MAYBE proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern append3(a,a,a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0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) PrologToTRSTransformerProof [SOUND, 0 ms] (38) QTRS (39) QTRSRRRProof [EQUIVALENT, 84 ms] (40) QTRS (41) QTRSRRRProof [EQUIVALENT, 0 ms] (42) QTRS (43) Overlay + Local Confluence [EQUIVALENT, 0 ms] (44) QTRS (45) DependencyPairsProof [EQUIVALENT, 0 ms] (46) QDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) QDP (49) QReductionProof [EQUIVALENT, 0 ms] (50) QDP (51) PrologToDTProblemTransformerProof [SOUND, 0 ms] (52) TRIPLES (53) TriplesToPiDPProof [SOUND, 0 ms] (54) PiDP (55) DependencyGraphProof [EQUIVALENT, 0 ms] (56) AND (57) PiDP (58) UsableRulesProof [EQUIVALENT, 0 ms] (59) PiDP (60) PiDPToQDPProof [SOUND, 0 ms] (61) QDP (62) PiDP (63) UsableRulesProof [EQUIVALENT, 0 ms] (64) PiDP (65) PrologToIRSwTTransformerProof [SOUND, 0 ms] (66) AND (67) IRSwT (68) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (69) TRUE (70) IRSwT (71) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (72) IRSwT (73) IntTRSCompressionProof [EQUIVALENT, 19 ms] (74) IRSwT (75) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (76) IRSwT (77) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (78) IRSwT (79) FilterProof [EQUIVALENT, 0 ms] (80) IntTRS (81) IntTRSNonPeriodicNontermProof [COMPLETE, 0 ms] (82) NO ---------------------------------------- (0) Obligation: Clauses: append([], L, L). append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3). append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)). Query: append3(a,a,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: append3_in_4: (f,f,f,b) append_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: APPEND3_IN_AAAG(A, B, C, D) -> U2_AAAG(A, B, C, D, append_in_aaa(A, B, E)) APPEND3_IN_AAAG(A, B, C, D) -> APPEND_IN_AAA(A, B, E) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> U3_AAAG(A, B, C, D, append_in_aag(E, C, D)) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> APPEND_IN_AAG(E, C, D) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> U1_AAG(H, L1, L2, L3, append_in_aag(L1, L2, L3)) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) APPEND3_IN_AAAG(x1, x2, x3, x4) = APPEND3_IN_AAAG(x4) U2_AAAG(x1, x2, x3, x4, x5) = U2_AAAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U3_AAAG(x1, x2, x3, x4, x5) = U3_AAAG(x1, x4, x5) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND3_IN_AAAG(A, B, C, D) -> U2_AAAG(A, B, C, D, append_in_aaa(A, B, E)) APPEND3_IN_AAAG(A, B, C, D) -> APPEND_IN_AAA(A, B, E) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> U3_AAAG(A, B, C, D, append_in_aag(E, C, D)) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> APPEND_IN_AAG(E, C, D) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> U1_AAG(H, L1, L2, L3, append_in_aag(L1, L2, L3)) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) APPEND3_IN_AAAG(x1, x2, x3, x4) = APPEND3_IN_AAAG(x4) U2_AAAG(x1, x2, x3, x4, x5) = U2_AAAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U3_AAAG(x1, x2, x3, x4, x5) = U3_AAAG(x1, x4, x5) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 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: APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 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: APPEND_IN_AAG(.(L3)) -> APPEND_IN_AAG(L3) 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: *APPEND_IN_AAG(.(L3)) -> APPEND_IN_AAG(L3) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x4, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2, x3) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x4, x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3, x4) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. 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: append3_in_4: (f,f,f,b) append_in_3: (f,f,f) (f,f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (20) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) ---------------------------------------- (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: APPEND3_IN_AAAG(A, B, C, D) -> U2_AAAG(A, B, C, D, append_in_aaa(A, B, E)) APPEND3_IN_AAAG(A, B, C, D) -> APPEND_IN_AAA(A, B, E) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> U3_AAAG(A, B, C, D, append_in_aag(E, C, D)) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> APPEND_IN_AAG(E, C, D) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> U1_AAG(H, L1, L2, L3, append_in_aag(L1, L2, L3)) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) APPEND3_IN_AAAG(x1, x2, x3, x4) = APPEND3_IN_AAAG(x4) U2_AAAG(x1, x2, x3, x4, x5) = U2_AAAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U3_AAAG(x1, x2, x3, x4, x5) = U3_AAAG(x1, x5) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND3_IN_AAAG(A, B, C, D) -> U2_AAAG(A, B, C, D, append_in_aaa(A, B, E)) APPEND3_IN_AAAG(A, B, C, D) -> APPEND_IN_AAA(A, B, E) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> U1_AAA(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> U3_AAAG(A, B, C, D, append_in_aag(E, C, D)) U2_AAAG(A, B, C, D, append_out_aaa(A, B, E)) -> APPEND_IN_AAG(E, C, D) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> U1_AAG(H, L1, L2, L3, append_in_aag(L1, L2, L3)) APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) APPEND3_IN_AAAG(x1, x2, x3, x4) = APPEND3_IN_AAAG(x4) U2_AAAG(x1, x2, x3, x4, x5) = U2_AAAG(x4, x5) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA U1_AAA(x1, x2, x3, x4, x5) = U1_AAA(x5) U3_AAAG(x1, x2, x3, x4, x5) = U3_AAAG(x1, x5) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (24) Complex Obligation (AND) ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 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: APPEND_IN_AAG(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAG(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAG(x1, x2, x3) = APPEND_IN_AAG(x3) 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: APPEND_IN_AAG(.(L3)) -> APPEND_IN_AAG(L3) 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: *APPEND_IN_AAG(.(L3)) -> APPEND_IN_AAG(L3) The graph contains the following edges 1 > 1 ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) The TRS R consists of the following rules: append3_in_aaag(A, B, C, D) -> U2_aaag(A, B, C, D, append_in_aaa(A, B, E)) append_in_aaa([], L, L) -> append_out_aaa([], L, L) append_in_aaa(.(H, L1), L2, .(H, L3)) -> U1_aaa(H, L1, L2, L3, append_in_aaa(L1, L2, L3)) U1_aaa(H, L1, L2, L3, append_out_aaa(L1, L2, L3)) -> append_out_aaa(.(H, L1), L2, .(H, L3)) U2_aaag(A, B, C, D, append_out_aaa(A, B, E)) -> U3_aaag(A, B, C, D, append_in_aag(E, C, D)) append_in_aag([], L, L) -> append_out_aag([], L, L) append_in_aag(.(H, L1), L2, .(H, L3)) -> U1_aag(H, L1, L2, L3, append_in_aag(L1, L2, L3)) U1_aag(H, L1, L2, L3, append_out_aag(L1, L2, L3)) -> append_out_aag(.(H, L1), L2, .(H, L3)) U3_aaag(A, B, C, D, append_out_aag(E, C, D)) -> append3_out_aaag(A, B, C, D) The argument filtering Pi contains the following mapping: append3_in_aaag(x1, x2, x3, x4) = append3_in_aaag(x4) U2_aaag(x1, x2, x3, x4, x5) = U2_aaag(x4, x5) append_in_aaa(x1, x2, x3) = append_in_aaa append_out_aaa(x1, x2, x3) = append_out_aaa(x1) U1_aaa(x1, x2, x3, x4, x5) = U1_aaa(x5) .(x1, x2) = .(x2) U3_aaag(x1, x2, x3, x4, x5) = U3_aaag(x1, x5) append_in_aag(x1, x2, x3) = append_in_aag(x3) append_out_aag(x1, x2, x3) = append_out_aag(x1, x2) U1_aag(x1, x2, x3, x4, x5) = U1_aag(x5) append3_out_aaag(x1, x2, x3, x4) = append3_out_aaag(x1, x3) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_AAA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_AAA(x1, x2, x3) = APPEND_IN_AAA 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: APPEND_IN_AAA -> APPEND_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (37) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 2, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append3 A B C D)", "(',' (append A B E) (append E C D))" ] ] }, "graph": { "nodes": { "190": { "goal": [ { "clause": 0, "scope": 2, "term": "(append T22 T23 X16)" }, { "clause": 1, "scope": 2, "term": "(append T22 T23 X16)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "191": { "goal": [{ "clause": 0, "scope": 2, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "192": { "goal": [{ "clause": 1, "scope": 2, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "type": "Nodes", "195": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "185": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T22 T23 X16) (append X16 T24 T21))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": ["X16"], "exprvars": [] } }, "196": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "197": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "188": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(append3 T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "189": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "200": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T45 T46 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X40"], "exprvars": [] } }, "201": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "212": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T68 T69 T67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T67"], "free": [], "exprvars": [] } }, "202": { "goal": [ { "clause": 0, "scope": 3, "term": "(append T27 T28 T21)" }, { "clause": 1, "scope": 3, "term": "(append T27 T28 T21)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "213": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "203": { "goal": [{ "clause": 0, "scope": 3, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "204": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "8": { "goal": [{ "clause": 2, "scope": 1, "term": "(append3 T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "206": { "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": 2, "to": 8, "label": "CASE" }, { "from": 8, "to": 185, "label": "ONLY EVAL with clause\nappend3(X12, X13, X14, X15) :- ','(append(X12, X13, X16), append(X16, X14, X15)).\nand substitutionT1 -> T22,\nX12 -> T22,\nT2 -> T23,\nX13 -> T23,\nT3 -> T24,\nX14 -> T24,\nT4 -> T21,\nX15 -> T21,\nT18 -> T22,\nT19 -> T23,\nT20 -> T24" }, { "from": 185, "to": 188, "label": "SPLIT 1" }, { "from": 185, "to": 189, "label": "SPLIT 2\nreplacements:X16 -> T27,\nT24 -> T28" }, { "from": 188, "to": 190, "label": "CASE" }, { "from": 189, "to": 202, "label": "CASE" }, { "from": 190, "to": 191, "label": "PARALLEL" }, { "from": 190, "to": 192, "label": "PARALLEL" }, { "from": 191, "to": 195, "label": "EVAL with clause\nappend([], X25, X25).\nand substitutionT22 -> [],\nT23 -> T35,\nX25 -> T35,\nX16 -> T35" }, { "from": 191, "to": 196, "label": "EVAL-BACKTRACK" }, { "from": 192, "to": 200, "label": "EVAL with clause\nappend(.(X36, X37), X38, .(X36, X39)) :- append(X37, X38, X39).\nand substitutionX36 -> T42,\nX37 -> T45,\nT22 -> .(T42, T45),\nT23 -> T46,\nX38 -> T46,\nX39 -> X40,\nX16 -> .(T42, X40),\nT43 -> T45,\nT44 -> T46" }, { "from": 192, "to": 201, "label": "EVAL-BACKTRACK" }, { "from": 195, "to": 197, "label": "SUCCESS" }, { "from": 200, "to": 188, "label": "INSTANCE with matching:\nT22 -> T45\nT23 -> T46\nX16 -> X40" }, { "from": 202, "to": 203, "label": "PARALLEL" }, { "from": 202, "to": 204, "label": "PARALLEL" }, { "from": 203, "to": 206, "label": "EVAL with clause\nappend([], X49, X49).\nand substitutionT27 -> [],\nT28 -> T55,\nX49 -> T55,\nT21 -> T55" }, { "from": 203, "to": 208, "label": "EVAL-BACKTRACK" }, { "from": 204, "to": 212, "label": "EVAL with clause\nappend(.(X58, X59), X60, .(X58, X61)) :- append(X59, X60, X61).\nand substitutionX58 -> T64,\nX59 -> T68,\nT27 -> .(T64, T68),\nT28 -> T69,\nX60 -> T69,\nX61 -> T67,\nT21 -> .(T64, T67),\nT65 -> T68,\nT66 -> T69" }, { "from": 204, "to": 213, "label": "EVAL-BACKTRACK" }, { "from": 206, "to": 209, "label": "SUCCESS" }, { "from": 212, "to": 189, "label": "INSTANCE with matching:\nT27 -> T68\nT28 -> T69\nT21 -> T67" } ], "type": "Graph" } } ---------------------------------------- (38) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f2_in(T21) -> U1(f185_in(T21), T21) U1(f185_out1(X16, T24), T21) -> f2_out1(T24) f188_in -> f188_out1 f188_in -> U2(f188_in) U2(f188_out1) -> f188_out1 f189_in(T55) -> f189_out1([], T55) f189_in(.(T64, T67)) -> U3(f189_in(T67), .(T64, T67)) U3(f189_out1(T68, T69), .(T64, T67)) -> f189_out1(.(T64, T68), T69) f185_in(T21) -> U4(f188_in, T21) U4(f188_out1, T21) -> U5(f189_in(T21), T21) U5(f189_out1(T27, T28), T21) -> f185_out1(T27, T28) Q is empty. ---------------------------------------- (39) QTRSRRRProof (EQUIVALENT) Used ordering: f2_in/1(YES) U1/2(YES,YES) f185_in/1(YES) f185_out1/2(YES,YES) f2_out1/1)YES( f188_in/0) f188_out1/0) U2/1)YES( f189_in/1(YES) f189_out1/2(YES,YES) []/0) ./2(YES,YES) U3/2(YES,YES) U4/2(YES,YES) U5/2(YES,YES) Quasi precedence: f2_in_1 > [f185_in_1, f188_in] > U4_2 > [f188_out1, f189_in_1] > [] > [U1_2, U5_2] f2_in_1 > [f185_in_1, f188_in] > U4_2 > [f188_out1, f189_in_1] > U3_2 > ._2 > [f185_out1_2, f189_out1_2] > [U1_2, U5_2] Status: f2_in_1: multiset status U1_2: multiset status f185_in_1: [1] f185_out1_2: multiset status f188_in: multiset status f188_out1: multiset status f189_in_1: multiset status f189_out1_2: multiset status []: multiset status ._2: multiset status U3_2: multiset status U4_2: multiset status U5_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: f2_in(T21) -> U1(f185_in(T21), T21) U1(f185_out1(X16, T24), T21) -> f2_out1(T24) f188_in -> f188_out1 f189_in(T55) -> f189_out1([], T55) f189_in(.(T64, T67)) -> U3(f189_in(T67), .(T64, T67)) U3(f189_out1(T68, T69), .(T64, T67)) -> f189_out1(.(T64, T68), T69) f185_in(T21) -> U4(f188_in, T21) U4(f188_out1, T21) -> U5(f189_in(T21), T21) U5(f189_out1(T27, T28), T21) -> f185_out1(T27, T28) ---------------------------------------- (40) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f188_in -> U2(f188_in) U2(f188_out1) -> f188_out1 Q is empty. ---------------------------------------- (41) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U2(x_1)) = 2*x_1 POL(f188_in) = 0 POL(f188_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U2(f188_out1) -> f188_out1 ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f188_in -> U2(f188_in) Q is empty. ---------------------------------------- (43) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f188_in -> U2(f188_in) The set Q consists of the following terms: f188_in ---------------------------------------- (45) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: F188_IN -> F188_IN The TRS R consists of the following rules: f188_in -> U2(f188_in) The set Q consists of the following terms: f188_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F188_IN -> F188_IN R is empty. The set Q consists of the following terms: f188_in We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. f188_in ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F188_IN -> F188_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 3, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append3 A B C D)", "(',' (append A B E) (append E C D))" ] ] }, "graph": { "nodes": { "193": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T42 T43 T41)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T41"], "free": [], "exprvars": [] } }, "type": "Nodes", "194": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "152": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T13 T14 X9) (append X9 T15 T12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": ["X9"], "exprvars": [] } }, "154": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (append T13 T14 X9) (append X9 T15 T12))" }, { "clause": 1, "scope": 2, "term": "(',' (append T13 T14 X9) (append X9 T15 T12))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": ["X9"], "exprvars": [] } }, "210": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T55 T56 X50)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X50"], "exprvars": [] } }, "178": { "goal": [ { "clause": 0, "scope": 3, "term": "(append T21 T22 T12)" }, { "clause": 1, "scope": 3, "term": "(append T21 T22 T12)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": [], "exprvars": [] } }, "211": { "goal": [{ "clause": -1, "scope": -1, "term": "(append (. 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T58 X50) T57 T12))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T12"], "free": ["X50"], "exprvars": [] } }, "207": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 19, "label": "CASE" }, { "from": 19, "to": 152, "label": "ONLY EVAL with clause\nappend3(X5, X6, X7, X8) :- ','(append(X5, X6, X9), append(X9, X7, X8)).\nand substitutionT1 -> T13,\nX5 -> T13,\nT2 -> T14,\nX6 -> T14,\nT3 -> T15,\nX7 -> T15,\nT4 -> T12,\nX8 -> T12,\nT9 -> T13,\nT10 -> T14,\nT11 -> T15" }, { "from": 152, "to": 154, "label": "CASE" }, { "from": 154, "to": 160, "label": "PARALLEL" }, { "from": 154, "to": 161, "label": "PARALLEL" }, { "from": 160, "to": 163, "label": "EVAL with clause\nappend([], X14, X14).\nand substitutionT13 -> [],\nT14 -> T21,\nX14 -> T21,\nX9 -> T21,\nT20 -> T21,\nT15 -> T22" }, { "from": 160, "to": 166, "label": "EVAL-BACKTRACK" }, { "from": 161, "to": 205, "label": "EVAL with clause\nappend(.(X46, X47), X48, .(X46, X49)) :- append(X47, X48, X49).\nand substitutionX46 -> T58,\nX47 -> T55,\nT13 -> .(T58, T55),\nT14 -> T56,\nX48 -> T56,\nX49 -> X50,\nX9 -> .(T58, X50),\nT53 -> T55,\nT54 -> T56,\nT15 -> T57,\nT52 -> T58" }, { "from": 161, "to": 207, "label": "EVAL-BACKTRACK" }, { "from": 163, "to": 178, "label": "CASE" }, { "from": 178, "to": 182, "label": "PARALLEL" }, { "from": 178, "to": 183, "label": "PARALLEL" }, { "from": 182, "to": 184, "label": "EVAL with clause\nappend([], X21, X21).\nand substitutionT21 -> [],\nT22 -> T29,\nX21 -> T29,\nT12 -> T29" }, { "from": 182, "to": 186, "label": "EVAL-BACKTRACK" }, { "from": 183, "to": 193, "label": "EVAL with clause\nappend(.(X30, X31), X32, .(X30, X33)) :- append(X31, X32, X33).\nand substitutionX30 -> T38,\nX31 -> T42,\nT21 -> .(T38, T42),\nT22 -> T43,\nX32 -> T43,\nX33 -> T41,\nT12 -> .(T38, T41),\nT39 -> T42,\nT40 -> T43" }, { "from": 183, "to": 194, "label": "EVAL-BACKTRACK" }, { "from": 184, "to": 187, "label": "SUCCESS" }, { "from": 193, "to": 163, "label": "INSTANCE with matching:\nT21 -> T42\nT22 -> T43\nT12 -> T41" }, { "from": 205, "to": 210, "label": "SPLIT 1" }, { "from": 205, "to": 211, "label": "SPLIT 2\nreplacements:X50 -> T61,\nT58 -> T62,\nT57 -> T63" }, { "from": 210, "to": 214, "label": "CASE" }, { "from": 211, "to": 163, "label": "INSTANCE with matching:\nT21 -> .(T62, T61)\nT22 -> T63" }, { "from": 214, "to": 215, "label": "PARALLEL" }, { "from": 214, "to": 216, "label": "PARALLEL" }, { "from": 215, "to": 217, "label": "EVAL with clause\nappend([], X59, X59).\nand substitutionT55 -> [],\nT56 -> T70,\nX59 -> T70,\nX50 -> T70" }, { "from": 215, "to": 218, "label": "EVAL-BACKTRACK" }, { "from": 216, "to": 220, "label": "EVAL with clause\nappend(.(X70, X71), X72, .(X70, X73)) :- append(X71, X72, X73).\nand substitutionX70 -> T77,\nX71 -> T80,\nT55 -> .(T77, T80),\nT56 -> T81,\nX72 -> T81,\nX73 -> X74,\nX50 -> .(T77, X74),\nT78 -> T80,\nT79 -> T81" }, { "from": 216, "to": 221, "label": "EVAL-BACKTRACK" }, { "from": 217, "to": 219, "label": "SUCCESS" }, { "from": 220, "to": 210, "label": "INSTANCE with matching:\nT55 -> T80\nT56 -> T81\nX50 -> X74" } ], "type": "Graph" } } ---------------------------------------- (52) Obligation: Triples: appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4). appendB(.(X1, X2), X3, .(X1, X4)) :- appendB(X2, X3, X4). append3C([], X1, X2, X3) :- appendA(X1, X2, X3). append3C(.(X1, X2), X3, X4, X5) :- appendB(X2, X3, X6). append3C(.(X1, X2), X3, X4, X5) :- ','(appendcB(X2, X3, X6), appendA(.(X1, X6), X4, X5)). Clauses: appendcA([], X1, X1). appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4). appendcB([], X1, X1). appendcB(.(X1, X2), X3, .(X1, X4)) :- appendcB(X2, X3, X4). Afs: append3C(x1, x2, x3, x4) = append3C(x4) ---------------------------------------- (53) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: append3C_in_4: (f,f,f,b) appendA_in_3: (f,f,b) appendB_in_3: (f,f,f) appendcB_in_3: (f,f,f) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: APPEND3C_IN_AAAG([], X1, X2, X3) -> U3_AAAG(X1, X2, X3, appendA_in_aag(X1, X2, X3)) APPEND3C_IN_AAAG([], X1, X2, X3) -> APPENDA_IN_AAG(X1, X2, X3) APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> U1_AAG(X1, X2, X3, X4, appendA_in_aag(X2, X3, X4)) APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_AAG(X2, X3, X4) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> U4_AAAG(X1, X2, X3, X4, X5, appendB_in_aaa(X2, X3, X6)) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> APPENDB_IN_AAA(X2, X3, X6) APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U2_AAA(X1, X2, X3, X4, appendB_in_aaa(X2, X3, X4)) APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AAA(X2, X3, X4) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> U5_AAAG(X1, X2, X3, X4, X5, appendcB_in_aaa(X2, X3, X6)) U5_AAAG(X1, X2, X3, X4, X5, appendcB_out_aaa(X2, X3, X6)) -> U6_AAAG(X1, X2, X3, X4, X5, appendA_in_aag(.(X1, X6), X4, X5)) U5_AAAG(X1, X2, X3, X4, X5, appendcB_out_aaa(X2, X3, X6)) -> APPENDA_IN_AAG(.(X1, X6), X4, X5) The TRS R consists of the following rules: appendcB_in_aaa([], X1, X1) -> appendcB_out_aaa([], X1, X1) appendcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U9_aaa(X1, X2, X3, X4, appendcB_in_aaa(X2, X3, X4)) U9_aaa(X1, X2, X3, X4, appendcB_out_aaa(X2, X3, X4)) -> appendcB_out_aaa(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: appendA_in_aag(x1, x2, x3) = appendA_in_aag(x3) .(x1, x2) = .(x2) appendB_in_aaa(x1, x2, x3) = appendB_in_aaa appendcB_in_aaa(x1, x2, x3) = appendcB_in_aaa appendcB_out_aaa(x1, x2, x3) = appendcB_out_aaa(x1) U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) [] = [] APPEND3C_IN_AAAG(x1, x2, x3, x4) = APPEND3C_IN_AAAG(x4) U3_AAAG(x1, x2, x3, x4) = U3_AAAG(x3, x4) APPENDA_IN_AAG(x1, x2, x3) = APPENDA_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) U4_AAAG(x1, x2, x3, x4, x5, x6) = U4_AAAG(x5, x6) APPENDB_IN_AAA(x1, x2, x3) = APPENDB_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) U5_AAAG(x1, x2, x3, x4, x5, x6) = U5_AAAG(x5, x6) U6_AAAG(x1, x2, x3, x4, x5, x6) = U6_AAAG(x2, x5, x6) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND3C_IN_AAAG([], X1, X2, X3) -> U3_AAAG(X1, X2, X3, appendA_in_aag(X1, X2, X3)) APPEND3C_IN_AAAG([], X1, X2, X3) -> APPENDA_IN_AAG(X1, X2, X3) APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> U1_AAG(X1, X2, X3, X4, appendA_in_aag(X2, X3, X4)) APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_AAG(X2, X3, X4) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> U4_AAAG(X1, X2, X3, X4, X5, appendB_in_aaa(X2, X3, X6)) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> APPENDB_IN_AAA(X2, X3, X6) APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> U2_AAA(X1, X2, X3, X4, appendB_in_aaa(X2, X3, X4)) APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AAA(X2, X3, X4) APPEND3C_IN_AAAG(.(X1, X2), X3, X4, X5) -> U5_AAAG(X1, X2, X3, X4, X5, appendcB_in_aaa(X2, X3, X6)) U5_AAAG(X1, X2, X3, X4, X5, appendcB_out_aaa(X2, X3, X6)) -> U6_AAAG(X1, X2, X3, X4, X5, appendA_in_aag(.(X1, X6), X4, X5)) U5_AAAG(X1, X2, X3, X4, X5, appendcB_out_aaa(X2, X3, X6)) -> APPENDA_IN_AAG(.(X1, X6), X4, X5) The TRS R consists of the following rules: appendcB_in_aaa([], X1, X1) -> appendcB_out_aaa([], X1, X1) appendcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U9_aaa(X1, X2, X3, X4, appendcB_in_aaa(X2, X3, X4)) U9_aaa(X1, X2, X3, X4, appendcB_out_aaa(X2, X3, X4)) -> appendcB_out_aaa(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: appendA_in_aag(x1, x2, x3) = appendA_in_aag(x3) .(x1, x2) = .(x2) appendB_in_aaa(x1, x2, x3) = appendB_in_aaa appendcB_in_aaa(x1, x2, x3) = appendcB_in_aaa appendcB_out_aaa(x1, x2, x3) = appendcB_out_aaa(x1) U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) [] = [] APPEND3C_IN_AAAG(x1, x2, x3, x4) = APPEND3C_IN_AAAG(x4) U3_AAAG(x1, x2, x3, x4) = U3_AAAG(x3, x4) APPENDA_IN_AAG(x1, x2, x3) = APPENDA_IN_AAG(x3) U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5) U4_AAAG(x1, x2, x3, x4, x5, x6) = U4_AAAG(x5, x6) APPENDB_IN_AAA(x1, x2, x3) = APPENDB_IN_AAA U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5) U5_AAAG(x1, x2, x3, x4, x5, x6) = U5_AAAG(x5, x6) U6_AAAG(x1, x2, x3, x4, x5, x6) = U6_AAAG(x2, x5, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes. ---------------------------------------- (56) Complex Obligation (AND) ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AAA(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aaa([], X1, X1) -> appendcB_out_aaa([], X1, X1) appendcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U9_aaa(X1, X2, X3, X4, appendcB_in_aaa(X2, X3, X4)) U9_aaa(X1, X2, X3, X4, appendcB_out_aaa(X2, X3, X4)) -> appendcB_out_aaa(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) appendcB_in_aaa(x1, x2, x3) = appendcB_in_aaa appendcB_out_aaa(x1, x2, x3) = appendcB_out_aaa(x1) U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) [] = [] APPENDB_IN_AAA(x1, x2, x3) = APPENDB_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (59) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDB_IN_AAA(.(X1, X2), X3, .(X1, X4)) -> APPENDB_IN_AAA(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDB_IN_AAA(x1, x2, x3) = APPENDB_IN_AAA We have to consider all (P,R,Pi)-chains ---------------------------------------- (60) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (61) Obligation: Q DP problem: The TRS P consists of the following rules: APPENDB_IN_AAA -> APPENDB_IN_AAA R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (62) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_AAG(X2, X3, X4) The TRS R consists of the following rules: appendcB_in_aaa([], X1, X1) -> appendcB_out_aaa([], X1, X1) appendcB_in_aaa(.(X1, X2), X3, .(X1, X4)) -> U9_aaa(X1, X2, X3, X4, appendcB_in_aaa(X2, X3, X4)) U9_aaa(X1, X2, X3, X4, appendcB_out_aaa(X2, X3, X4)) -> appendcB_out_aaa(.(X1, X2), X3, .(X1, X4)) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) appendcB_in_aaa(x1, x2, x3) = appendcB_in_aaa appendcB_out_aaa(x1, x2, x3) = appendcB_out_aaa(x1) U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5) [] = [] APPENDA_IN_AAG(x1, x2, x3) = APPENDA_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (63) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (64) Obligation: Pi DP problem: The TRS P consists of the following rules: APPENDA_IN_AAG(.(X1, X2), X3, .(X1, X4)) -> APPENDA_IN_AAG(X2, X3, X4) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPENDA_IN_AAG(x1, x2, x3) = APPENDA_IN_AAG(x3) We have to consider all (P,R,Pi)-chains ---------------------------------------- (65) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(append ([]) L L)", null ], [ "(append (. H L1) L2 (. H L3))", "(append L1 L2 L3)" ], [ "(append3 A B C D)", "(',' (append A B E) (append E C D))" ] ] }, "graph": { "nodes": { "type": "Nodes", "150": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "172": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "151": { "goal": [ { "clause": 0, "scope": 2, "term": "(append T22 T23 X16)" }, { "clause": 1, "scope": 2, "term": "(append T22 T23 X16)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "162": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "141": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (append T22 T23 X16) (append X16 T24 T21))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": ["X16"], "exprvars": [] } }, "165": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "198": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T45 T46 X40)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X40"], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(append3 T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } }, "199": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "222": { "goal": [ { "clause": 0, "scope": 3, "term": "(append T27 T28 T21)" }, { "clause": 1, "scope": 3, "term": "(append T27 T28 T21)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "157": { "goal": [{ "clause": 0, "scope": 2, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "223": { "goal": [{ "clause": 0, "scope": 3, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "158": { "goal": [{ "clause": 1, "scope": 2, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "224": { "goal": [{ "clause": 1, "scope": 3, "term": "(append T27 T28 T21)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T21"], "free": [], "exprvars": [] } }, "225": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "149": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T22 T23 X16)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X16"], "exprvars": [] } }, "226": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "227": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "228": { "goal": [{ "clause": -1, "scope": -1, "term": "(append T68 T69 T67)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T67"], "free": [], "exprvars": [] } }, "229": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "20": { "goal": [{ "clause": 2, "scope": 1, "term": "(append3 T1 T2 T3 T4)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T4"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 20, "label": "CASE" }, { "from": 20, "to": 141, "label": "ONLY EVAL with clause\nappend3(X12, X13, X14, X15) :- ','(append(X12, X13, X16), append(X16, X14, X15)).\nand substitutionT1 -> T22,\nX12 -> T22,\nT2 -> T23,\nX13 -> T23,\nT3 -> T24,\nX14 -> T24,\nT4 -> T21,\nX15 -> T21,\nT18 -> T22,\nT19 -> T23,\nT20 -> T24" }, { "from": 141, "to": 149, "label": "SPLIT 1" }, { "from": 141, "to": 150, "label": "SPLIT 2\nreplacements:X16 -> T27,\nT24 -> T28" }, { "from": 149, "to": 151, "label": "CASE" }, { "from": 150, "to": 222, "label": "CASE" }, { "from": 151, "to": 157, "label": "PARALLEL" }, { "from": 151, "to": 158, "label": "PARALLEL" }, { "from": 157, "to": 162, "label": "EVAL with clause\nappend([], X25, X25).\nand substitutionT22 -> [],\nT23 -> T35,\nX25 -> T35,\nX16 -> T35" }, { "from": 157, "to": 165, "label": "EVAL-BACKTRACK" }, { "from": 158, "to": 198, "label": "EVAL with clause\nappend(.(X36, X37), X38, .(X36, X39)) :- append(X37, X38, X39).\nand substitutionX36 -> T42,\nX37 -> T45,\nT22 -> .(T42, T45),\nT23 -> T46,\nX38 -> T46,\nX39 -> X40,\nX16 -> .(T42, X40),\nT43 -> T45,\nT44 -> T46" }, { "from": 158, "to": 199, "label": "EVAL-BACKTRACK" }, { "from": 162, "to": 172, "label": "SUCCESS" }, { "from": 198, "to": 149, "label": "INSTANCE with matching:\nT22 -> T45\nT23 -> T46\nX16 -> X40" }, { "from": 222, "to": 223, "label": "PARALLEL" }, { "from": 222, "to": 224, "label": "PARALLEL" }, { "from": 223, "to": 225, "label": "EVAL with clause\nappend([], X49, X49).\nand substitutionT27 -> [],\nT28 -> T55,\nX49 -> T55,\nT21 -> T55" }, { "from": 223, "to": 226, "label": "EVAL-BACKTRACK" }, { "from": 224, "to": 228, "label": "EVAL with clause\nappend(.(X58, X59), X60, .(X58, X61)) :- append(X59, X60, X61).\nand substitutionX58 -> T64,\nX59 -> T68,\nT27 -> .(T64, T68),\nT28 -> T69,\nX60 -> T69,\nX61 -> T67,\nT21 -> .(T64, T67),\nT65 -> T68,\nT66 -> T69" }, { "from": 224, "to": 229, "label": "EVAL-BACKTRACK" }, { "from": 225, "to": 227, "label": "SUCCESS" }, { "from": 228, "to": 150, "label": "INSTANCE with matching:\nT27 -> T68\nT28 -> T69\nT21 -> T67" } ], "type": "Graph" } } ---------------------------------------- (66) Complex Obligation (AND) ---------------------------------------- (67) Obligation: Rules: f222_in(T21) -> f223_in(T21) :|: TRUE f223_out(x) -> f222_out(x) :|: TRUE f222_in(x1) -> f224_in(x1) :|: TRUE f224_out(x2) -> f222_out(x2) :|: TRUE f150_out(T67) -> f228_out(T67) :|: TRUE f228_in(x3) -> f150_in(x3) :|: TRUE f150_in(x4) -> f222_in(x4) :|: TRUE f222_out(x5) -> f150_out(x5) :|: TRUE f224_in(.(x6, x7)) -> f228_in(x7) :|: TRUE f228_out(x8) -> f224_out(.(x9, x8)) :|: TRUE f224_in(x10) -> f229_in :|: TRUE f229_out -> f224_out(x11) :|: TRUE f20_out(T4) -> f1_out(T4) :|: TRUE f1_in(x12) -> f20_in(x12) :|: TRUE f20_in(x13) -> f141_in(x13) :|: TRUE f141_out(x14) -> f20_out(x14) :|: TRUE f141_in(x15) -> f149_in :|: TRUE f150_out(x16) -> f141_out(x16) :|: TRUE f149_out -> f150_in(x17) :|: TRUE Start term: f1_in(T4) ---------------------------------------- (68) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: ---------------------------------------- (69) TRUE ---------------------------------------- (70) Obligation: Rules: f149_out -> f198_out :|: TRUE f198_in -> f149_in :|: TRUE f151_in -> f157_in :|: TRUE f158_out -> f151_out :|: TRUE f151_in -> f158_in :|: TRUE f157_out -> f151_out :|: TRUE f158_in -> f198_in :|: TRUE f198_out -> f158_out :|: TRUE f199_out -> f158_out :|: TRUE f158_in -> f199_in :|: TRUE f149_in -> f151_in :|: TRUE f151_out -> f149_out :|: TRUE f20_out(T4) -> f1_out(T4) :|: TRUE f1_in(x) -> f20_in(x) :|: TRUE f20_in(T21) -> f141_in(T21) :|: TRUE f141_out(x1) -> f20_out(x1) :|: TRUE f141_in(x2) -> f149_in :|: TRUE f150_out(x3) -> f141_out(x3) :|: TRUE f149_out -> f150_in(x4) :|: TRUE Start term: f1_in(T4) ---------------------------------------- (71) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f198_in -> f149_in :|: TRUE f151_in -> f158_in :|: TRUE f158_in -> f198_in :|: TRUE f149_in -> f151_in :|: TRUE ---------------------------------------- (72) Obligation: Rules: f198_in -> f149_in :|: TRUE f151_in -> f158_in :|: TRUE f158_in -> f198_in :|: TRUE f149_in -> f151_in :|: TRUE ---------------------------------------- (73) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (74) Obligation: Rules: f151_in -> f151_in :|: TRUE ---------------------------------------- (75) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (76) Obligation: Rules: f151_in -> f151_in :|: TRUE ---------------------------------------- (77) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f151_in -> f151_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (78) Obligation: Termination digraph: Nodes: (1) f151_in -> f151_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (79) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f151_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (80) Obligation: Rules: f151_in -> f151_in :|: TRUE ---------------------------------------- (81) IntTRSNonPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Proved unsatisfiability of the following formula, indicating that the system is never left after entering: ((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T)) Proved satisfiability of the following formula, indicating that the system is entered at least once: (run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) ---------------------------------------- (82) NO