules: SUBSET1A_IN_AG(.(X1, X2), .(X3, X4)) -> U3_AG(X1, X2, X3, X4, member1cB_in_ag(X1, X4)) U3_AG(X1, X2, X3, X4, member1cB_out_ag(X1, X4)) -> SUBSET1A_IN_AG(X2, .(X3, X4)) SUBSET1A_IN_AG(.(X1, X2), .(X1, X3)) -> SUBSET1A_IN_AG(X2, .(X1, X3)) The TRS R consists of the following rules: member1cB_in_ag(X1, .(X2, X3)) -> U10_ag(X1, X2, X3, member1cB_in_ag(X1, X3)) member1cB_in_ag(X1, .(X1, X2)) -> member1cB_out_ag(X1, .(X1, X2)) U10_ag(X1, X2, X3, member1cB_out_ag(X1, X3)) -> member1cB_out_ag(X1, .(X2, X3)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) member1cB_in_ag(x1, x2) = member1cB_in_ag(x2) U10_ag(x1, x2, x3, x4) = U10_ag(x2, x3, x4) member1cB_out_ag(x1, x2) = member1cB_out_ag(x1, x2) SUBSET1A_IN_AG(x1, x2) = SUBSET1A_IN_AG(x2) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (95) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: SUBSET1A_IN_AG(.(X3, X4)) -> U3_AG(X3, X4, member1cB_in_ag(X4)) U3_AG(X3, X4, member1cB_out_ag(X1, X4)) -> SUBSET1A_IN_AG(.(X3, X4)) SUBSET1A_IN_AG(.(X1, X3)) -> SUBSET1A_IN_AG(.(X1, X3)) The TRS R consists of the following rules: member1cB_in_ag(.(X2, X3)) -> U10_ag(X2, X3, member1cB_in_ag(X3)) member1cB_in_ag(.(X1, X2)) -> member1cB_out_ag(X1, .(X1, X2)) U10_ag(X2, X3, member1cB_out_ag(X1, X3)) -> member1cB_out_ag(X1, .(X2, X3)) The set Q consists of the following terms: member1cB_in_ag(x0) U10_ag(x0, x1, x2) We have to consider all (P,Q,R)-chains.