/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern e(g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) UnifyTransformerProof [EQUIVALENT, 0 ms] (4) Prolog (5) PrologToPiTRSProof [SOUND, 0 ms] (6) PiTRS (7) DependencyPairsProof [EQUIVALENT, 0 ms] (8) PiDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) PiDP (12) UsableRulesProof [EQUIVALENT, 0 ms] (13) PiDP (14) PiDPToQDPProof [SOUND, 0 ms] (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) PiDP (19) UsableRulesProof [EQUIVALENT, 0 ms] (20) PiDP (21) PiDPToQDPProof [EQUIVALENT, 0 ms] (22) QDP (23) PrologToPiTRSProof [SOUND, 0 ms] (24) PiTRS (25) DependencyPairsProof [EQUIVALENT, 0 ms] (26) PiDP (27) DependencyGraphProof [EQUIVALENT, 1 ms] (28) AND (29) PiDP (30) UsableRulesProof [EQUIVALENT, 0 ms] (31) PiDP (32) PiDPToQDPProof [SOUND, 0 ms] (33) QDP (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] (35) YES (36) PiDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) PiDP (39) PiDPToQDPProof [EQUIVALENT, 0 ms] (40) QDP (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) Overlay + Local Confluence [EQUIVALENT, 0 ms] (44) QTRS (45) DependencyPairsProof [EQUIVALENT, 0 ms] (46) QDP (47) DependencyGraphProof [EQUIVALENT, 0 ms] (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) QReductionProof [EQUIVALENT, 0 ms] (52) QDP (53) PrologToDTProblemTransformerProof [SOUND, 0 ms] (54) TRIPLES (55) TriplesToPiDPProof [SOUND, 0 ms] (56) PiDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) PiDP (59) PiDPToQDPProof [EQUIVALENT, 0 ms] (60) QDP (61) PrologToIRSwTTransformerProof [SOUND, 0 ms] (62) IRSwT (63) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (64) IRSwT (65) IntTRSCompressionProof [EQUIVALENT, 1 ms] (66) IRSwT (67) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (68) IRSwT (69) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (70) IRSwT (71) FilterProof [EQUIVALENT, 0 ms] (72) IntTRS (73) IntTRSPeriodicNontermProof [COMPLETE, 5 ms] (74) NO ---------------------------------------- (0) Obligation: Clauses: e(X) :- ','(=(Y, f(e, f(o, Y))), ','(!, c(Y, X))). e(X) :- e(X). c(f(e, X1), 0). c(f(X2, X), s(Y)) :- c(X, Y). Query: e(g) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: e(X) :- ','(=(Y, f(e, f(o, Y))), c(Y, X)). e(X) :- e(X). c(f(e, X1), 0). c(f(X2, X), s(Y)) :- c(X, Y). Query: e(g) ---------------------------------------- (3) UnifyTransformerProof (EQUIVALENT) Added a fact for the built-in = predicate [PROLOG]. ---------------------------------------- (4) Obligation: Clauses: e(X) :- ','(=(Y, f(e, f(o, Y))), c(Y, X)). e(X) :- e(X). c(f(e, X1), 0). c(f(X2, X), s(Y)) :- c(X, Y). =(X, X). Query: e(g) ---------------------------------------- (5) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: e_in_1: (b) c_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (6) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) ---------------------------------------- (7) 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: E_IN_G(X) -> U1_G(X, =_in_aa(Y, f(e, f(o, Y)))) E_IN_G(X) -> =_IN_AA(Y, f(e, f(o, Y))) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_G(X, c_in_ag(Y, X)) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> C_IN_AG(Y, X) C_IN_AG(f(X2, X), s(Y)) -> U4_AG(X2, X, Y, c_in_ag(X, Y)) C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) E_IN_G(X) -> U3_G(X, e_in_g(X)) E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) E_IN_G(x1) = E_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) =_IN_AA(x1, x2) = =_IN_AA U2_G(x1, x2) = U2_G(x1, x2) C_IN_AG(x1, x2) = C_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) U3_G(x1, x2) = U3_G(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> U1_G(X, =_in_aa(Y, f(e, f(o, Y)))) E_IN_G(X) -> =_IN_AA(Y, f(e, f(o, Y))) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_G(X, c_in_ag(Y, X)) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> C_IN_AG(Y, X) C_IN_AG(f(X2, X), s(Y)) -> U4_AG(X2, X, Y, c_in_ag(X, Y)) C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) E_IN_G(X) -> U3_G(X, e_in_g(X)) E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) E_IN_G(x1) = E_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) =_IN_AA(x1, x2) = =_IN_AA U2_G(x1, x2) = U2_G(x1, x2) C_IN_AG(x1, x2) = C_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4) U3_G(x1, x2) = U3_G(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Pi DP problem: The TRS P consists of the following rules: C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) C_IN_AG(x1, x2) = C_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (12) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (13) Obligation: Pi DP problem: The TRS P consists of the following rules: C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) C_IN_AG(x1, x2) = C_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (14) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: C_IN_AG(s(Y)) -> C_IN_AG(Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (16) 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: *C_IN_AG(s(Y)) -> C_IN_AG(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x1, x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag(x2) s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4) e_out_g(x1) = e_out_g(x1) U3_g(x1, x2) = U3_g(x1, x2) E_IN_G(x1) = E_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (20) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (21) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (23) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: e_in_1: (b) c_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (24) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) ---------------------------------------- (25) 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: E_IN_G(X) -> U1_G(X, =_in_aa(Y, f(e, f(o, Y)))) E_IN_G(X) -> =_IN_AA(Y, f(e, f(o, Y))) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_G(X, c_in_ag(Y, X)) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> C_IN_AG(Y, X) C_IN_AG(f(X2, X), s(Y)) -> U4_AG(X2, X, Y, c_in_ag(X, Y)) C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) E_IN_G(X) -> U3_G(X, e_in_g(X)) E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) E_IN_G(x1) = E_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) =_IN_AA(x1, x2) = =_IN_AA U2_G(x1, x2) = U2_G(x2) C_IN_AG(x1, x2) = C_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) U3_G(x1, x2) = U3_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (26) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> U1_G(X, =_in_aa(Y, f(e, f(o, Y)))) E_IN_G(X) -> =_IN_AA(Y, f(e, f(o, Y))) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_G(X, c_in_ag(Y, X)) U1_G(X, =_out_aa(Y, f(e, f(o, Y)))) -> C_IN_AG(Y, X) C_IN_AG(f(X2, X), s(Y)) -> U4_AG(X2, X, Y, c_in_ag(X, Y)) C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) E_IN_G(X) -> U3_G(X, e_in_g(X)) E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) E_IN_G(x1) = E_IN_G(x1) U1_G(x1, x2) = U1_G(x1, x2) =_IN_AA(x1, x2) = =_IN_AA U2_G(x1, x2) = U2_G(x2) C_IN_AG(x1, x2) = C_IN_AG(x2) U4_AG(x1, x2, x3, x4) = U4_AG(x4) U3_G(x1, x2) = U3_G(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (27) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (28) Complex Obligation (AND) ---------------------------------------- (29) Obligation: Pi DP problem: The TRS P consists of the following rules: C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) C_IN_AG(x1, x2) = C_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (30) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (31) Obligation: Pi DP problem: The TRS P consists of the following rules: C_IN_AG(f(X2, X), s(Y)) -> C_IN_AG(X, Y) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) C_IN_AG(x1, x2) = C_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (32) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: C_IN_AG(s(Y)) -> C_IN_AG(Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (34) 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: *C_IN_AG(s(Y)) -> C_IN_AG(Y) The graph contains the following edges 1 > 1 ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) The TRS R consists of the following rules: e_in_g(X) -> U1_g(X, =_in_aa(Y, f(e, f(o, Y)))) =_in_aa(X, X) -> =_out_aa(X, X) U1_g(X, =_out_aa(Y, f(e, f(o, Y)))) -> U2_g(X, c_in_ag(Y, X)) c_in_ag(f(e, X1), 0) -> c_out_ag(f(e, X1), 0) c_in_ag(f(X2, X), s(Y)) -> U4_ag(X2, X, Y, c_in_ag(X, Y)) U4_ag(X2, X, Y, c_out_ag(X, Y)) -> c_out_ag(f(X2, X), s(Y)) U2_g(X, c_out_ag(Y, X)) -> e_out_g(X) e_in_g(X) -> U3_g(X, e_in_g(X)) U3_g(X, e_out_g(X)) -> e_out_g(X) The argument filtering Pi contains the following mapping: e_in_g(x1) = e_in_g(x1) U1_g(x1, x2) = U1_g(x1, x2) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_g(x1, x2) = U2_g(x2) c_in_ag(x1, x2) = c_in_ag(x2) 0 = 0 c_out_ag(x1, x2) = c_out_ag s(x1) = s(x1) U4_ag(x1, x2, x3, x4) = U4_ag(x4) e_out_g(x1) = e_out_g U3_g(x1, x2) = U3_g(x2) E_IN_G(x1) = E_IN_G(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (39) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: E_IN_G(X) -> E_IN_G(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 13, "program": { "directives": [], "clauses": [ [ "(e X)", "(',' (= Y (f (e) (f (o) Y))) (',' (!) (c Y X)))" ], [ "(e X)", "(e X)" ], [ "(c (f (e) X1) (0))", null ], [ "(c (f X2 X) (s Y))", "(c X Y)" ] ] }, "graph": { "nodes": { "154": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": 0, "scope": 1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "24": { "goal": [{ "clause": 1, "scope": 1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "14": { "goal": [ { "clause": 0, "scope": 1, "term": "(e T1)" }, { "clause": 1, "scope": 1, "term": "(e T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "28": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X10 (f (e) (f (o) X10))) (',' (!_1) (c X10 T6)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X10"], "exprvars": [] } }, "type": "Nodes", "31": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 13, "to": 14, "label": "CASE" }, { "from": 14, "to": 23, "label": "PARALLEL" }, { "from": 14, "to": 24, "label": "PARALLEL" }, { "from": 23, "to": 28, "label": "ONLY EVAL with clause\ne(X9) :- ','(=(X10, f(e, f(o, X10))), ','(!_1, c(X10, X9))).\nand substitutionT1 -> T6,\nX9 -> T6" }, { "from": 24, "to": 154, "label": "ONLY EVAL with clause\ne(X16) :- e(X16).\nand substitutionT1 -> T10,\nX16 -> T10" }, { "from": 28, "to": 31, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 154, "to": 13, "label": "INSTANCE with matching:\nT1 -> T10" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f13_in(T10) -> U1(f13_in(T10), T10) U1(f13_out1, T10) -> f13_out1 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: f13_in(T10) -> U1(f13_in(T10), T10) U1(f13_out1, T10) -> f13_out1 The set Q consists of the following terms: f13_in(x0) U1(f13_out1, x0) ---------------------------------------- (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: F13_IN(T10) -> U1^1(f13_in(T10), T10) F13_IN(T10) -> F13_IN(T10) The TRS R consists of the following rules: f13_in(T10) -> U1(f13_in(T10), T10) U1(f13_out1, T10) -> f13_out1 The set Q consists of the following terms: f13_in(x0) U1(f13_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F13_IN(T10) -> F13_IN(T10) The TRS R consists of the following rules: f13_in(T10) -> U1(f13_in(T10), T10) U1(f13_out1, T10) -> f13_out1 The set Q consists of the following terms: f13_in(x0) U1(f13_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) 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. ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: F13_IN(T10) -> F13_IN(T10) R is empty. The set Q consists of the following terms: f13_in(x0) U1(f13_out1, x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) 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]. f13_in(x0) U1(f13_out1, x0) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F13_IN(T10) -> F13_IN(T10) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(e X)", "(',' (= Y (f (e) (f (o) Y))) (',' (!) (c Y X)))" ], [ "(e X)", "(e X)" ], [ "(c (f (e) X1) (0))", null ], [ "(c (f X2 X) (s Y))", "(c X Y)" ] ] }, "graph": { "nodes": { "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "2": { "goal": [ { "clause": 0, "scope": 1, "term": "(e T1)" }, { "clause": 1, "scope": 1, "term": "(e T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "79": { "goal": [{ "clause": 0, "scope": 2, "term": "(e T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "17": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (= X5 (f (e) (f (o) X5))) (',' (!_1) (c X5 T3)))" }, { "clause": 1, "scope": 1, "term": "(e T3)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": ["X5"], "exprvars": [] } }, "18": { "goal": [{ "clause": 1, "scope": 1, "term": "(e T3)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T3"], "free": [], "exprvars": [] } }, "80": { "goal": [{ "clause": 1, "scope": 2, "term": "(e T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "81": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X16 (f (e) (f (o) X16))) (',' (!_2) (c X16 T10)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": ["X16"], "exprvars": [] } }, "82": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "83": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T14)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T14"], "free": [], "exprvars": [] } }, "type": "Nodes", "73": { "goal": [ { "clause": 0, "scope": 2, "term": "(e T5)" }, { "clause": 1, "scope": 2, "term": "(e T5)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } }, "64": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T5)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T5"], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 17, "label": "ONLY EVAL with clause\ne(X4) :- ','(=(X5, f(e, f(o, X5))), ','(!_1, c(X5, X4))).\nand substitutionT1 -> T3,\nX4 -> T3" }, { "from": 17, "to": 18, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 18, "to": 64, "label": "ONLY EVAL with clause\ne(X8) :- e(X8).\nand substitutionT3 -> T5,\nX8 -> T5" }, { "from": 64, "to": 73, "label": "CASE" }, { "from": 73, "to": 79, "label": "PARALLEL" }, { "from": 73, "to": 80, "label": "PARALLEL" }, { "from": 79, "to": 81, "label": "ONLY EVAL with clause\ne(X15) :- ','(=(X16, f(e, f(o, X16))), ','(!_2, c(X16, X15))).\nand substitutionT5 -> T10,\nX15 -> T10" }, { "from": 80, "to": 83, "label": "ONLY EVAL with clause\ne(X22) :- e(X22).\nand substitutionT5 -> T14,\nX22 -> T14" }, { "from": 81, "to": 82, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 83, "to": 1, "label": "INSTANCE with matching:\nT1 -> T14" } ], "type": "Graph" } } ---------------------------------------- (54) Obligation: Triples: eA(X1) :- eA(X1). Clauses: ecA(X1) :- ecA(X1). Afs: eA(x1) = eA(x1) ---------------------------------------- (55) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: eA_in_1: (b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: EA_IN_G(X1) -> U1_G(X1, eA_in_g(X1)) EA_IN_G(X1) -> EA_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (56) Obligation: Pi DP problem: The TRS P consists of the following rules: EA_IN_G(X1) -> U1_G(X1, eA_in_g(X1)) EA_IN_G(X1) -> EA_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (58) Obligation: Pi DP problem: The TRS P consists of the following rules: EA_IN_G(X1) -> EA_IN_G(X1) R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (59) PiDPToQDPProof (EQUIVALENT) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: EA_IN_G(X1) -> EA_IN_G(X1) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (61) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 15, "program": { "directives": [], "clauses": [ [ "(e X)", "(',' (= Y (f (e) (f (o) Y))) (',' (!) (c Y X)))" ], [ "(e X)", "(e X)" ], [ "(c (f (e) X1) (0))", null ], [ "(c (f X2 X) (s Y))", "(c X Y)" ] ] }, "graph": { "nodes": { "22": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "155": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T10)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T10"], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "16": { "goal": [ { "clause": 0, "scope": 1, "term": "(e T1)" }, { "clause": 1, "scope": 1, "term": "(e T1)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 0, "scope": 1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "type": "Nodes", "20": { "goal": [{ "clause": 1, "scope": 1, "term": "(e T1)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T1"], "free": [], "exprvars": [] } }, "21": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X10 (f (e) (f (o) X10))) (',' (!_1) (c X10 T6)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T6"], "free": ["X10"], "exprvars": [] } } }, "edges": [ { "from": 15, "to": 16, "label": "CASE" }, { "from": 16, "to": 19, "label": "PARALLEL" }, { "from": 16, "to": 20, "label": "PARALLEL" }, { "from": 19, "to": 21, "label": "ONLY EVAL with clause\ne(X9) :- ','(=(X10, f(e, f(o, X10))), ','(!_1, c(X10, X9))).\nand substitutionT1 -> T6,\nX9 -> T6" }, { "from": 20, "to": 155, "label": "ONLY EVAL with clause\ne(X16) :- e(X16).\nand substitutionT1 -> T10,\nX16 -> T10" }, { "from": 21, "to": 22, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 155, "to": 15, "label": "INSTANCE with matching:\nT1 -> T10" } ], "type": "Graph" } } ---------------------------------------- (62) Obligation: Rules: f15_out(T10) -> f155_out(T10) :|: TRUE f155_in(x) -> f15_in(x) :|: TRUE f19_out(T1) -> f16_out(T1) :|: TRUE f16_in(x1) -> f20_in(x1) :|: TRUE f16_in(x2) -> f19_in(x2) :|: TRUE f20_out(x3) -> f16_out(x3) :|: TRUE f20_in(x4) -> f155_in(x4) :|: TRUE f155_out(x5) -> f20_out(x5) :|: TRUE f15_in(x6) -> f16_in(x6) :|: TRUE f16_out(x7) -> f15_out(x7) :|: TRUE Start term: f15_in(T1) ---------------------------------------- (63) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f155_in(x) -> f15_in(x) :|: TRUE f16_in(x1) -> f20_in(x1) :|: TRUE f20_in(x4) -> f155_in(x4) :|: TRUE f15_in(x6) -> f16_in(x6) :|: TRUE ---------------------------------------- (64) Obligation: Rules: f155_in(x) -> f15_in(x) :|: TRUE f16_in(x1) -> f20_in(x1) :|: TRUE f20_in(x4) -> f155_in(x4) :|: TRUE f15_in(x6) -> f16_in(x6) :|: TRUE ---------------------------------------- (65) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (66) Obligation: Rules: f16_in(x1:0) -> f16_in(x1:0) :|: TRUE ---------------------------------------- (67) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (68) Obligation: Rules: f16_in(x1:0) -> f16_in(x1:0) :|: TRUE ---------------------------------------- (69) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f16_in(x1:0) -> f16_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (70) Obligation: Termination digraph: Nodes: (1) f16_in(x1:0) -> f16_in(x1:0) :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (71) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f16_in(VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (72) Obligation: Rules: f16_in(x1:0) -> f16_in(x1:0) :|: TRUE ---------------------------------------- (73) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, x1:0) -> f(1, x1:0) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1, -8) ---------------------------------------- (74) NO