/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern t() w.r.t. the given Prolog program could not be shown: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog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) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (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) PiDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) PiDP (33) PrologToDTProblemTransformerProof [SOUND, 0 ms] (34) TRIPLES (35) TriplesToPiDPProof [SOUND, 0 ms] (36) PiDP (37) DependencyGraphProof [EQUIVALENT, 0 ms] (38) PiDP (39) PiDPToQDPProof [EQUIVALENT, 0 ms] (40) QDP (41) PrologToTRSTransformerProof [SOUND, 0 ms] (42) QTRS (43) QTRSRRRProof [EQUIVALENT, 0 ms] (44) QTRS (45) Overlay + Local Confluence [EQUIVALENT, 0 ms] (46) QTRS (47) DependencyPairsProof [EQUIVALENT, 0 ms] (48) QDP (49) UsableRulesProof [EQUIVALENT, 0 ms] (50) QDP (51) QReductionProof [EQUIVALENT, 0 ms] (52) QDP (53) PrologToIRSwTTransformerProof [SOUND, 0 ms] (54) IRSwT (55) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (56) IRSwT (57) IntTRSCompressionProof [EQUIVALENT, 0 ms] (58) IRSwT (59) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (60) IRSwT (61) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (62) IRSwT (63) FilterProof [EQUIVALENT, 0 ms] (64) IntTRS (65) IntTRSNonPeriodicNontermProof [COMPLETE, 7 ms] (66) NO ---------------------------------------- (0) Obligation: Clauses: t :- ','(=(X, f(0, f(1, f(0, X)))), ','(!, g(X))). t :- t. g(f(X, f(X, X1))) :- !. g(f(X2, X)) :- g(X). Query: t() ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: t :- ','(=(X, f(0, f(1, f(0, X)))), g(X)). t :- t. g(f(X, f(X, X1))). g(f(X2, X)) :- g(X). Query: t() ---------------------------------------- (3) UnifyTransformerProof (EQUIVALENT) Added a fact for the built-in = predicate [PROLOG]. ---------------------------------------- (4) Obligation: Clauses: t :- ','(=(X, f(0, f(1, f(0, X)))), g(X)). t :- t. g(f(X, f(X, X1))). g(f(X2, X)) :- g(X). =(X, X). Query: t() ---------------------------------------- (5) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: g_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (6) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) ---------------------------------------- (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: T_IN_ -> U1_^1(=_in_aa(X, f(0, f(1, f(0, X))))) T_IN_ -> =_IN_AA(X, f(0, f(1, f(0, X)))) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_^1(g_in_a(X)) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> G_IN_A(X) G_IN_A(f(X2, X)) -> U4_A(X2, X, g_in_a(X)) G_IN_A(f(X2, X)) -> G_IN_A(X) T_IN_ -> U3_^1(t_in_) T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ U1_^1(x1) = U1_^1(x1) =_IN_AA(x1, x2) = =_IN_AA U2_^1(x1) = U2_^1(x1) G_IN_A(x1) = G_IN_A U4_A(x1, x2, x3) = U4_A(x3) U3_^1(x1) = U3_^1(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> U1_^1(=_in_aa(X, f(0, f(1, f(0, X))))) T_IN_ -> =_IN_AA(X, f(0, f(1, f(0, X)))) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_^1(g_in_a(X)) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> G_IN_A(X) G_IN_A(f(X2, X)) -> U4_A(X2, X, g_in_a(X)) G_IN_A(f(X2, X)) -> G_IN_A(X) T_IN_ -> U3_^1(t_in_) T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ U1_^1(x1) = U1_^1(x1) =_IN_AA(x1, x2) = =_IN_AA U2_^1(x1) = U2_^1(x1) G_IN_A(x1) = G_IN_A U4_A(x1, x2, x3) = U4_A(x3) U3_^1(x1) = U3_^1(x1) 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: G_IN_A(f(X2, X)) -> G_IN_A(X) The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) G_IN_A(x1) = G_IN_A 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: G_IN_A(f(X2, X)) -> G_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: f(x1, x2) = f(x2) G_IN_A(x1) = G_IN_A 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: G_IN_A -> G_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (18) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> T_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-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: g_in_1: (f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (20) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) ---------------------------------------- (21) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> U1_^1(=_in_aa(X, f(0, f(1, f(0, X))))) T_IN_ -> =_IN_AA(X, f(0, f(1, f(0, X)))) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_^1(g_in_a(X)) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> G_IN_A(X) G_IN_A(f(X2, X)) -> U4_A(X2, X, g_in_a(X)) G_IN_A(f(X2, X)) -> G_IN_A(X) T_IN_ -> U3_^1(t_in_) T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ U1_^1(x1) = U1_^1(x1) =_IN_AA(x1, x2) = =_IN_AA U2_^1(x1) = U2_^1(x1) G_IN_A(x1) = G_IN_A U4_A(x1, x2, x3) = U4_A(x3) U3_^1(x1) = U3_^1(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> U1_^1(=_in_aa(X, f(0, f(1, f(0, X))))) T_IN_ -> =_IN_AA(X, f(0, f(1, f(0, X)))) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_^1(g_in_a(X)) U1_^1(=_out_aa(X, f(0, f(1, f(0, X))))) -> G_IN_A(X) G_IN_A(f(X2, X)) -> U4_A(X2, X, g_in_a(X)) G_IN_A(f(X2, X)) -> G_IN_A(X) T_IN_ -> U3_^1(t_in_) T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ U1_^1(x1) = U1_^1(x1) =_IN_AA(x1, x2) = =_IN_AA U2_^1(x1) = U2_^1(x1) G_IN_A(x1) = G_IN_A U4_A(x1, x2, x3) = U4_A(x3) U3_^1(x1) = U3_^1(x1) 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: G_IN_A(f(X2, X)) -> G_IN_A(X) The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) G_IN_A(x1) = G_IN_A 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: G_IN_A(f(X2, X)) -> G_IN_A(X) R is empty. The argument filtering Pi contains the following mapping: f(x1, x2) = f(x2) G_IN_A(x1) = G_IN_A 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: G_IN_A -> G_IN_A R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> T_IN_ The TRS R consists of the following rules: t_in_ -> U1_(=_in_aa(X, f(0, f(1, f(0, X))))) =_in_aa(X, X) -> =_out_aa(X, X) U1_(=_out_aa(X, f(0, f(1, f(0, X))))) -> U2_(g_in_a(X)) g_in_a(f(X, f(X, X1))) -> g_out_a(f(X, f(X, X1))) g_in_a(f(X2, X)) -> U4_a(X2, X, g_in_a(X)) U4_a(X2, X, g_out_a(X)) -> g_out_a(f(X2, X)) U2_(g_out_a(X)) -> t_out_ t_in_ -> U3_(t_in_) U3_(t_out_) -> t_out_ The argument filtering Pi contains the following mapping: t_in_ = t_in_ U1_(x1) = U1_(x1) =_in_aa(x1, x2) = =_in_aa =_out_aa(x1, x2) = =_out_aa U2_(x1) = U2_(x1) g_in_a(x1) = g_in_a g_out_a(x1) = g_out_a f(x1, x2) = f(x2) U4_a(x1, x2, x3) = U4_a(x3) t_out_ = t_out_ U3_(x1) = U3_(x1) T_IN_ = T_IN_ We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (32) Obligation: Pi DP problem: The TRS P consists of the following rules: T_IN_ -> T_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (33) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 1, "program": { "directives": [], "clauses": [ [ "(t)", "(',' (= X (f (0) (f (1) (f (0) X)))) (',' (!) (g X)))" ], [ "(t)", "(t)" ], [ "(g (f X (f X X1)))", "(!)" ], [ "(g (f X2 X))", "(g X)" ] ] }, "graph": { "nodes": { "22": { "goal": [{ "clause": 1, "scope": 2, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "23": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X7 (f (0) (f (1) (f (0) X7)))) (',' (!_2) (g X7)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X7"], "exprvars": [] } }, "24": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "14": { "goal": [{ "clause": 1, "scope": 1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "25": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "15": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [ { "clause": 0, "scope": 1, "term": "(t)" }, { "clause": 1, "scope": 1, "term": "(t)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "16": { "goal": [ { "clause": 0, "scope": 2, "term": "(t)" }, { "clause": 1, "scope": 2, "term": "(t)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "8": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (= X3 (f (0) (f (1) (f (0) X3)))) (',' (!_1) (g X3)))" }, { "clause": 1, "scope": 1, "term": "(t)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X3"], "exprvars": [] } }, "type": "Nodes", "21": { "goal": [{ "clause": 0, "scope": 2, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 5, "label": "CASE" }, { "from": 5, "to": 8, "label": "ONLY EVAL with clause\nt :- ','(=(X3, f(0, f(1, f(0, X3)))), ','(!_1, g(X3))).\nand substitution" }, { "from": 8, "to": 14, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 14, "to": 15, "label": "ONLY EVAL with clause\nt :- t.\nand substitution" }, { "from": 15, "to": 16, "label": "CASE" }, { "from": 16, "to": 21, "label": "PARALLEL" }, { "from": 16, "to": 22, "label": "PARALLEL" }, { "from": 21, "to": 23, "label": "ONLY EVAL with clause\nt :- ','(=(X7, f(0, f(1, f(0, X7)))), ','(!_2, g(X7))).\nand substitution" }, { "from": 22, "to": 25, "label": "ONLY EVAL with clause\nt :- t.\nand substitution" }, { "from": 23, "to": 24, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 25, "to": 1, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (34) Obligation: Triples: tA :- tA. Clauses: tcA :- tcA. Afs: tA = tA ---------------------------------------- (35) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: TA_IN_ -> U1_^1(tA_in_) TA_IN_ -> TA_IN_ 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 ---------------------------------------- (36) Obligation: Pi DP problem: The TRS P consists of the following rules: TA_IN_ -> U1_^1(tA_in_) TA_IN_ -> TA_IN_ R is empty. Pi is empty. We have to consider all (P,R,Pi)-chains ---------------------------------------- (37) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node. ---------------------------------------- (38) Obligation: Pi DP problem: The TRS P consists of the following rules: TA_IN_ -> TA_IN_ 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: TA_IN_ -> TA_IN_ R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (41) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 3, "program": { "directives": [], "clauses": [ [ "(t)", "(',' (= X (f (0) (f (1) (f (0) X)))) (',' (!) (g X)))" ], [ "(t)", "(t)" ], [ "(g (f X (f X X1)))", "(!)" ], [ "(g (f X2 X))", "(g X)" ] ] }, "graph": { "nodes": { "11": { "goal": [{ "clause": 1, "scope": 1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "12": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X5 (f (0) (f (1) (f (0) X5)))) (',' (!_1) (g X5)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X5"], "exprvars": [] } }, "3": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "6": { "goal": [ { "clause": 0, "scope": 1, "term": "(t)" }, { "clause": 1, "scope": 1, "term": "(t)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "17": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "10": { "goal": [{ "clause": 0, "scope": 1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 6, "label": "CASE" }, { "from": 6, "to": 10, "label": "PARALLEL" }, { "from": 6, "to": 11, "label": "PARALLEL" }, { "from": 10, "to": 12, "label": "ONLY EVAL with clause\nt :- ','(=(X5, f(0, f(1, f(0, X5)))), ','(!_1, g(X5))).\nand substitution" }, { "from": 11, "to": 19, "label": "ONLY EVAL with clause\nt :- t.\nand substitution" }, { "from": 12, "to": 17, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 19, "to": 3, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (42) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in -> U1(f3_in) U1(f3_out1) -> f3_out1 Q is empty. ---------------------------------------- (43) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(U1(x_1)) = 2*x_1 POL(f3_in) = 0 POL(f3_out1) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U1(f3_out1) -> f3_out1 ---------------------------------------- (44) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in -> U1(f3_in) Q is empty. ---------------------------------------- (45) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (46) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f3_in -> U1(f3_in) The set Q consists of the following terms: f3_in ---------------------------------------- (47) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN -> F3_IN The TRS R consists of the following rules: f3_in -> U1(f3_in) The set Q consists of the following terms: f3_in 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: F3_IN -> F3_IN R is empty. The set Q consists of the following terms: f3_in 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]. f3_in ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: F3_IN -> F3_IN R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 2, "program": { "directives": [], "clauses": [ [ "(t)", "(',' (= X (f (0) (f (1) (f (0) X)))) (',' (!) (g X)))" ], [ "(t)", "(t)" ], [ "(g (f X (f X X1)))", "(!)" ], [ "(g (f X2 X))", "(g X)" ] ] }, "graph": { "nodes": { "2": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "13": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (= X5 (f (0) (f (1) (f (0) X5)))) (',' (!_1) (g X5)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X5"], "exprvars": [] } }, "4": { "goal": [ { "clause": 0, "scope": 1, "term": "(t)" }, { "clause": 1, "scope": 1, "term": "(t)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "7": { "goal": [{ "clause": 0, "scope": 1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "18": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "9": { "goal": [{ "clause": 1, "scope": 1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "20": { "goal": [{ "clause": -1, "scope": -1, "term": "(t)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 2, "to": 4, "label": "CASE" }, { "from": 4, "to": 7, "label": "PARALLEL" }, { "from": 4, "to": 9, "label": "PARALLEL" }, { "from": 7, "to": 13, "label": "ONLY EVAL with clause\nt :- ','(=(X5, f(0, f(1, f(0, X5)))), ','(!_1, g(X5))).\nand substitution" }, { "from": 9, "to": 20, "label": "ONLY EVAL with clause\nt :- t.\nand substitution" }, { "from": 13, "to": 18, "label": "UNIFY-FAIL\nbecause of non-unification" }, { "from": 20, "to": 2, "label": "INSTANCE" } ], "type": "Graph" } } ---------------------------------------- (54) Obligation: Rules: f4_in -> f7_in :|: TRUE f4_in -> f9_in :|: TRUE f7_out -> f4_out :|: TRUE f9_out -> f4_out :|: TRUE f9_in -> f20_in :|: TRUE f20_out -> f9_out :|: TRUE f2_in -> f4_in :|: TRUE f4_out -> f2_out :|: TRUE f2_out -> f20_out :|: TRUE f20_in -> f2_in :|: TRUE Start term: f2_in ---------------------------------------- (55) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f4_in -> f9_in :|: TRUE f9_in -> f20_in :|: TRUE f2_in -> f4_in :|: TRUE f20_in -> f2_in :|: TRUE ---------------------------------------- (56) Obligation: Rules: f4_in -> f9_in :|: TRUE f9_in -> f20_in :|: TRUE f2_in -> f4_in :|: TRUE f20_in -> f2_in :|: TRUE ---------------------------------------- (57) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (58) Obligation: Rules: f2_in -> f2_in :|: TRUE ---------------------------------------- (59) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (60) Obligation: Rules: f2_in -> f2_in :|: TRUE ---------------------------------------- (61) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f2_in -> f2_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (62) Obligation: Termination digraph: Nodes: (1) f2_in -> f2_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (63) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f2_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (64) Obligation: Rules: f2_in -> f2_in :|: TRUE ---------------------------------------- (65) 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)) ---------------------------------------- (66) NO