/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern p(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2)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, 24 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 1 ms] (11) QDP (12) TransformationProof [SOUND, 0 ms] (13) QDP (14) TransformationProof [SOUND, 0 ms] (15) QDP (16) PiDP (17) UsableRulesProof [EQUIVALENT, 0 ms] (18) PiDP (19) PiDP (20) UsableRulesProof [EQUIVALENT, 0 ms] (21) PiDP (22) PiDP (23) UsableRulesProof [EQUIVALENT, 0 ms] (24) PiDP (25) PiDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) PiDP (28) PiDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) PiDP (31) PrologToPiTRSProof [SOUND, 0 ms] (32) PiTRS (33) DependencyPairsProof [EQUIVALENT, 55 ms] (34) PiDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) AND (37) PiDP (38) UsableRulesProof [EQUIVALENT, 0 ms] (39) PiDP (40) PiDPToQDPProof [SOUND, 5 ms] (41) QDP (42) TransformationProof [SOUND, 0 ms] (43) QDP (44) TransformationProof [SOUND, 0 ms] (45) QDP (46) PiDP (47) UsableRulesProof [EQUIVALENT, 0 ms] (48) PiDP (49) PiDP (50) UsableRulesProof [EQUIVALENT, 0 ms] (51) PiDP (52) PiDP (53) UsableRulesProof [EQUIVALENT, 0 ms] (54) PiDP (55) PiDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) PiDP (58) PiDP (59) UsableRulesProof [EQUIVALENT, 0 ms] (60) PiDP (61) PrologToTRSTransformerProof [SOUND, 45 ms] (62) QTRS (63) DependencyPairsProof [EQUIVALENT, 0 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) AND (67) QDP (68) NonLoopProof [COMPLETE, 29 ms] (69) NO (70) QDP (71) QDPSizeChangeProof [EQUIVALENT, 0 ms] (72) YES (73) QDP (74) NonLoopProof [COMPLETE, 1170 ms] (75) NO (76) PrologToIRSwTTransformerProof [SOUND, 54 ms] (77) AND (78) IRSwT (79) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (80) IRSwT (81) IntTRSCompressionProof [EQUIVALENT, 24 ms] (82) IRSwT (83) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (84) IRSwT (85) IRSwTTerminationDigraphProof [EQUIVALENT, 2 ms] (86) IRSwT (87) FilterProof [EQUIVALENT, 0 ms] (88) IntTRS (89) IntTRSNonPeriodicNontermProof [COMPLETE, 7 ms] (90) NO (91) IRSwT (92) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (93) IRSwT (94) IntTRSCompressionProof [EQUIVALENT, 7 ms] (95) IRSwT (96) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (97) IRSwT (98) IRSwTTerminationDigraphProof [EQUIVALENT, 34 ms] (99) IRSwT (100) IntTRSCompressionProof [EQUIVALENT, 0 ms] (101) IRSwT (102) IRSwTToIntTRSProof [SOUND, 0 ms] (103) IRSwT (104) IntTRSCompressionProof [EQUIVALENT, 0 ms] (105) IRSwT (106) IntTRSPeriodicNontermProof [COMPLETE, 8 ms] (107) NO (108) IRSwT (109) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (110) IRSwT (111) IntTRSCompressionProof [EQUIVALENT, 24 ms] (112) IRSwT (113) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (114) IRSwT (115) IRSwTTerminationDigraphProof [EQUIVALENT, 27 ms] (116) IRSwT (117) IntTRSCompressionProof [EQUIVALENT, 9 ms] (118) IRSwT (119) PrologToDTProblemTransformerProof [SOUND, 133 ms] (120) TRIPLES (121) TriplesToPiDPProof [SOUND, 1615 ms] (122) PiDP (123) DependencyGraphProof [EQUIVALENT, 0 ms] (124) AND (125) PiDP (126) UsableRulesProof [EQUIVALENT, 0 ms] (127) PiDP (128) PiDPToQDPProof [SOUND, 0 ms] (129) QDP (130) TransformationProof [SOUND, 0 ms] (131) QDP (132) PiDP (133) UsableRulesProof [EQUIVALENT, 0 ms] (134) PiDP (135) PiDP (136) UsableRulesProof [EQUIVALENT, 0 ms] (137) PiDP (138) PiDP (139) UsableRulesProof [EQUIVALENT, 0 ms] (140) PiDP (141) PiDP (142) UsableRulesProof [EQUIVALENT, 0 ms] (143) PiDP (144) PiDP (145) UsableRulesProof [EQUIVALENT, 0 ms] (146) PiDP ---------------------------------------- (0) Obligation: Clauses: p(d(e(t)), const(1)). p(d(e(const(A))), const(0)). p(d(e(+(X, Y))), +(DX, DY)) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY)). p(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY)). p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX)). Query: p(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: p_in_2: (f,b) (b,b) (b,f) (f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) ---------------------------------------- (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: P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(d(X)), DDX) -> U5_GG(X, DDX, p_in_ga(d(X), DX)) P_IN_GG(d(d(X)), DDX) -> P_IN_GA(d(X), DX) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(d(X)), DDX) -> U5_GA(X, DDX, p_in_ga(d(X), DX)) P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_AG(d(d(X)), DDX) -> U5_AG(X, DDX, p_in_aa(d(X), DX)) P_IN_AG(d(d(X)), DDX) -> P_IN_AA(d(X), DX) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(d(X)), DDX) -> U5_AA(X, DDX, p_in_aa(d(X), DX)) P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, x5) U5_GG(x1, x2, x3) = U5_GG(x2, x3) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) U5_GA(x1, x2, x3) = U5_GA(x3) U6_GA(x1, x2, x3, x4) = U6_GA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x4, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x4) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x5) U5_AG(x1, x2, x3) = U5_AG(x2, x3) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U5_AA(x1, x2, x3) = U5_AA(x3) U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5) U6_AG(x1, x2, x3, x4) = U6_AG(x1, x4) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(d(X)), DDX) -> U5_GG(X, DDX, p_in_ga(d(X), DX)) P_IN_GG(d(d(X)), DDX) -> P_IN_GA(d(X), DX) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(d(X)), DDX) -> U5_GA(X, DDX, p_in_ga(d(X), DX)) P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_AG(d(d(X)), DDX) -> U5_AG(X, DDX, p_in_aa(d(X), DX)) P_IN_AG(d(d(X)), DDX) -> P_IN_AA(d(X), DX) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(d(X)), DDX) -> U5_AA(X, DDX, p_in_aa(d(X), DX)) P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, x5) U5_GG(x1, x2, x3) = U5_GG(x2, x3) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) U5_GA(x1, x2, x3) = U5_GA(x3) U6_GA(x1, x2, x3, x4) = U6_GA(x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x4, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x4) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x5) U5_AG(x1, x2, x3) = U5_AG(x2, x3) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U5_AA(x1, x2, x3) = U5_AA(x3) U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5) U6_AG(x1, x2, x3, x4) = U6_AG(x1, x4) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 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: U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) The TRS R consists of the following rules: p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) 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: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_in_aa) P_IN_AA -> P_IN_AA P_IN_AA -> U3_AA(p_in_aa) U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) TransformationProof (SOUND) By narrowing [LPAR04] the rule P_IN_AA -> U1_AA(p_in_aa) at position [0] we obtained the following new rules [LPAR04]: (P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)),P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const))) (P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)),P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const))) (P_IN_AA -> U1_AA(U1_aa(p_in_aa)),P_IN_AA -> U1_AA(U1_aa(p_in_aa))) (P_IN_AA -> U1_AA(U3_aa(p_in_aa)),P_IN_AA -> U1_AA(U3_aa(p_in_aa))) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> P_IN_AA P_IN_AA -> U3_AA(p_in_aa) U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U1_AA(U1_aa(p_in_aa)) P_IN_AA -> U1_AA(U3_aa(p_in_aa)) The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (14) TransformationProof (SOUND) By narrowing [LPAR04] the rule P_IN_AA -> U3_AA(p_in_aa) at position [0] we obtained the following new rules [LPAR04]: (P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const)),P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const))) (P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const)),P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const))) (P_IN_AA -> U3_AA(U1_aa(p_in_aa)),P_IN_AA -> U3_AA(U1_aa(p_in_aa))) (P_IN_AA -> U3_AA(U3_aa(p_in_aa)),P_IN_AA -> U3_AA(U3_aa(p_in_aa))) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> P_IN_AA U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U1_AA(U1_aa(p_in_aa)) P_IN_AA -> U1_AA(U3_aa(p_in_aa)) P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U3_AA(U1_aa(p_in_aa)) P_IN_AA -> U3_AA(U3_aa(p_in_aa)) The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) 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: U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (19) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AA(x1, x2) = P_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (20) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) P_IN_AA(x1, x2) = P_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (23) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (24) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) P_IN_GA(x1, x2) = P_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (25) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, x5) 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: U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) The TRS R consists of the following rules: p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (28) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U5_gg(x1, x2, x3) = U5_gg(x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x3) U6_ga(x1, x2, x3, x4) = U6_ga(x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (30) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (31) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: p_in_2: (f,b) (b,b) (b,f) (f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (32) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) ---------------------------------------- (33) 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: P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(d(X)), DDX) -> U5_GG(X, DDX, p_in_ga(d(X), DX)) P_IN_GG(d(d(X)), DDX) -> P_IN_GA(d(X), DX) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(d(X)), DDX) -> U5_GA(X, DDX, p_in_ga(d(X), DX)) P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_AG(d(d(X)), DDX) -> U5_AG(X, DDX, p_in_aa(d(X), DX)) P_IN_AG(d(d(X)), DDX) -> P_IN_AA(d(X), DX) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(d(X)), DDX) -> U5_AA(X, DDX, p_in_aa(d(X), DX)) P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x4, x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x4, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) U5_GA(x1, x2, x3) = U5_GA(x1, x3) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x4, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x4) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x1, x2, x3, x4, x5) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x1, x2, x3, x4, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x3, x4, x5) U5_AG(x1, x2, x3) = U5_AG(x2, x3) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U5_AA(x1, x2, x3) = U5_AA(x3) U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5) U6_AG(x1, x2, x3, x4) = U6_AG(x1, x2, x4) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (34) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(d(X)), DDX) -> U5_GG(X, DDX, p_in_ga(d(X), DX)) P_IN_GG(d(d(X)), DDX) -> P_IN_GA(d(X), DX) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(d(X)), DDX) -> U5_GA(X, DDX, p_in_ga(d(X), DX)) P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_GA(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_GG(X, DDX, p_out_ga(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_AG(d(d(X)), DDX) -> U5_AG(X, DDX, p_in_aa(d(X), DX)) P_IN_AG(d(d(X)), DDX) -> P_IN_AA(d(X), DX) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(d(X)), DDX) -> U5_AA(X, DDX, p_in_aa(d(X), DX)) P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U5_AA(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GA(d(e(DX)), DDX) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U5_AG(X, DDX, p_out_aa(d(X), DX)) -> P_IN_GG(d(e(DX)), DDX) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x4, x5) U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x4, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5) U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) U5_GA(x1, x2, x3) = U5_GA(x1, x3) U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4) U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x4, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x3, x5) U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x4) U4_GG(x1, x2, x3, x4, x5) = U4_GG(x1, x2, x3, x4, x5) U2_GG(x1, x2, x3, x4, x5) = U2_GG(x1, x2, x3, x4, x5) U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x3, x4, x5) U5_AG(x1, x2, x3) = U5_AG(x2, x3) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) U5_AA(x1, x2, x3) = U5_AA(x3) U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4) U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5) U6_AG(x1, x2, x3, x4) = U6_AG(x1, x2, x4) U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes. ---------------------------------------- (36) Complex Obligation (AND) ---------------------------------------- (37) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (38) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (39) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) P_IN_AA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AA(d(e(X)), DX) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> P_IN_AA(d(e(Y)), DY) P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_AA(d(e(X)), DX) The TRS R consists of the following rules: p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) P_IN_AA(x1, x2) = P_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (40) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_in_aa) P_IN_AA -> P_IN_AA P_IN_AA -> U3_AA(p_in_aa) U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (42) TransformationProof (SOUND) By narrowing [LPAR04] the rule P_IN_AA -> U1_AA(p_in_aa) at position [0] we obtained the following new rules [LPAR04]: (P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)),P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const))) (P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)),P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const))) (P_IN_AA -> U1_AA(U1_aa(p_in_aa)),P_IN_AA -> U1_AA(U1_aa(p_in_aa))) (P_IN_AA -> U1_AA(U3_aa(p_in_aa)),P_IN_AA -> U1_AA(U3_aa(p_in_aa))) ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> P_IN_AA P_IN_AA -> U3_AA(p_in_aa) U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U1_AA(U1_aa(p_in_aa)) P_IN_AA -> U1_AA(U3_aa(p_in_aa)) The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (44) TransformationProof (SOUND) By narrowing [LPAR04] the rule P_IN_AA -> U3_AA(p_in_aa) at position [0] we obtained the following new rules [LPAR04]: (P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const)),P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const))) (P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const)),P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const))) (P_IN_AA -> U3_AA(U1_aa(p_in_aa)),P_IN_AA -> U3_AA(U1_aa(p_in_aa))) (P_IN_AA -> U3_AA(U3_aa(p_in_aa)),P_IN_AA -> U3_AA(U3_aa(p_in_aa))) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U1_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> P_IN_AA U3_AA(p_out_aa(d(e(X)), DX)) -> P_IN_AA P_IN_AA -> U1_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U1_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U1_AA(U1_aa(p_in_aa)) P_IN_AA -> U1_AA(U3_aa(p_in_aa)) P_IN_AA -> U3_AA(p_out_aa(d(e(t)), const)) P_IN_AA -> U3_AA(p_out_aa(d(e(const)), const)) P_IN_AA -> U3_AA(U1_aa(p_in_aa)) P_IN_AA -> U3_AA(U3_aa(p_in_aa)) The TRS R consists of the following rules: p_in_aa -> p_out_aa(d(e(t)), const) p_in_aa -> p_out_aa(d(e(const)), const) p_in_aa -> U1_aa(p_in_aa) p_in_aa -> U3_aa(p_in_aa) U1_aa(p_out_aa(d(e(X)), DX)) -> U2_aa(X, DX, p_in_aa) U3_aa(p_out_aa(d(e(X)), DX)) -> U4_aa(X, DX, p_in_aa) U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The set Q consists of the following terms: p_in_aa U1_aa(x0) U3_aa(x0) U2_aa(x0, x1, x2) U4_aa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (46) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (47) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (48) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) P_IN_GA(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GA(d(e(X)), DX) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> P_IN_GA(d(e(Y)), DY) P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GA(d(e(X)), DX) The TRS R consists of the following rules: p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (49) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AA(x1, x2) = P_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (50) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (51) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_AA(d(d(X)), DDX) -> P_IN_AA(d(X), DX) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) P_IN_AA(x1, x2) = P_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (52) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_GA(x1, x2) = P_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (53) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (54) Obligation: Pi DP problem: The TRS P consists of the following rules: P_IN_GA(d(d(X)), DDX) -> P_IN_GA(d(X), DX) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) P_IN_GA(x1, x2) = P_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (55) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (57) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) P_IN_GG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_GG(d(e(X)), DX) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> P_IN_GG(d(e(Y)), DY) P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> P_IN_GG(d(e(X)), DX) The TRS R consists of the following rules: p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) P_IN_GG(x1, x2) = P_IN_GG(x1, x2) U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5) U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (58) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) p_in_gg(d(d(X)), DDX) -> U5_gg(X, DDX, p_in_ga(d(X), DX)) p_in_ga(d(e(t)), const(1)) -> p_out_ga(d(e(t)), const(1)) p_in_ga(d(e(const(A))), const(0)) -> p_out_ga(d(e(const(A))), const(0)) p_in_ga(d(e(+(X, Y))), +(DX, DY)) -> U1_ga(X, Y, DX, DY, p_in_ga(d(e(X)), DX)) p_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ga(X, Y, DY, DX, p_in_ga(d(e(X)), DX)) p_in_ga(d(d(X)), DDX) -> U5_ga(X, DDX, p_in_ga(d(X), DX)) U5_ga(X, DDX, p_out_ga(d(X), DX)) -> U6_ga(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_ga(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_ga(d(d(X)), DDX) U3_ga(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) -> U4_ga(X, Y, DY, DX, p_in_ga(d(e(Y)), DY)) U4_ga(X, Y, DY, DX, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_ga(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) -> U2_ga(X, Y, DX, DY, p_in_ga(d(e(Y)), DY)) U2_ga(X, Y, DX, DY, p_out_ga(d(e(Y)), DY)) -> p_out_ga(d(e(+(X, Y))), +(DX, DY)) U5_gg(X, DDX, p_out_ga(d(X), DX)) -> U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_gg(d(d(X)), DDX) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) p_in_ag(d(d(X)), DDX) -> U5_ag(X, DDX, p_in_aa(d(X), DX)) p_in_aa(d(e(t)), const(1)) -> p_out_aa(d(e(t)), const(1)) p_in_aa(d(e(const(A))), const(0)) -> p_out_aa(d(e(const(A))), const(0)) p_in_aa(d(e(+(X, Y))), +(DX, DY)) -> U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX)) p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX)) p_in_aa(d(d(X)), DDX) -> U5_aa(X, DDX, p_in_aa(d(X), DX)) U5_aa(X, DDX, p_out_aa(d(X), DX)) -> U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX)) U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) -> p_out_aa(d(d(X)), DDX) U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) -> U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY)) U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) -> U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY)) U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) -> p_out_aa(d(e(+(X, Y))), +(DX, DY)) U5_ag(X, DDX, p_out_aa(d(X), DX)) -> U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX)) U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) -> p_out_ag(d(d(X)), DDX) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) p_in_ga(x1, x2) = p_in_ga(x1) p_out_ga(x1, x2) = p_out_ga(x1, x2) U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5) U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x5) U5_ga(x1, x2, x3) = U5_ga(x1, x3) U6_ga(x1, x2, x3, x4) = U6_ga(x1, x4) U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x4, x5) U2_ga(x1, x2, x3, x4, x5) = U2_ga(x1, x2, x3, x5) U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U5_ag(x1, x2, x3) = U5_ag(x2, x3) p_in_aa(x1, x2) = p_in_aa p_out_aa(x1, x2) = p_out_aa(x1, x2) U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5) U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5) U5_aa(x1, x2, x3) = U5_aa(x3) U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4) U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5) U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5) U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (59) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (60) Obligation: Pi DP problem: The TRS P consists of the following rules: U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> P_IN_AG(d(e(Y)), DY) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) P_IN_AG(d(e(+(X, Y))), +(DX, DY)) -> P_IN_AG(d(e(X)), DX) The TRS R consists of the following rules: p_in_ag(d(e(t)), const(1)) -> p_out_ag(d(e(t)), const(1)) p_in_ag(d(e(const(A))), const(0)) -> p_out_ag(d(e(const(A))), const(0)) p_in_ag(d(e(+(X, Y))), +(DX, DY)) -> U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX)) p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) -> U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY)) U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) -> p_out_ag(d(e(+(X, Y))), +(DX, DY)) p_in_gg(d(e(t)), const(1)) -> p_out_gg(d(e(t)), const(1)) p_in_gg(d(e(const(A))), const(0)) -> p_out_gg(d(e(const(A))), const(0)) p_in_gg(d(e(+(X, Y))), +(DX, DY)) -> U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX)) p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) -> U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX)) U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) -> U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY)) U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) -> U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY)) U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(+(X, Y))), +(DX, DY)) U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) -> p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) The argument filtering Pi contains the following mapping: p_in_ag(x1, x2) = p_in_ag(x2) const(x1) = const 1 = 1 p_out_ag(x1, x2) = p_out_ag(x1, x2) 0 = 0 +(x1, x2) = +(x1, x2) U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5) *(x1, x2) = *(x1, x2) U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5) p_in_gg(x1, x2) = p_in_gg(x1, x2) d(x1) = d(x1) e(x1) = e(x1) t = t p_out_gg(x1, x2) = p_out_gg(x1, x2) U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5) U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5) U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5) U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5) U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5) U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5) P_IN_AG(x1, x2) = P_IN_AG(x2) U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (61) PrologToTRSTransformerProof (SOUND) Transformed Prolog program to TRS. { "root": 8, "program": { "directives": [], "clauses": [ [ "(p (d (e (t))) (const (1)))", null ], [ "(p (d (e (const A))) (const (0)))", null ], [ "(p (d (e (+ X Y))) (+ DX DY))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (e (* X Y))) (+ (* X DY) (* Y DX)))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (d X)) DDX)", "(',' (p (d X) DX) (p (d (e DX)) DDX))" ] ] }, "graph": { "nodes": { "390": { "goal": [{ "clause": 3, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "391": { "goal": [{ "clause": 4, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "type": "Nodes", "473": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T72)) T74) (p (d (e T73)) T75))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T72", "T73", "T74", "T75" ], "free": [], "exprvars": [] } }, "550": { "goal": [ { "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 4, "scope": 3, "term": "(p (d T102) X100)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "474": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "234": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "432": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T47)) T50)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T47", "T50" ], "free": [], "exprvars": [] } }, "433": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "510": { "goal": [{ "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "511": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T92)) T95) (p (d (e T93)) T94))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T92", "T93", "T94", "T95" ], "free": [], "exprvars": [] } }, "555": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T119)) X134) (p (d (e T120)) X135))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X134", "X135" ], "exprvars": [] } }, "556": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "316": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T28)) T26)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": [], "exprvars": [] } }, "318": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T30)) T27)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T27"], "free": [], "exprvars": [] } }, "516": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "517": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "19": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "561": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T119)) X134)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X134"], "exprvars": [] } }, "122": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "562": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T122)) X135)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X135"], "exprvars": [] } }, "123": { "goal": [ { "clause": 2, "scope": 1, "term": "(p T1 T2)" }, { "clause": 3, "scope": 1, "term": "(p T1 T2)" }, { "clause": 4, "scope": 1, "term": "(p T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "124": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "125": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "522": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d T102) X100) (p (d (e X100)) T101))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T101"], "free": ["X100"], "exprvars": [] } }, "127": { "goal": [{ "clause": 2, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "446": { "goal": [ { "clause": 0, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 1, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 2, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "523": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "128": { "goal": [ { "clause": 3, "scope": 1, "term": "(p T1 T2)" }, { "clause": 4, "scope": 1, "term": "(p T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "524": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "8": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "525": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T103)) T101)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T101"], "free": [], "exprvars": [] } }, "9": { "goal": [ { "clause": 0, "scope": 1, "term": "(p T1 T2)" }, { "clause": 1, "scope": 1, "term": "(p T1 T2)" }, { "clause": 2, "scope": 1, "term": "(p T1 T2)" }, { "clause": 3, "scope": 1, "term": "(p T1 T2)" }, { "clause": 4, "scope": 1, "term": "(p T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "449": { "goal": [{ "clause": 0, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "526": { "goal": [ { "clause": 0, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 1, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 2, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 4, "scope": 3, "term": "(p (d T102) X100)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "527": { "goal": [{ "clause": 0, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "529": { "goal": [ { "clause": 1, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 2, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 4, "scope": 3, "term": "(p (d T102) X100)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "20": { "goal": [ { "clause": 1, "scope": 1, "term": "(p T1 T2)" }, { "clause": 2, "scope": 1, "term": "(p T1 T2)" }, { "clause": 3, "scope": 1, "term": "(p T1 T2)" }, { "clause": 4, "scope": 1, "term": "(p T1 T2)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "450": { "goal": [ { "clause": 1, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 2, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "453": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "530": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "410": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T47)) T50) (p (d (e T48)) T49))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T47", "T49", "T48", "T50" ], "free": [], "exprvars": [] } }, "454": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "531": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "575": { "goal": [{ "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "411": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "455": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "532": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "576": { "goal": [{ "clause": 4, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "456": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "457": { "goal": [ { "clause": 2, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "537": { "goal": [{ "clause": 1, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "538": { "goal": [ { "clause": 2, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }, { "clause": 4, "scope": 3, "term": "(p (d T102) X100)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "72": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "78": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "460": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "581": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T133)) X165) (p (d (e T134)) X164))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X164", "X165" ], "exprvars": [] } }, "461": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "582": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "462": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "583": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d T138) X175) (p (d (e X175)) X176))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X176", "X175" ], "exprvars": [] } }, "584": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "222": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T28)) T26) (p (d (e T29)) T27))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T26", "T27" ], "free": [], "exprvars": [] } }, "541": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "585": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T138) X175)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X175"], "exprvars": [] } }, "586": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T139)) X176)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X176"], "exprvars": [] } }, "466": { "goal": [{ "clause": 2, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "543": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "104": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "467": { "goal": [ { "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }, { "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "545": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "548": { "goal": [{ "clause": 2, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "509": { "goal": [{ "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } } }, "edges": [ { "from": 8, "to": 9, "label": "CASE" }, { "from": 9, "to": 19, "label": "PARALLEL" }, { "from": 9, "to": 20, "label": "PARALLEL" }, { "from": 19, "to": 72, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 19, "to": 78, "label": "EVAL-BACKTRACK" }, { "from": 20, "to": 122, "label": "PARALLEL" }, { "from": 20, "to": 123, "label": "PARALLEL" }, { "from": 72, "to": 104, "label": "SUCCESS" }, { "from": 122, "to": 124, "label": "EVAL with clause\np(d(e(const(X5))), const(0)).\nand substitutionX5 -> T7,\nT1 -> d(e(const(T7))),\nT2 -> const(0)" }, { "from": 122, "to": 125, "label": "EVAL-BACKTRACK" }, { "from": 123, "to": 127, "label": "PARALLEL" }, { "from": 123, "to": 128, "label": "PARALLEL" }, { "from": 124, "to": 126, "label": "SUCCESS" }, { "from": 127, "to": 222, "label": "EVAL with clause\np(d(e(+(X22, X23))), +(X24, X25)) :- ','(p(d(e(X22)), X24), p(d(e(X23)), X25)).\nand substitutionX22 -> T28,\nX23 -> T29,\nT1 -> d(e(+(T28, T29))),\nX24 -> T26,\nX25 -> T27,\nT2 -> +(T26, T27),\nT24 -> T28,\nT25 -> T29" }, { "from": 127, "to": 234, "label": "EVAL-BACKTRACK" }, { "from": 128, "to": 390, "label": "PARALLEL" }, { "from": 128, "to": 391, "label": "PARALLEL" }, { "from": 222, "to": 316, "label": "SPLIT 1" }, { "from": 222, "to": 318, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:T29 -> T30" }, { "from": 316, "to": 8, "label": "INSTANCE with matching:\nT1 -> d(e(T28))\nT2 -> T26" }, { "from": 318, "to": 8, "label": "INSTANCE with matching:\nT1 -> d(e(T30))\nT2 -> T27" }, { "from": 390, "to": 410, "label": "EVAL with clause\np(d(e(*(X42, X43))), +(*(X42, X44), *(X43, X45))) :- ','(p(d(e(X42)), X45), p(d(e(X43)), X44)).\nand substitutionX42 -> T47,\nX43 -> T48,\nT1 -> d(e(*(T47, T48))),\nX44 -> T49,\nX45 -> T50,\nT2 -> +(*(T47, T49), *(T48, T50))" }, { "from": 390, "to": 411, "label": "EVAL-BACKTRACK" }, { "from": 391, "to": 522, "label": "EVAL with clause\np(d(d(X98)), X99) :- ','(p(d(X98), X100), p(d(e(X100)), X99)).\nand substitutionX98 -> T102,\nT1 -> d(d(T102)),\nT2 -> T101,\nX99 -> T101,\nT100 -> T102" }, { "from": 391, "to": 523, "label": "EVAL-BACKTRACK" }, { "from": 410, "to": 432, "label": "SPLIT 1" }, { "from": 410, "to": 433, "label": "SPLIT 2\nnew knowledge:\nT47 is ground\nT50 is ground" }, { "from": 432, "to": 433, "label": "INSTANCE with matching:\nT48 -> T47\nT49 -> T50" }, { "from": 433, "to": 446, "label": "CASE" }, { "from": 446, "to": 449, "label": "PARALLEL" }, { "from": 446, "to": 450, "label": "PARALLEL" }, { "from": 449, "to": 453, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT48 -> t,\nT49 -> const(1)" }, { "from": 449, "to": 454, "label": "EVAL-BACKTRACK" }, { "from": 450, "to": 456, "label": "PARALLEL" }, { "from": 450, "to": 457, "label": "PARALLEL" }, { "from": 453, "to": 455, "label": "SUCCESS" }, { "from": 456, "to": 460, "label": "EVAL with clause\np(d(e(const(X50))), const(0)).\nand substitutionX50 -> T55,\nT48 -> const(T55),\nT49 -> const(0)" }, { "from": 456, "to": 461, "label": "EVAL-BACKTRACK" }, { "from": 457, "to": 466, "label": "PARALLEL" }, { "from": 457, "to": 467, "label": "PARALLEL" }, { "from": 460, "to": 462, "label": "SUCCESS" }, { "from": 466, "to": 473, "label": "EVAL with clause\np(d(e(+(X67, X68))), +(X69, X70)) :- ','(p(d(e(X67)), X69), p(d(e(X68)), X70)).\nand substitutionX67 -> T72,\nX68 -> T73,\nT48 -> +(T72, T73),\nX69 -> T74,\nX70 -> T75,\nT49 -> +(T74, T75)" }, { "from": 466, "to": 474, "label": "EVAL-BACKTRACK" }, { "from": 467, "to": 509, "label": "PARALLEL" }, { "from": 467, "to": 510, "label": "PARALLEL" }, { "from": 473, "to": 410, "label": "INSTANCE with matching:\nT47 -> T72\nT50 -> T74\nT48 -> T73\nT49 -> T75" }, { "from": 509, "to": 511, "label": "EVAL with clause\np(d(e(*(X87, X88))), +(*(X87, X89), *(X88, X90))) :- ','(p(d(e(X87)), X90), p(d(e(X88)), X89)).\nand substitutionX87 -> T92,\nX88 -> T93,\nT48 -> *(T92, T93),\nX89 -> T94,\nX90 -> T95,\nT49 -> +(*(T92, T94), *(T93, T95))" }, { "from": 509, "to": 516, "label": "EVAL-BACKTRACK" }, { "from": 510, "to": 517, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 511, "to": 410, "label": "INSTANCE with matching:\nT47 -> T92\nT50 -> T95\nT48 -> T93\nT49 -> T94" }, { "from": 522, "to": 524, "label": "SPLIT 1" }, { "from": 522, "to": 525, "label": "SPLIT 2\nreplacements:X100 -> T103" }, { "from": 524, "to": 526, "label": "CASE" }, { "from": 525, "to": 8, "label": "INSTANCE with matching:\nT1 -> d(e(T103))\nT2 -> T101" }, { "from": 526, "to": 527, "label": "PARALLEL" }, { "from": 526, "to": 529, "label": "PARALLEL" }, { "from": 527, "to": 530, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT102 -> e(t),\nX100 -> const(1)" }, { "from": 527, "to": 531, "label": "EVAL-BACKTRACK" }, { "from": 529, "to": 537, "label": "PARALLEL" }, { "from": 529, "to": 538, "label": "PARALLEL" }, { "from": 530, "to": 532, "label": "SUCCESS" }, { "from": 537, "to": 541, "label": "EVAL with clause\np(d(e(const(X105))), const(0)).\nand substitutionX105 -> T108,\nT102 -> e(const(T108)),\nX100 -> const(0)" }, { "from": 537, "to": 543, "label": "EVAL-BACKTRACK" }, { "from": 538, "to": 548, "label": "PARALLEL" }, { "from": 538, "to": 550, "label": "PARALLEL" }, { "from": 541, "to": 545, "label": "SUCCESS" }, { "from": 548, "to": 555, "label": "EVAL with clause\np(d(e(+(X130, X131))), +(X132, X133)) :- ','(p(d(e(X130)), X132), p(d(e(X131)), X133)).\nand substitutionX130 -> T119,\nX131 -> T120,\nT102 -> e(+(T119, T120)),\nX132 -> X134,\nX133 -> X135,\nX100 -> +(X134, X135),\nT117 -> T119,\nT118 -> T120" }, { "from": 548, "to": 556, "label": "EVAL-BACKTRACK" }, { "from": 550, "to": 575, "label": "PARALLEL" }, { "from": 550, "to": 576, "label": "PARALLEL" }, { "from": 555, "to": 561, "label": "SPLIT 1" }, { "from": 555, "to": 562, "label": "SPLIT 2\nreplacements:X134 -> T121,\nT120 -> T122" }, { "from": 561, "to": 524, "label": "INSTANCE with matching:\nT102 -> e(T119)\nX100 -> X134" }, { "from": 562, "to": 524, "label": "INSTANCE with matching:\nT102 -> e(T122)\nX100 -> X135" }, { "from": 575, "to": 581, "label": "EVAL with clause\np(d(e(*(X160, X161))), +(*(X160, X162), *(X161, X163))) :- ','(p(d(e(X160)), X163), p(d(e(X161)), X162)).\nand substitutionX160 -> T133,\nX161 -> T134,\nT102 -> e(*(T133, T134)),\nX162 -> X164,\nX163 -> X165,\nX100 -> +(*(T133, X164), *(T134, X165)),\nT131 -> T133,\nT132 -> T134" }, { "from": 575, "to": 582, "label": "EVAL-BACKTRACK" }, { "from": 576, "to": 583, "label": "EVAL with clause\np(d(d(X173)), X174) :- ','(p(d(X173), X175), p(d(e(X175)), X174)).\nand substitutionX173 -> T138,\nT102 -> d(T138),\nX100 -> X176,\nX174 -> X176,\nT137 -> T138" }, { "from": 576, "to": 584, "label": "EVAL-BACKTRACK" }, { "from": 581, "to": 555, "label": "INSTANCE with matching:\nT119 -> T133\nX134 -> X165\nT120 -> T134\nX135 -> X164" }, { "from": 583, "to": 585, "label": "SPLIT 1" }, { "from": 583, "to": 586, "label": "SPLIT 2\nreplacements:X175 -> T139" }, { "from": 585, "to": 524, "label": "INSTANCE with matching:\nT102 -> T138\nX100 -> X175" }, { "from": 586, "to": 524, "label": "INSTANCE with matching:\nT102 -> e(T139)\nX100 -> X176" } ], "type": "Graph" } } ---------------------------------------- (62) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f8_in(const(1)) -> f8_out1 f8_in(const(0)) -> f8_out1 f8_in(+(T26, T27)) -> U1(f222_in(T26, T27), +(T26, T27)) U1(f222_out1, +(T26, T27)) -> f8_out1 f8_in(+(*(T47, T49), *(T48, T50))) -> U2(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f410_out1, +(*(T47, T49), *(T48, T50))) -> f8_out1 f8_in(T101) -> U3(f522_in(T101), T101) U3(f522_out1, T101) -> f8_out1 f433_in(t, const(1)) -> f433_out1 f433_in(const(T55), const(0)) -> f433_out1 f433_in(+(T72, T73), +(T74, T75)) -> U4(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f410_out1, +(T72, T73), +(T74, T75)) -> f433_out1 f433_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f410_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f433_out1 f524_in -> f524_out1 f524_in -> U6(f555_in) U6(f555_out1) -> f524_out1 f524_in -> U7(f555_in) U7(f555_out1) -> f524_out1 f524_in -> U8(f583_in) U8(f583_out1) -> f524_out1 f222_in(T26, T27) -> U9(f8_in(T26), T26, T27) U9(f8_out1, T26, T27) -> U10(f8_in(T27), T26, T27) U10(f8_out1, T26, T27) -> f222_out1 f410_in(T47, T50, T48, T49) -> U11(f433_in(T47, T50), T47, T50, T48, T49) U11(f433_out1, T47, T50, T48, T49) -> U12(f433_in(T48, T49), T47, T50, T48, T49) U12(f433_out1, T47, T50, T48, T49) -> f410_out1 f522_in(T101) -> U13(f524_in, T101) U13(f524_out1, T101) -> U14(f8_in(T101), T101) U14(f8_out1, T101) -> f522_out1 f555_in -> U15(f524_in) U15(f524_out1) -> U16(f524_in) U16(f524_out1) -> f555_out1 f583_in -> U17(f524_in) U17(f524_out1) -> U18(f524_in) U18(f524_out1) -> f583_out1 Q is empty. ---------------------------------------- (63) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN(+(T26, T27)) -> U1^1(f222_in(T26, T27), +(T26, T27)) F8_IN(+(T26, T27)) -> F222_IN(T26, T27) F8_IN(+(*(T47, T49), *(T48, T50))) -> U2^1(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) F8_IN(+(*(T47, T49), *(T48, T50))) -> F410_IN(T47, T50, T48, T49) F8_IN(T101) -> U3^1(f522_in(T101), T101) F8_IN(T101) -> F522_IN(T101) F433_IN(+(T72, T73), +(T74, T75)) -> U4^1(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) F433_IN(+(T72, T73), +(T74, T75)) -> F410_IN(T72, T74, T73, T75) F433_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5^1(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) F433_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F410_IN(T92, T95, T93, T94) F524_IN -> U6^1(f555_in) F524_IN -> F555_IN F524_IN -> U7^1(f555_in) F524_IN -> U8^1(f583_in) F524_IN -> F583_IN F222_IN(T26, T27) -> U9^1(f8_in(T26), T26, T27) F222_IN(T26, T27) -> F8_IN(T26) U9^1(f8_out1, T26, T27) -> U10^1(f8_in(T27), T26, T27) U9^1(f8_out1, T26, T27) -> F8_IN(T27) F410_IN(T47, T50, T48, T49) -> U11^1(f433_in(T47, T50), T47, T50, T48, T49) F410_IN(T47, T50, T48, T49) -> F433_IN(T47, T50) U11^1(f433_out1, T47, T50, T48, T49) -> U12^1(f433_in(T48, T49), T47, T50, T48, T49) U11^1(f433_out1, T47, T50, T48, T49) -> F433_IN(T48, T49) F522_IN(T101) -> U13^1(f524_in, T101) F522_IN(T101) -> F524_IN U13^1(f524_out1, T101) -> U14^1(f8_in(T101), T101) U13^1(f524_out1, T101) -> F8_IN(T101) F555_IN -> U15^1(f524_in) F555_IN -> F524_IN U15^1(f524_out1) -> U16^1(f524_in) U15^1(f524_out1) -> F524_IN F583_IN -> U17^1(f524_in) F583_IN -> F524_IN U17^1(f524_out1) -> U18^1(f524_in) U17^1(f524_out1) -> F524_IN The TRS R consists of the following rules: f8_in(const(1)) -> f8_out1 f8_in(const(0)) -> f8_out1 f8_in(+(T26, T27)) -> U1(f222_in(T26, T27), +(T26, T27)) U1(f222_out1, +(T26, T27)) -> f8_out1 f8_in(+(*(T47, T49), *(T48, T50))) -> U2(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f410_out1, +(*(T47, T49), *(T48, T50))) -> f8_out1 f8_in(T101) -> U3(f522_in(T101), T101) U3(f522_out1, T101) -> f8_out1 f433_in(t, const(1)) -> f433_out1 f433_in(const(T55), const(0)) -> f433_out1 f433_in(+(T72, T73), +(T74, T75)) -> U4(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f410_out1, +(T72, T73), +(T74, T75)) -> f433_out1 f433_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f410_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f433_out1 f524_in -> f524_out1 f524_in -> U6(f555_in) U6(f555_out1) -> f524_out1 f524_in -> U7(f555_in) U7(f555_out1) -> f524_out1 f524_in -> U8(f583_in) U8(f583_out1) -> f524_out1 f222_in(T26, T27) -> U9(f8_in(T26), T26, T27) U9(f8_out1, T26, T27) -> U10(f8_in(T27), T26, T27) U10(f8_out1, T26, T27) -> f222_out1 f410_in(T47, T50, T48, T49) -> U11(f433_in(T47, T50), T47, T50, T48, T49) U11(f433_out1, T47, T50, T48, T49) -> U12(f433_in(T48, T49), T47, T50, T48, T49) U12(f433_out1, T47, T50, T48, T49) -> f410_out1 f522_in(T101) -> U13(f524_in, T101) U13(f524_out1, T101) -> U14(f8_in(T101), T101) U14(f8_out1, T101) -> f522_out1 f555_in -> U15(f524_in) U15(f524_out1) -> U16(f524_in) U16(f524_out1) -> f555_out1 f583_in -> U17(f524_in) U17(f524_out1) -> U18(f524_in) U18(f524_out1) -> f583_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 15 less nodes. ---------------------------------------- (66) Complex Obligation (AND) ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: F524_IN -> F555_IN F555_IN -> U15^1(f524_in) U15^1(f524_out1) -> F524_IN F524_IN -> F583_IN F583_IN -> U17^1(f524_in) U17^1(f524_out1) -> F524_IN F583_IN -> F524_IN F555_IN -> F524_IN The TRS R consists of the following rules: f8_in(const(1)) -> f8_out1 f8_in(const(0)) -> f8_out1 f8_in(+(T26, T27)) -> U1(f222_in(T26, T27), +(T26, T27)) U1(f222_out1, +(T26, T27)) -> f8_out1 f8_in(+(*(T47, T49), *(T48, T50))) -> U2(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f410_out1, +(*(T47, T49), *(T48, T50))) -> f8_out1 f8_in(T101) -> U3(f522_in(T101), T101) U3(f522_out1, T101) -> f8_out1 f433_in(t, const(1)) -> f433_out1 f433_in(const(T55), const(0)) -> f433_out1 f433_in(+(T72, T73), +(T74, T75)) -> U4(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f410_out1, +(T72, T73), +(T74, T75)) -> f433_out1 f433_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f410_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f433_out1 f524_in -> f524_out1 f524_in -> U6(f555_in) U6(f555_out1) -> f524_out1 f524_in -> U7(f555_in) U7(f555_out1) -> f524_out1 f524_in -> U8(f583_in) U8(f583_out1) -> f524_out1 f222_in(T26, T27) -> U9(f8_in(T26), T26, T27) U9(f8_out1, T26, T27) -> U10(f8_in(T27), T26, T27) U10(f8_out1, T26, T27) -> f222_out1 f410_in(T47, T50, T48, T49) -> U11(f433_in(T47, T50), T47, T50, T48, T49) U11(f433_out1, T47, T50, T48, T49) -> U12(f433_in(T48, T49), T47, T50, T48, T49) U12(f433_out1, T47, T50, T48, T49) -> f410_out1 f522_in(T101) -> U13(f524_in, T101) U13(f524_out1, T101) -> U14(f8_in(T101), T101) U14(f8_out1, T101) -> f522_out1 f555_in -> U15(f524_in) U15(f524_out1) -> U16(f524_in) U16(f524_out1) -> f555_out1 f583_in -> U17(f524_in) U17(f524_out1) -> U18(f524_in) U18(f524_out1) -> f583_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) NonLoopProof (COMPLETE) By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. We apply the theorem with m = 1, b = 0, σ' = [ ], and μ' = [ ] on the rule F555_IN[ ]^n[ ] -> F555_IN[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: F555_IN[ ]^n[ ] -> F555_IN[ ]^n[ ] by Narrowing at position: [] F555_IN[ ]^n[ ] -> F524_IN[ ]^n[ ] by Rule from TRS P F524_IN[ ]^n[ ] -> F555_IN[ ]^n[ ] by Rule from TRS P ---------------------------------------- (69) NO ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: F433_IN(+(T72, T73), +(T74, T75)) -> F410_IN(T72, T74, T73, T75) F410_IN(T47, T50, T48, T49) -> U11^1(f433_in(T47, T50), T47, T50, T48, T49) U11^1(f433_out1, T47, T50, T48, T49) -> F433_IN(T48, T49) F433_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F410_IN(T92, T95, T93, T94) F410_IN(T47, T50, T48, T49) -> F433_IN(T47, T50) The TRS R consists of the following rules: f8_in(const(1)) -> f8_out1 f8_in(const(0)) -> f8_out1 f8_in(+(T26, T27)) -> U1(f222_in(T26, T27), +(T26, T27)) U1(f222_out1, +(T26, T27)) -> f8_out1 f8_in(+(*(T47, T49), *(T48, T50))) -> U2(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f410_out1, +(*(T47, T49), *(T48, T50))) -> f8_out1 f8_in(T101) -> U3(f522_in(T101), T101) U3(f522_out1, T101) -> f8_out1 f433_in(t, const(1)) -> f433_out1 f433_in(const(T55), const(0)) -> f433_out1 f433_in(+(T72, T73), +(T74, T75)) -> U4(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f410_out1, +(T72, T73), +(T74, T75)) -> f433_out1 f433_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f410_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f433_out1 f524_in -> f524_out1 f524_in -> U6(f555_in) U6(f555_out1) -> f524_out1 f524_in -> U7(f555_in) U7(f555_out1) -> f524_out1 f524_in -> U8(f583_in) U8(f583_out1) -> f524_out1 f222_in(T26, T27) -> U9(f8_in(T26), T26, T27) U9(f8_out1, T26, T27) -> U10(f8_in(T27), T26, T27) U10(f8_out1, T26, T27) -> f222_out1 f410_in(T47, T50, T48, T49) -> U11(f433_in(T47, T50), T47, T50, T48, T49) U11(f433_out1, T47, T50, T48, T49) -> U12(f433_in(T48, T49), T47, T50, T48, T49) U12(f433_out1, T47, T50, T48, T49) -> f410_out1 f522_in(T101) -> U13(f524_in, T101) U13(f524_out1, T101) -> U14(f8_in(T101), T101) U14(f8_out1, T101) -> f522_out1 f555_in -> U15(f524_in) U15(f524_out1) -> U16(f524_in) U16(f524_out1) -> f555_out1 f583_in -> U17(f524_in) U17(f524_out1) -> U18(f524_in) U18(f524_out1) -> f583_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) 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: *F410_IN(T47, T50, T48, T49) -> F433_IN(T47, T50) The graph contains the following edges 1 >= 1, 2 >= 2 *F410_IN(T47, T50, T48, T49) -> U11^1(f433_in(T47, T50), T47, T50, T48, T49) The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5 *U11^1(f433_out1, T47, T50, T48, T49) -> F433_IN(T48, T49) The graph contains the following edges 4 >= 1, 5 >= 2 *F433_IN(+(T72, T73), +(T74, T75)) -> F410_IN(T72, T74, T73, T75) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *F433_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F410_IN(T92, T95, T93, T94) The graph contains the following edges 1 > 1, 2 > 1, 2 > 2, 1 > 3, 2 > 3, 2 > 4 ---------------------------------------- (72) YES ---------------------------------------- (73) Obligation: Q DP problem: The TRS P consists of the following rules: F8_IN(+(T26, T27)) -> F222_IN(T26, T27) F222_IN(T26, T27) -> U9^1(f8_in(T26), T26, T27) U9^1(f8_out1, T26, T27) -> F8_IN(T27) F8_IN(T101) -> F522_IN(T101) F522_IN(T101) -> U13^1(f524_in, T101) U13^1(f524_out1, T101) -> F8_IN(T101) F222_IN(T26, T27) -> F8_IN(T26) The TRS R consists of the following rules: f8_in(const(1)) -> f8_out1 f8_in(const(0)) -> f8_out1 f8_in(+(T26, T27)) -> U1(f222_in(T26, T27), +(T26, T27)) U1(f222_out1, +(T26, T27)) -> f8_out1 f8_in(+(*(T47, T49), *(T48, T50))) -> U2(f410_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f410_out1, +(*(T47, T49), *(T48, T50))) -> f8_out1 f8_in(T101) -> U3(f522_in(T101), T101) U3(f522_out1, T101) -> f8_out1 f433_in(t, const(1)) -> f433_out1 f433_in(const(T55), const(0)) -> f433_out1 f433_in(+(T72, T73), +(T74, T75)) -> U4(f410_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f410_out1, +(T72, T73), +(T74, T75)) -> f433_out1 f433_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f410_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f410_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f433_out1 f524_in -> f524_out1 f524_in -> U6(f555_in) U6(f555_out1) -> f524_out1 f524_in -> U7(f555_in) U7(f555_out1) -> f524_out1 f524_in -> U8(f583_in) U8(f583_out1) -> f524_out1 f222_in(T26, T27) -> U9(f8_in(T26), T26, T27) U9(f8_out1, T26, T27) -> U10(f8_in(T27), T26, T27) U10(f8_out1, T26, T27) -> f222_out1 f410_in(T47, T50, T48, T49) -> U11(f433_in(T47, T50), T47, T50, T48, T49) U11(f433_out1, T47, T50, T48, T49) -> U12(f433_in(T48, T49), T47, T50, T48, T49) U12(f433_out1, T47, T50, T48, T49) -> f410_out1 f522_in(T101) -> U13(f524_in, T101) U13(f524_out1, T101) -> U14(f8_in(T101), T101) U14(f8_out1, T101) -> f522_out1 f555_in -> U15(f524_in) U15(f524_out1) -> U16(f524_in) U16(f524_out1) -> f555_out1 f583_in -> U17(f524_in) U17(f524_out1) -> U18(f524_in) U18(f524_out1) -> f583_out1 Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (74) NonLoopProof (COMPLETE) By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP. We apply the theorem with m = 1, b = 0, σ' = [ ], and μ' = [ ] on the rule U13^1(f524_out1, x0)[ ]^n[ ] -> U13^1(f524_out1, x0)[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: U13^1(f524_out1, x0)[ ]^n[ ] -> U13^1(f524_out1, x0)[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation U13^1(f524_out1, T101)[ ]^n[ ] -> F8_IN(T101)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation F8_IN(x0)[ ]^n[ ] -> U13^1(f524_out1, x0)[ ]^n[ ] by Narrowing at position: [0] F8_IN(x0)[ ]^n[ ] -> U13^1(f524_in, x0)[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation F8_IN(T101)[ ]^n[ ] -> F522_IN(T101)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation F522_IN(T101)[ ]^n[ ] -> U13^1(f524_in, T101)[ ]^n[ ] by Rule from TRS P f524_in[ ]^n[ ] -> f524_out1[ ]^n[ ] by Rule from TRS R ---------------------------------------- (75) NO ---------------------------------------- (76) PrologToIRSwTTransformerProof (SOUND) Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert { "root": 1, "program": { "directives": [], "clauses": [ [ "(p (d (e (t))) (const (1)))", null ], [ "(p (d (e (const A))) (const (0)))", null ], [ "(p (d (e (+ X Y))) (+ DX DY))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (e (* X Y))) (+ (* X DY) (* Y DX)))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (d X)) DDX)", "(',' (p (d X) DX) (p (d (e DX)) DDX))" ] ] }, "graph": { "nodes": { "type": "Nodes", "393": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T28)) T26)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": [], "exprvars": [] } }, "394": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T30)) T27)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T27"], "free": [], "exprvars": [] } }, "471": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, 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"(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "429": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T92)) T95) (p (d (e T93)) T94))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T92", "T93", "T94", "T95" ], "free": [], "exprvars": [] } }, "507": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T119)) X134)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X134"], "exprvars": [] } }, "508": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T122)) X135)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X135"], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 7, "label": "CASE" }, { "from": 7, "to": 33, "label": "PARALLEL" }, { "from": 7, "to": 34, "label": "PARALLEL" }, { "from": 33, "to": 131, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 33, "to": 132, "label": "EVAL-BACKTRACK" }, { "from": 34, "to": 134, "label": "PARALLEL" }, { "from": 34, "to": 135, "label": "PARALLEL" }, { "from": 131, "to": 133, "label": "SUCCESS" }, { "from": 134, "to": 136, "label": "EVAL with clause\np(d(e(const(X5))), const(0)).\nand substitutionX5 -> T7,\nT1 -> d(e(const(T7))),\nT2 -> const(0)" }, { "from": 134, "to": 137, "label": "EVAL-BACKTRACK" }, { "from": 135, "to": 139, "label": "PARALLEL" }, { "from": 135, "to": 140, "label": "PARALLEL" }, { "from": 136, "to": 138, "label": "SUCCESS" }, { "from": 139, "to": 383, "label": "EVAL with clause\np(d(e(+(X22, X23))), +(X24, X25)) :- ','(p(d(e(X22)), X24), p(d(e(X23)), X25)).\nand substitutionX22 -> T28,\nX23 -> T29,\nT1 -> d(e(+(T28, T29))),\nX24 -> T26,\nX25 -> T27,\nT2 -> +(T26, T27),\nT24 -> T28,\nT25 -> T29" }, { "from": 139, "to": 384, "label": "EVAL-BACKTRACK" }, { "from": 140, "to": 398, "label": "PARALLEL" }, { "from": 140, "to": 399, "label": "PARALLEL" }, { "from": 383, "to": 393, "label": "SPLIT 1" }, { "from": 383, "to": 394, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:T29 -> T30" }, { "from": 393, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T28))\nT2 -> T26" }, { "from": 394, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T30))\nT2 -> T27" }, { "from": 398, "to": 406, "label": "EVAL with clause\np(d(e(*(X42, X43))), +(*(X42, X44), *(X43, X45))) :- ','(p(d(e(X42)), X45), p(d(e(X43)), X44)).\nand substitutionX42 -> T47,\nX43 -> T48,\nT1 -> d(e(*(T47, T48))),\nX44 -> T49,\nX45 -> T50,\nT2 -> +(*(T47, T49), *(T48, T50))" }, { "from": 398, "to": 407, "label": "EVAL-BACKTRACK" }, { "from": 399, "to": 447, "label": "EVAL with clause\np(d(d(X98)), X99) :- ','(p(d(X98), X100), p(d(e(X100)), X99)).\nand substitutionX98 -> T102,\nT1 -> d(d(T102)),\nT2 -> T101,\nX99 -> T101,\nT100 -> T102" }, { "from": 399, "to": 448, "label": "EVAL-BACKTRACK" }, { "from": 406, "to": 408, "label": "SPLIT 1" }, { "from": 406, "to": 409, "label": "SPLIT 2\nnew knowledge:\nT47 is ground\nT50 is ground" }, { "from": 408, "to": 409, "label": "INSTANCE with matching:\nT48 -> T47\nT49 -> T50" }, { "from": 409, "to": 412, "label": "CASE" }, { "from": 412, "to": 413, "label": "PARALLEL" }, { "from": 412, "to": 414, "label": "PARALLEL" }, { "from": 413, "to": 415, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT48 -> t,\nT49 -> const(1)" }, { "from": 413, "to": 416, "label": "EVAL-BACKTRACK" }, { "from": 414, "to": 418, "label": "PARALLEL" }, { "from": 414, "to": 419, "label": "PARALLEL" }, { "from": 415, "to": 417, "label": "SUCCESS" }, { "from": 418, "to": 420, "label": "EVAL with clause\np(d(e(const(X50))), const(0)).\nand substitutionX50 -> T55,\nT48 -> const(T55),\nT49 -> const(0)" }, { "from": 418, "to": 421, "label": "EVAL-BACKTRACK" }, { "from": 419, "to": 423, "label": "PARALLEL" }, { "from": 419, "to": 424, "label": "PARALLEL" }, { "from": 420, "to": 422, "label": "SUCCESS" }, { "from": 423, "to": 425, "label": "EVAL with clause\np(d(e(+(X67, X68))), +(X69, X70)) :- ','(p(d(e(X67)), X69), p(d(e(X68)), X70)).\nand substitutionX67 -> T72,\nX68 -> T73,\nT48 -> +(T72, T73),\nX69 -> T74,\nX70 -> T75,\nT49 -> +(T74, T75)" }, { "from": 423, "to": 426, "label": "EVAL-BACKTRACK" }, { "from": 424, "to": 427, "label": "PARALLEL" }, { "from": 424, "to": 428, "label": "PARALLEL" }, { "from": 425, "to": 406, "label": "INSTANCE with matching:\nT47 -> T72\nT50 -> T74\nT48 -> T73\nT49 -> T75" }, { "from": 427, "to": 429, "label": "EVAL with clause\np(d(e(*(X87, X88))), +(*(X87, X89), *(X88, X90))) :- ','(p(d(e(X87)), X90), p(d(e(X88)), X89)).\nand substitutionX87 -> T92,\nX88 -> T93,\nT48 -> *(T92, T93),\nX89 -> T94,\nX90 -> T95,\nT49 -> +(*(T92, T94), *(T93, T95))" }, { "from": 427, "to": 430, "label": "EVAL-BACKTRACK" }, { "from": 428, "to": 431, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 429, "to": 406, "label": "INSTANCE with matching:\nT47 -> T92\nT50 -> T95\nT48 -> T93\nT49 -> T94" }, { "from": 447, "to": 458, "label": "SPLIT 1" }, { "from": 447, "to": 459, "label": "SPLIT 2\nreplacements:X100 -> T103" }, { "from": 458, "to": 463, "label": "CASE" }, { "from": 459, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T103))\nT2 -> T101" }, { "from": 463, "to": 464, "label": "PARALLEL" }, { "from": 463, "to": 465, "label": "PARALLEL" }, { "from": 464, "to": 468, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT102 -> e(t),\nX100 -> const(1)" }, { "from": 464, "to": 469, "label": "EVAL-BACKTRACK" }, { "from": 465, "to": 475, "label": "PARALLEL" }, { "from": 465, "to": 476, "label": "PARALLEL" }, { "from": 468, "to": 471, "label": "SUCCESS" }, { "from": 475, "to": 477, "label": "EVAL with clause\np(d(e(const(X105))), const(0)).\nand substitutionX105 -> T108,\nT102 -> e(const(T108)),\nX100 -> const(0)" }, { "from": 475, "to": 478, "label": "EVAL-BACKTRACK" }, { "from": 476, "to": 499, "label": "PARALLEL" }, { "from": 476, "to": 500, "label": "PARALLEL" }, { "from": 477, "to": 479, "label": "SUCCESS" }, { "from": 499, "to": 503, "label": "EVAL with clause\np(d(e(+(X130, X131))), +(X132, X133)) :- ','(p(d(e(X130)), X132), p(d(e(X131)), X133)).\nand substitutionX130 -> T119,\nX131 -> T120,\nT102 -> e(+(T119, T120)),\nX132 -> X134,\nX133 -> X135,\nX100 -> +(X134, X135),\nT117 -> T119,\nT118 -> T120" }, { "from": 499, "to": 504, "label": "EVAL-BACKTRACK" }, { "from": 500, "to": 512, "label": "PARALLEL" }, { "from": 500, "to": 513, "label": "PARALLEL" }, { "from": 503, "to": 507, "label": "SPLIT 1" }, { "from": 503, "to": 508, "label": "SPLIT 2\nreplacements:X134 -> T121,\nT120 -> T122" }, { "from": 507, "to": 458, "label": "INSTANCE with matching:\nT102 -> e(T119)\nX100 -> X134" }, { "from": 508, "to": 458, "label": "INSTANCE with matching:\nT102 -> e(T122)\nX100 -> X135" }, { "from": 512, "to": 520, "label": "EVAL with clause\np(d(e(*(X160, X161))), +(*(X160, X162), *(X161, X163))) :- ','(p(d(e(X160)), X163), p(d(e(X161)), X162)).\nand substitutionX160 -> T133,\nX161 -> T134,\nT102 -> e(*(T133, T134)),\nX162 -> X164,\nX163 -> X165,\nX100 -> +(*(T133, X164), *(T134, X165)),\nT131 -> T133,\nT132 -> T134" }, { "from": 512, "to": 521, "label": "EVAL-BACKTRACK" }, { "from": 513, "to": 535, "label": "EVAL with clause\np(d(d(X173)), X174) :- ','(p(d(X173), X175), p(d(e(X175)), X174)).\nand substitutionX173 -> T138,\nT102 -> d(T138),\nX100 -> X176,\nX174 -> X176,\nT137 -> T138" }, { "from": 513, "to": 536, "label": "EVAL-BACKTRACK" }, { "from": 520, "to": 503, "label": "INSTANCE with matching:\nT119 -> T133\nX134 -> X165\nT120 -> T134\nX135 -> X164" }, { "from": 535, "to": 542, "label": "SPLIT 1" }, { "from": 535, "to": 544, "label": "SPLIT 2\nreplacements:X175 -> T139" }, { "from": 542, "to": 458, "label": "INSTANCE with matching:\nT102 -> T138\nX100 -> X175" }, { "from": 544, "to": 458, "label": "INSTANCE with matching:\nT102 -> e(T139)\nX100 -> X176" } ], "type": "Graph" } } ---------------------------------------- (77) Complex Obligation (AND) ---------------------------------------- (78) Obligation: Rules: f512_out -> f500_out :|: TRUE f513_out -> f500_out :|: TRUE f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f476_out -> f465_out :|: TRUE f475_out -> f465_out :|: TRUE f465_in -> f475_in :|: TRUE f465_in -> f476_in :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f500_out -> f476_out :|: TRUE f499_out -> f476_out :|: TRUE f513_in -> f535_in :|: TRUE f535_out -> f513_out :|: TRUE f513_in -> f536_in :|: TRUE f536_out -> f513_out :|: TRUE f458_out -> f544_out :|: TRUE f544_in -> f458_in :|: TRUE f520_in -> f503_in :|: TRUE f503_out -> f520_out :|: TRUE f458_out -> f542_out :|: TRUE f542_in -> f458_in :|: TRUE f463_out -> f458_out :|: TRUE f458_in -> f463_in :|: TRUE f463_in -> f464_in :|: TRUE f464_out -> f463_out :|: TRUE f463_in -> f465_in :|: TRUE f465_out -> f463_out :|: TRUE f542_out -> f544_in :|: TRUE f544_out -> f535_out :|: TRUE f535_in -> f542_in :|: TRUE f507_in -> f458_in :|: TRUE f458_out -> f507_out :|: TRUE f499_in -> f504_in :|: TRUE f499_in -> f503_in :|: TRUE f503_out -> f499_out :|: TRUE f504_out -> f499_out :|: TRUE f512_in -> f521_in :|: TRUE f512_in -> f520_in :|: TRUE f521_out -> f512_out :|: TRUE f520_out -> f512_out :|: TRUE f458_out -> f508_out :|: TRUE f508_in -> f458_in :|: TRUE f503_in -> f507_in :|: TRUE f508_out -> f503_out :|: TRUE f507_out -> f508_in :|: TRUE f7_out(T2) -> f1_out(T2) :|: TRUE f1_in(x) -> f7_in(x) :|: TRUE f7_in(x1) -> f33_in(x1) :|: TRUE f34_out(x2) -> f7_out(x2) :|: TRUE f33_out(x3) -> f7_out(x3) :|: TRUE f7_in(x4) -> f34_in(x4) :|: TRUE f34_in(x5) -> f135_in(x5) :|: TRUE f34_in(x6) -> f134_in(x6) :|: TRUE f135_out(x7) -> f34_out(x7) :|: TRUE f134_out(x8) -> f34_out(x8) :|: TRUE f135_in(x9) -> f140_in(x9) :|: TRUE f140_out(x10) -> f135_out(x10) :|: TRUE f139_out(x11) -> f135_out(x11) :|: TRUE f135_in(x12) -> f139_in(x12) :|: TRUE f398_out(x13) -> f140_out(x13) :|: TRUE f399_out(x14) -> f140_out(x14) :|: TRUE f140_in(x15) -> f399_in(x15) :|: TRUE f140_in(x16) -> f398_in(x16) :|: TRUE f447_out(T101) -> f399_out(T101) :|: TRUE f399_in(x17) -> f448_in :|: TRUE f399_in(x18) -> f447_in(x18) :|: TRUE f448_out -> f399_out(x19) :|: TRUE f458_out -> f459_in(x20) :|: TRUE f459_out(x21) -> f447_out(x21) :|: TRUE f447_in(x22) -> f458_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (79) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f465_in -> f476_in :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f513_in -> f535_in :|: TRUE f520_in -> f503_in :|: TRUE f542_in -> f458_in :|: TRUE f458_in -> f463_in :|: TRUE f463_in -> f465_in :|: TRUE f535_in -> f542_in :|: TRUE f507_in -> f458_in :|: TRUE f499_in -> f503_in :|: TRUE f512_in -> f520_in :|: TRUE f503_in -> f507_in :|: TRUE ---------------------------------------- (80) Obligation: Rules: f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f465_in -> f476_in :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f513_in -> f535_in :|: TRUE f520_in -> f503_in :|: TRUE f542_in -> f458_in :|: TRUE f458_in -> f463_in :|: TRUE f463_in -> f465_in :|: TRUE f535_in -> f542_in :|: TRUE f507_in -> f458_in :|: TRUE f499_in -> f503_in :|: TRUE f512_in -> f520_in :|: TRUE f503_in -> f507_in :|: TRUE ---------------------------------------- (81) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (82) Obligation: Rules: f465_in -> f465_in :|: TRUE ---------------------------------------- (83) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (84) Obligation: Rules: f465_in -> f465_in :|: TRUE ---------------------------------------- (85) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f465_in -> f465_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (86) Obligation: Termination digraph: Nodes: (1) f465_in -> f465_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (87) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f465_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (88) Obligation: Rules: f465_in -> f465_in :|: TRUE ---------------------------------------- (89) 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)) ---------------------------------------- (90) NO ---------------------------------------- (91) Obligation: Rules: f425_in(T72, T74, T73, T75) -> f406_in(T72, T74, T73, T75) :|: TRUE f406_out(x, x1, x2, x3) -> f425_out(x, x1, x2, x3) :|: TRUE f427_in(T92 * T93, T92 * T94 + T93 * T95) -> f429_in(T92, T95, T93, T94) :|: TRUE f430_out -> f427_out(T48, T49) :|: TRUE f429_out(x4, x5, x6, x7) -> f427_out(x4 * x6, x4 * x7 + x6 * x5) :|: TRUE f427_in(x8, x9) -> f430_in :|: TRUE f412_in(x10, x11) -> f414_in(x10, x11) :|: TRUE f413_out(x12, x13) -> f412_out(x12, x13) :|: TRUE f412_in(x14, x15) -> f413_in(x14, x15) :|: TRUE f414_out(x16, x17) -> f412_out(x16, x17) :|: TRUE f425_out(x18, x19, x20, x21) -> f423_out(x18 + x20, x19 + x21) :|: TRUE f423_in(x22, x23) -> f426_in :|: TRUE f426_out -> f423_out(x24, x25) :|: TRUE f423_in(x26 + x27, x28 + x29) -> f425_in(x26, x28, x27, x29) :|: TRUE f409_out(T47, T50) -> f408_out(T47, T50) :|: TRUE f408_in(x30, x31) -> f409_in(x30, x31) :|: TRUE f414_in(x32, x33) -> f419_in(x32, x33) :|: TRUE f419_out(x34, x35) -> f414_out(x34, x35) :|: TRUE f418_out(x36, x37) -> f414_out(x36, x37) :|: TRUE f414_in(x38, x39) -> f418_in(x38, x39) :|: TRUE f409_in(x40, x41) -> f412_in(x40, x41) :|: TRUE f412_out(x42, x43) -> f409_out(x42, x43) :|: TRUE f427_out(x44, x45) -> f424_out(x44, x45) :|: TRUE f428_out(x46, x47) -> f424_out(x46, x47) :|: TRUE f424_in(x48, x49) -> f427_in(x48, x49) :|: TRUE f424_in(x50, x51) -> f428_in(x50, x51) :|: TRUE f419_in(x52, x53) -> f423_in(x52, x53) :|: TRUE f423_out(x54, x55) -> f419_out(x54, x55) :|: TRUE f424_out(x56, x57) -> f419_out(x56, x57) :|: TRUE f419_in(x58, x59) -> f424_in(x58, x59) :|: TRUE f409_out(x60, x61) -> f406_out(x62, x63, x60, x61) :|: TRUE f406_in(x64, x65, x66, x67) -> f408_in(x64, x65) :|: TRUE f408_out(x68, x69) -> f409_in(x70, x71) :|: TRUE f406_out(x72, x73, x74, x75) -> f429_out(x72, x73, x74, x75) :|: TRUE f429_in(x76, x77, x78, x79) -> f406_in(x76, x77, x78, x79) :|: TRUE f7_out(T2) -> f1_out(T2) :|: TRUE f1_in(x80) -> f7_in(x80) :|: TRUE f7_in(x81) -> f33_in(x81) :|: TRUE f34_out(x82) -> f7_out(x82) :|: TRUE f33_out(x83) -> f7_out(x83) :|: TRUE f7_in(x84) -> f34_in(x84) :|: TRUE f34_in(x85) -> f135_in(x85) :|: TRUE f34_in(x86) -> f134_in(x86) :|: TRUE f135_out(x87) -> f34_out(x87) :|: TRUE f134_out(x88) -> f34_out(x88) :|: TRUE f135_in(x89) -> f140_in(x89) :|: TRUE f140_out(x90) -> f135_out(x90) :|: TRUE f139_out(x91) -> f135_out(x91) :|: TRUE f135_in(x92) -> f139_in(x92) :|: TRUE f398_out(x93) -> f140_out(x93) :|: TRUE f399_out(x94) -> f140_out(x94) :|: TRUE f140_in(x95) -> f399_in(x95) :|: TRUE f140_in(x96) -> f398_in(x96) :|: TRUE f398_in(x97) -> f407_in :|: TRUE f406_out(x98, x99, x100, x101) -> f398_out(x98 * x101 + x100 * x99) :|: TRUE f407_out -> f398_out(x102) :|: TRUE f398_in(x103 * x104 + x105 * x106) -> f406_in(x103, x106, x105, x104) :|: TRUE Start term: f1_in(T2) ---------------------------------------- (92) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f425_in(T72, T74, T73, T75) -> f406_in(T72, T74, T73, T75) :|: TRUE f427_in(T92 * T93, T92 * T94 + T93 * T95) -> f429_in(T92, T95, T93, T94) :|: TRUE f412_in(x10, x11) -> f414_in(x10, x11) :|: TRUE f423_in(x26 + x27, x28 + x29) -> f425_in(x26, x28, x27, x29) :|: TRUE f408_in(x30, x31) -> f409_in(x30, x31) :|: TRUE f414_in(x32, x33) -> f419_in(x32, x33) :|: TRUE f409_in(x40, x41) -> f412_in(x40, x41) :|: TRUE f424_in(x48, x49) -> f427_in(x48, x49) :|: TRUE f419_in(x52, x53) -> f423_in(x52, x53) :|: TRUE f419_in(x58, x59) -> f424_in(x58, x59) :|: TRUE f406_in(x64, x65, x66, x67) -> f408_in(x64, x65) :|: TRUE f429_in(x76, x77, x78, x79) -> f406_in(x76, x77, x78, x79) :|: TRUE ---------------------------------------- (93) Obligation: Rules: f425_in(T72, T74, T73, T75) -> f406_in(T72, T74, T73, T75) :|: TRUE f427_in(T92 * T93, T92 * T94 + T93 * T95) -> f429_in(T92, T95, T93, T94) :|: TRUE f412_in(x10, x11) -> f414_in(x10, x11) :|: TRUE f423_in(x26 + x27, x28 + x29) -> f425_in(x26, x28, x27, x29) :|: TRUE f408_in(x30, x31) -> f409_in(x30, x31) :|: TRUE f414_in(x32, x33) -> f419_in(x32, x33) :|: TRUE f409_in(x40, x41) -> f412_in(x40, x41) :|: TRUE f424_in(x48, x49) -> f427_in(x48, x49) :|: TRUE f419_in(x52, x53) -> f423_in(x52, x53) :|: TRUE f419_in(x58, x59) -> f424_in(x58, x59) :|: TRUE f406_in(x64, x65, x66, x67) -> f408_in(x64, x65) :|: TRUE f429_in(x76, x77, x78, x79) -> f406_in(x76, x77, x78, x79) :|: TRUE ---------------------------------------- (94) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (95) Obligation: Rules: f408_in(times~T92:0~T93:0, sum~times~T92:0~T94:0~times~T93:0~T95:0) -> f408_in(T92:0, T95:0) :|: TRUE && times~T92:0~T93:0 = T92:0 * T93:0 && sum~times~T92:0~T94:0~times~T93:0~T95:0 = T92:0 * T94:0 + T93:0 * T95:0 f408_in(sum~x26:0~x27:0, sum~x28:0~x29:0) -> f408_in(x26:0, x28:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 && sum~x28:0~x29:0 = x28:0 + x29:0 ---------------------------------------- (96) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (97) Obligation: Rules: f408_in(times~T92:0~T93:0, sum~times~T92:0~T94:0~times~T93:0~T95:0) -> f408_in(T92:0, T95:0) :|: TRUE && times~T92:0~T93:0 = T92:0 * T93:0 && sum~times~T92:0~T94:0~times~T93:0~T95:0 = T92:0 * T94:0 + T93:0 * T95:0 f408_in(sum~x26:0~x27:0, sum~x28:0~x29:0) -> f408_in(x26:0, x28:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 && sum~x28:0~x29:0 = x28:0 + x29:0 ---------------------------------------- (98) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f408_in(times~T92:0~T93:0, sum~times~T92:0~T94:0~times~T93:0~T95:0) -> f408_in(T92:0, T95:0) :|: TRUE && times~T92:0~T93:0 = T92:0 * T93:0 && sum~times~T92:0~T94:0~times~T93:0~T95:0 = T92:0 * T94:0 + T93:0 * T95:0 (2) f408_in(sum~x26:0~x27:0, sum~x28:0~x29:0) -> f408_in(x26:0, x28:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 && sum~x28:0~x29:0 = x28:0 + x29:0 Arcs: (1) -> (1), (2) (2) -> (1), (2) This digraph is fully evaluated! ---------------------------------------- (99) Obligation: Termination digraph: Nodes: (1) f408_in(times~T92:0~T93:0, sum~times~T92:0~T94:0~times~T93:0~T95:0) -> f408_in(T92:0, T95:0) :|: TRUE && times~T92:0~T93:0 = T92:0 * T93:0 && sum~times~T92:0~T94:0~times~T93:0~T95:0 = T92:0 * T94:0 + T93:0 * T95:0 (2) f408_in(sum~x26:0~x27:0, sum~x28:0~x29:0) -> f408_in(x26:0, x28:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 && sum~x28:0~x29:0 = x28:0 + x29:0 Arcs: (1) -> (1), (2) (2) -> (1), (2) This digraph is fully evaluated! ---------------------------------------- (100) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (101) Obligation: Rules: f408_in(times~T92:0:0~T93:0:0, sum~times~T92:0:0~T94:0:0~times~T93:0:0~T95:0:0) -> f408_in(T92:0:0, T95:0:0) :|: TRUE && times~T92:0:0~T93:0:0 = T92:0:0 * T93:0:0 && sum~times~T92:0:0~T94:0:0~times~T93:0:0~T95:0:0 = T92:0:0 * T94:0:0 + T93:0:0 * T95:0:0 f408_in(sum~x26:0:0~x27:0:0, sum~x28:0:0~x29:0:0) -> f408_in(x26:0:0, x28:0:0) :|: TRUE && sum~x26:0:0~x27:0:0 = x26:0:0 + x27:0:0 && sum~x28:0:0~x29:0:0 = x28:0:0 + x29:0:0 ---------------------------------------- (102) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (103) Obligation: Rules: f408_in(times~x2~x4, sum~times~x2~x5~times~x4~x3) -> f408_in(x2, x3) :|: TRUE && times~x2~x4 = x2 * x4 && sum~times~x2~x5~times~x4~x3 = x2 * x5 + x4 * x3 f408_in(sum~x8~x10, sum~x9~x11) -> f408_in(x8, x9) :|: TRUE && sum~x8~x10 = x8 + x10 && sum~x9~x11 = x9 + x11 ---------------------------------------- (104) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (105) Obligation: Rules: f408_in(times~x2:0~x4:0, sum~times~x2:0~x5:0~times~x4:0~x3:0) -> f408_in(x2:0, x3:0) :|: TRUE && times~x2:0~x4:0 = x2:0 * x4:0 && sum~times~x2:0~x5:0~times~x4:0~x3:0 = x2:0 * x5:0 + x4:0 * x3:0 f408_in(sum~x8:0~x10:0, sum~x9:0~x11:0) -> f408_in(x8:0, x9:0) :|: TRUE && sum~x8:0~x10:0 = x8:0 + x10:0 && sum~x9:0~x11:0 = x9:0 + x11:0 ---------------------------------------- (106) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc, times~x2:0~x4:0, sum~times~x2:0~x5:0~times~x4:0~x3:0) -> f(1, x2:0, x3:0) :|: pc = 1 && (TRUE && times~x2:0~x4:0 = x2:0 * x4:0 && sum~times~x2:0~x5:0~times~x4:0~x3:0 = x2:0 * x5:0 + x4:0 * x3:0) f(pc, sum~x8:0~x10:0, sum~x9:0~x11:0) -> f(1, x8:0, x9:0) :|: pc = 1 && (TRUE && sum~x8:0~x10:0 = x8:0 + x10:0 && sum~x9:0~x11:0 = x9:0 + x11:0) Witness term starting non-terminating reduction: f(1, 0, 0) ---------------------------------------- (107) NO ---------------------------------------- (108) Obligation: Rules: f135_in(T2) -> f140_in(T2) :|: TRUE f140_out(x) -> f135_out(x) :|: TRUE f139_out(x1) -> f135_out(x1) :|: TRUE f135_in(x2) -> f139_in(x2) :|: TRUE f1_out(T101) -> f459_out(T101) :|: TRUE f459_in(x3) -> f1_in(x3) :|: TRUE f394_in(T27) -> f1_in(T27) :|: TRUE f1_out(x4) -> f394_out(x4) :|: TRUE f476_out -> f465_out :|: TRUE f475_out -> f465_out :|: TRUE f465_in -> f475_in :|: TRUE f465_in -> f476_in :|: TRUE f477_in -> f477_out :|: TRUE f7_in(x5) -> f33_in(x5) :|: TRUE f34_out(x6) -> f7_out(x6) :|: TRUE f33_out(x7) -> f7_out(x7) :|: TRUE f7_in(x8) -> f34_in(x8) :|: TRUE f447_out(x9) -> f399_out(x9) :|: TRUE f399_in(x10) -> f448_in :|: TRUE f399_in(x11) -> f447_in(x11) :|: TRUE f448_out -> f399_out(x12) :|: TRUE f513_in -> f535_in :|: TRUE f535_out -> f513_out :|: TRUE f513_in -> f536_in :|: TRUE f536_out -> f513_out :|: TRUE f520_in -> f503_in :|: TRUE f503_out -> f520_out :|: TRUE f1_out(T26) -> f393_out(T26) :|: TRUE f393_in(x13) -> f1_in(x13) :|: TRUE f463_out -> f458_out :|: TRUE f458_in -> f463_in :|: TRUE f507_in -> f458_in :|: TRUE f458_out -> f507_out :|: TRUE f499_in -> f504_in :|: TRUE f499_in -> f503_in :|: TRUE f503_out -> f499_out :|: TRUE f504_out -> f499_out :|: TRUE f512_in -> f521_in :|: TRUE f512_in -> f520_in :|: TRUE f521_out -> f512_out :|: TRUE f520_out -> f512_out :|: TRUE f7_out(x14) -> f1_out(x14) :|: TRUE f1_in(x15) -> f7_in(x15) :|: TRUE f458_out -> f508_out :|: TRUE f508_in -> f458_in :|: TRUE f475_in -> f478_in :|: TRUE f478_out -> f475_out :|: TRUE f477_out -> f475_out :|: TRUE f475_in -> f477_in :|: TRUE f458_out -> f459_in(x16) :|: TRUE f459_out(x17) -> f447_out(x17) :|: TRUE f447_in(x18) -> f458_in :|: TRUE f468_out -> f464_out :|: TRUE f464_in -> f469_in :|: TRUE f469_out -> f464_out :|: TRUE f464_in -> f468_in :|: TRUE f512_out -> f500_out :|: TRUE f513_out -> f500_out :|: TRUE f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f468_in -> f468_out :|: TRUE f394_out(x19) -> f383_out(x20, x19) :|: TRUE f383_in(x21, x22) -> f393_in(x21) :|: TRUE f393_out(x23) -> f394_in(x24) :|: TRUE f139_in(x25) -> f384_in :|: TRUE f139_in(x26 + x27) -> f383_in(x26, x27) :|: TRUE f383_out(x28, x29) -> f139_out(x28 + x29) :|: TRUE f384_out -> f139_out(x30) :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f500_out -> f476_out :|: TRUE f499_out -> f476_out :|: TRUE f458_out -> f544_out :|: TRUE f544_in -> f458_in :|: TRUE f458_out -> f542_out :|: TRUE f542_in -> f458_in :|: TRUE f463_in -> f464_in :|: TRUE f464_out -> f463_out :|: TRUE f463_in -> f465_in :|: TRUE f465_out -> f463_out :|: TRUE f542_out -> f544_in :|: TRUE f544_out -> f535_out :|: TRUE f535_in -> f542_in :|: TRUE f398_out(x31) -> f140_out(x31) :|: TRUE f399_out(x32) -> f140_out(x32) :|: TRUE f140_in(x33) -> f399_in(x33) :|: TRUE f140_in(x34) -> f398_in(x34) :|: TRUE f34_in(x35) -> f135_in(x35) :|: TRUE f34_in(x36) -> f134_in(x36) :|: TRUE f135_out(x37) -> f34_out(x37) :|: TRUE f134_out(x38) -> f34_out(x38) :|: TRUE f503_in -> f507_in :|: TRUE f508_out -> f503_out :|: TRUE f507_out -> f508_in :|: TRUE Start term: f1_in(T2) ---------------------------------------- (109) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f135_in(T2) -> f140_in(T2) :|: TRUE f135_in(x2) -> f139_in(x2) :|: TRUE f459_in(x3) -> f1_in(x3) :|: TRUE f476_out -> f465_out :|: TRUE f475_out -> f465_out :|: TRUE f465_in -> f475_in :|: TRUE f465_in -> f476_in :|: TRUE f477_in -> f477_out :|: TRUE f7_in(x8) -> f34_in(x8) :|: TRUE f399_in(x11) -> f447_in(x11) :|: TRUE f513_in -> f535_in :|: TRUE f535_out -> f513_out :|: TRUE f520_in -> f503_in :|: TRUE f503_out -> f520_out :|: TRUE f393_in(x13) -> f1_in(x13) :|: TRUE f463_out -> f458_out :|: TRUE f458_in -> f463_in :|: TRUE f507_in -> f458_in :|: TRUE f458_out -> f507_out :|: TRUE f499_in -> f503_in :|: TRUE f503_out -> f499_out :|: TRUE f512_in -> f520_in :|: TRUE f520_out -> f512_out :|: TRUE f1_in(x15) -> f7_in(x15) :|: TRUE f458_out -> f508_out :|: TRUE f508_in -> f458_in :|: TRUE f477_out -> f475_out :|: TRUE f475_in -> f477_in :|: TRUE f458_out -> f459_in(x16) :|: TRUE f447_in(x18) -> f458_in :|: TRUE f468_out -> f464_out :|: TRUE f464_in -> f468_in :|: TRUE f512_out -> f500_out :|: TRUE f513_out -> f500_out :|: TRUE f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f468_in -> f468_out :|: TRUE f383_in(x21, x22) -> f393_in(x21) :|: TRUE f139_in(x26 + x27) -> f383_in(x26, x27) :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f500_out -> f476_out :|: TRUE f499_out -> f476_out :|: TRUE f458_out -> f544_out :|: TRUE f544_in -> f458_in :|: TRUE f458_out -> f542_out :|: TRUE f542_in -> f458_in :|: TRUE f463_in -> f464_in :|: TRUE f464_out -> f463_out :|: TRUE f463_in -> f465_in :|: TRUE f465_out -> f463_out :|: TRUE f542_out -> f544_in :|: TRUE f544_out -> f535_out :|: TRUE f535_in -> f542_in :|: TRUE f140_in(x33) -> f399_in(x33) :|: TRUE f34_in(x35) -> f135_in(x35) :|: TRUE f503_in -> f507_in :|: TRUE f508_out -> f503_out :|: TRUE f507_out -> f508_in :|: TRUE ---------------------------------------- (110) Obligation: Rules: f135_in(T2) -> f140_in(T2) :|: TRUE f135_in(x2) -> f139_in(x2) :|: TRUE f459_in(x3) -> f1_in(x3) :|: TRUE f476_out -> f465_out :|: TRUE f475_out -> f465_out :|: TRUE f465_in -> f475_in :|: TRUE f465_in -> f476_in :|: TRUE f477_in -> f477_out :|: TRUE f7_in(x8) -> f34_in(x8) :|: TRUE f399_in(x11) -> f447_in(x11) :|: TRUE f513_in -> f535_in :|: TRUE f535_out -> f513_out :|: TRUE f520_in -> f503_in :|: TRUE f503_out -> f520_out :|: TRUE f393_in(x13) -> f1_in(x13) :|: TRUE f463_out -> f458_out :|: TRUE f458_in -> f463_in :|: TRUE f507_in -> f458_in :|: TRUE f458_out -> f507_out :|: TRUE f499_in -> f503_in :|: TRUE f503_out -> f499_out :|: TRUE f512_in -> f520_in :|: TRUE f520_out -> f512_out :|: TRUE f1_in(x15) -> f7_in(x15) :|: TRUE f458_out -> f508_out :|: TRUE f508_in -> f458_in :|: TRUE f477_out -> f475_out :|: TRUE f475_in -> f477_in :|: TRUE f458_out -> f459_in(x16) :|: TRUE f447_in(x18) -> f458_in :|: TRUE f468_out -> f464_out :|: TRUE f464_in -> f468_in :|: TRUE f512_out -> f500_out :|: TRUE f513_out -> f500_out :|: TRUE f500_in -> f513_in :|: TRUE f500_in -> f512_in :|: TRUE f468_in -> f468_out :|: TRUE f383_in(x21, x22) -> f393_in(x21) :|: TRUE f139_in(x26 + x27) -> f383_in(x26, x27) :|: TRUE f476_in -> f499_in :|: TRUE f476_in -> f500_in :|: TRUE f500_out -> f476_out :|: TRUE f499_out -> f476_out :|: TRUE f458_out -> f544_out :|: TRUE f544_in -> f458_in :|: TRUE f458_out -> f542_out :|: TRUE f542_in -> f458_in :|: TRUE f463_in -> f464_in :|: TRUE f464_out -> f463_out :|: TRUE f463_in -> f465_in :|: TRUE f465_out -> f463_out :|: TRUE f542_out -> f544_in :|: TRUE f544_out -> f535_out :|: TRUE f535_in -> f542_in :|: TRUE f140_in(x33) -> f399_in(x33) :|: TRUE f34_in(x35) -> f135_in(x35) :|: TRUE f503_in -> f507_in :|: TRUE f508_out -> f503_out :|: TRUE f507_out -> f508_in :|: TRUE ---------------------------------------- (111) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (112) Obligation: Rules: f135_in(T2:0) -> f458_in :|: TRUE f458_in -> f463_out :|: TRUE f463_out -> f463_out :|: TRUE f458_in -> f458_in :|: TRUE f463_out -> f135_in(x16:0) :|: TRUE f463_out -> f458_in :|: TRUE f135_in(sum~x26:0~x27:0) -> f135_in(x26:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 ---------------------------------------- (113) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (114) Obligation: Rules: f135_in(T2:0) -> f458_in :|: TRUE f458_in -> f463_out :|: TRUE f463_out -> f463_out :|: TRUE f458_in -> f458_in :|: TRUE f463_out -> f135_in(x16:0) :|: TRUE f463_out -> f458_in :|: TRUE f135_in(sum~x26:0~x27:0) -> f135_in(x26:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 ---------------------------------------- (115) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f135_in(T2:0) -> f458_in :|: TRUE (2) f458_in -> f463_out :|: TRUE (3) f463_out -> f463_out :|: TRUE (4) f458_in -> f458_in :|: TRUE (5) f463_out -> f135_in(x16:0) :|: TRUE (6) f463_out -> f458_in :|: TRUE (7) f135_in(sum~x26:0~x27:0) -> f135_in(x26:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 Arcs: (1) -> (2), (4) (2) -> (3), (5), (6) (3) -> (3), (5), (6) (4) -> (2), (4) (5) -> (1), (7) (6) -> (2), (4) (7) -> (1), (7) This digraph is fully evaluated! ---------------------------------------- (116) Obligation: Termination digraph: Nodes: (1) f135_in(T2:0) -> f458_in :|: TRUE (2) f135_in(sum~x26:0~x27:0) -> f135_in(x26:0) :|: TRUE && sum~x26:0~x27:0 = x26:0 + x27:0 (3) f463_out -> f135_in(x16:0) :|: TRUE (4) f458_in -> f463_out :|: TRUE (5) f458_in -> f458_in :|: TRUE (6) f463_out -> f458_in :|: TRUE (7) f463_out -> f463_out :|: TRUE Arcs: (1) -> (4), (5) (2) -> (1), (2) (3) -> (1), (2) (4) -> (3), (6), (7) (5) -> (4), (5) (6) -> (4), (5) (7) -> (3), (6), (7) This digraph is fully evaluated! ---------------------------------------- (117) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (118) Obligation: Rules: f135_in(T2:0:0) -> f458_in :|: TRUE f463_out -> f135_in(x16:0:0) :|: TRUE f463_out -> f463_out :|: TRUE f135_in(sum~x26:0:0~x27:0:0) -> f135_in(x26:0:0) :|: TRUE && sum~x26:0:0~x27:0:0 = x26:0:0 + x27:0:0 f458_in -> f463_out :|: TRUE f458_in -> f458_in :|: TRUE f463_out -> f458_in :|: TRUE ---------------------------------------- (119) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 5, "program": { "directives": [], "clauses": [ [ "(p (d (e (t))) (const (1)))", null ], [ "(p (d (e (const A))) (const (0)))", null ], [ "(p (d (e (+ X Y))) (+ DX DY))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (e (* X Y))) (+ (* X DY) (* Y DX)))", "(',' (p (d (e X)) DX) (p (d (e Y)) DY))" ], [ "(p (d (d X)) DDX)", "(',' (p (d X) DX) (p (d (e DX)) DDX))" ] ] }, "graph": { "nodes": { "590": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T174)) X324) (p (d (e X324)) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X324"], "exprvars": [] } }, "470": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T76)) X145)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X145"], "exprvars": [] } }, "591": { "goal": [ { "clause": -1, "scope": -1, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 3, "scope": 1, "term": "(p T1 (+ T181 T182))" }, { "clause": 4, "scope": 1, "term": "(p T1 (+ T181 T182))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "592": { "goal": [ { "clause": 3, "scope": 1, "term": "(p T1 T2)" }, { "clause": 4, "scope": 1, "term": "(p T1 T2)" } ], "kb": { "nonunifying": [ [ "(p T1 T2)", "(p (d (e (t))) (const (1)))" ], [ "(p T1 T2)", "(p (d (e (const X235))) (const (0)))" ], [ "(p T1 T2)", "(p (d (e (+ X329 X330))) (+ X331 X332))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [ "X235", "X329", "X330", "X331", "X332" ], "exprvars": [] } }, "472": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (+ (* T79 T78) (* T80 T81)))) (const (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "593": { "goal": [ { "clause": 0, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 1, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 2, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 3, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 4, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": -1, "scope": 6, "term": null }, { "clause": 3, "scope": 1, "term": "(p T1 (+ T181 T182))" }, { "clause": 4, "scope": 1, "term": "(p T1 (+ T181 T182))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "594": { "goal": [{ "clause": 0, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "595": { "goal": [ { "clause": 1, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 2, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 3, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 4, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": -1, "scope": 6, "term": null }, { "clause": 3, "scope": 1, "term": "(p T1 (+ T181 T182))" }, { "clause": 4, "scope": 1, "term": "(p T1 (+ T181 T182))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "596": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T185)) T182)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T182"], "free": [], "exprvars": [] } }, "597": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "235": { "goal": [{ "clause": 0, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "598": { "goal": [{ "clause": 1, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "599": { "goal": [ { "clause": 2, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 3, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": 4, "scope": 6, "term": "(',' (p (d (e T183)) T181) (p (d (e T184)) T182))" }, { "clause": -1, "scope": 6, "term": null }, { "clause": 3, "scope": 1, "term": "(p T1 (+ T181 T182))" }, { "clause": 4, "scope": 1, "term": "(p T1 (+ T181 T182))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T181", "T182" ], "free": [], "exprvars": [] } }, "480": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (p (d T85) X156) (p (d (e X156)) X157)) (p (d (e X157)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X157", "X156" ], "exprvars": [] } }, "481": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "482": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T85) X156)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X156"], "exprvars": [] } }, "483": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T86)) X157) (p (d (e X157)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X157"], "exprvars": [] } }, "242": { "goal": [ { "clause": 1, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 2, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 3, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 4, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "484": { "goal": [ { "clause": 0, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 1, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 2, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 3, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 4, "scope": 4, "term": "(p (d T85) X156)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X156"], "exprvars": [] } }, "364": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T26)) X48)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X48"], "exprvars": [] } }, "485": { "goal": [{ "clause": 0, "scope": 4, "term": "(p (d T85) X156)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X156"], "exprvars": [] } }, "486": { "goal": [ { "clause": 1, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 2, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 3, "scope": 4, "term": "(p (d T85) X156)" }, { "clause": 4, "scope": 4, "term": "(p (d T85) X156)" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X156"], "exprvars": [] } }, "366": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T29)) X49) (p (d (e (+ T28 X49))) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X49"], "exprvars": [] } }, "487": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "488": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "5": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", 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"207": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "209": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "570": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (+ T150 T149))) (const (0)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "571": { "goal": [{ "clause": 3, "scope": 5, "term": "(',' (p (d T129) X248) (p (d (e X248)) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X248"], "exprvars": [] } }, "692": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T275)) T276)" }], "kb": { "nonunifying": [[ "(p T1 (+ (* (t) T276) (* T275 (const (1)))))", "(p (d (e (+ X329 X330))) (+ X331 X332))" ]], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T276", "T275" ], "free": [ "X329", "X330", "X331", "X332" ], "exprvars": [] } }, "451": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T73)) X146)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X146"], "exprvars": [] } }, "572": { "goal": [{ "clause": 4, "scope": 5, "term": "(',' (p (d T129) X248) (p (d (e X248)) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X248"], "exprvars": [] } }, "452": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T76)) X145) (p (d (e (+ (* T77 X145) (* T76 T75)))) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X145"], "exprvars": [] } }, "573": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (p (d (e T161)) X313) (p (d (e T162)) X312)) (p (d (e (+ (* T161 X312) (* T162 X313)))) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X312", "X313" ], "exprvars": [] } }, "574": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "577": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T161)) X313)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X313"], "exprvars": [] } }, "578": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T164)) X312) (p (d (e (+ (* T165 X312) (* T164 T163)))) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X312"], "exprvars": [] } }, "579": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T164)) X312)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X312"], "exprvars": [] } }, "219": { "goal": [ { "clause": 0, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 1, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 2, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 3, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }, { "clause": 4, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" } ], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "580": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (+ (* T167 T166) (* T168 T169)))) (const (0)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "344": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (p (d (e T26)) X48) (p (d (e T27)) X49)) (p (d (e (+ X48 X49))) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X48", "X49" ], "exprvars": [] } }, "587": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (',' (p (d T173) X323) (p (d (e X323)) X324)) (p (d (e X324)) (const (0))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X324", "X323" ], "exprvars": [] } }, "588": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "347": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "589": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T173) X323)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X323"], "exprvars": [] } } }, "edges": [ { "from": 5, "to": 6, "label": "CASE" }, { "from": 6, "to": 77, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 6, "to": 91, "label": "EVAL-BACKTRACK" }, { "from": 77, "to": 189, "label": "SUCCESS" }, { "from": 91, "to": 528, "label": "EVAL with clause\np(d(e(const(X235))), const(0)).\nand substitutionX235 -> T122,\nT1 -> d(e(const(T122))),\nT2 -> const(0)" }, { "from": 91, "to": 533, "label": "EVAL-BACKTRACK" }, { "from": 189, "to": 192, "label": "BACKTRACK\nfor clause: p(d(e(const(A))), const(0))because of non-unification" }, { "from": 192, "to": 198, "label": "BACKTRACK\nfor clause: p(d(e(+(X, Y))), +(DX, DY)) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY))because of non-unification" }, { "from": 198, "to": 201, "label": "BACKTRACK\nfor clause: p(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY))because of non-unification" }, { "from": 201, "to": 207, "label": "EVAL with clause\np(d(d(X12)), X13) :- ','(p(d(X12), X14), p(d(e(X14)), X13)).\nand substitutionX12 -> T10,\nT1 -> d(d(T10)),\nX13 -> const(1),\nT9 -> T10" }, { "from": 201, "to": 209, "label": "EVAL-BACKTRACK" }, { "from": 207, "to": 219, "label": "CASE" }, { "from": 219, "to": 235, "label": "PARALLEL" }, { "from": 219, "to": 242, "label": "PARALLEL" }, { "from": 235, "to": 259, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT10 -> e(t),\nX14 -> const(1)" }, { "from": 235, "to": 264, "label": "EVAL-BACKTRACK" }, { "from": 242, "to": 288, "label": "PARALLEL" }, { "from": 242, "to": 290, "label": "PARALLEL" }, { "from": 259, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> const(1)" }, { "from": 288, "to": 298, "label": "EVAL with clause\np(d(e(const(X19))), const(0)).\nand substitutionX19 -> T15,\nT10 -> e(const(T15)),\nX14 -> const(0)" }, { "from": 288, "to": 301, "label": "EVAL-BACKTRACK" }, { "from": 290, "to": 320, "label": "PARALLEL" }, { "from": 290, "to": 322, "label": "PARALLEL" }, { "from": 298, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> const(1)" }, { "from": 320, "to": 344, "label": "EVAL with clause\np(d(e(+(X44, X45))), +(X46, X47)) :- ','(p(d(e(X44)), X46), p(d(e(X45)), X47)).\nand substitutionX44 -> T26,\nX45 -> T27,\nT10 -> e(+(T26, T27)),\nX46 -> X48,\nX47 -> X49,\nX14 -> +(X48, X49),\nT24 -> T26,\nT25 -> T27" }, { "from": 320, "to": 347, "label": "EVAL-BACKTRACK" }, { "from": 322, "to": 442, "label": "PARALLEL" }, { "from": 322, "to": 443, "label": "PARALLEL" }, { "from": 344, "to": 364, "label": "SPLIT 1" }, { "from": 344, "to": 366, "label": "SPLIT 2\nreplacements:X48 -> T28,\nT27 -> T29" }, { "from": 364, "to": 372, "label": "CASE" }, { "from": 366, "to": 440, "label": "SPLIT 1" }, { "from": 366, "to": 441, "label": "SPLIT 2\nreplacements:X49 -> T61,\nT28 -> T62" }, { "from": 372, "to": 375, "label": "PARALLEL" }, { "from": 372, "to": 377, "label": "PARALLEL" }, { "from": 375, "to": 378, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT26 -> t,\nX48 -> const(1)" }, { "from": 375, "to": 379, "label": "EVAL-BACKTRACK" }, { "from": 377, "to": 381, "label": "PARALLEL" }, { "from": 377, "to": 382, "label": "PARALLEL" }, { "from": 378, "to": 380, "label": "SUCCESS" }, { "from": 381, "to": 385, "label": "EVAL with clause\np(d(e(const(X54))), const(0)).\nand substitutionX54 -> T34,\nT26 -> const(T34),\nX48 -> const(0)" }, { "from": 381, "to": 386, "label": "EVAL-BACKTRACK" }, { "from": 382, "to": 388, "label": "PARALLEL" }, { "from": 382, "to": 389, "label": "PARALLEL" }, { "from": 385, "to": 387, "label": "SUCCESS" }, { "from": 388, "to": 392, "label": "EVAL with clause\np(d(e(+(X79, X80))), +(X81, X82)) :- ','(p(d(e(X79)), X81), p(d(e(X80)), X82)).\nand substitutionX79 -> T45,\nX80 -> T46,\nT26 -> +(T45, T46),\nX81 -> X83,\nX82 -> X84,\nX48 -> +(X83, X84),\nT43 -> T45,\nT44 -> T46" }, { "from": 388, "to": 395, "label": "EVAL-BACKTRACK" }, { "from": 389, "to": 435, "label": "PARALLEL" }, { "from": 389, "to": 436, "label": "PARALLEL" }, { "from": 392, "to": 396, "label": "SPLIT 1" }, { "from": 392, "to": 397, "label": "SPLIT 2\nreplacements:X83 -> T47,\nT46 -> T48" }, { "from": 396, "to": 364, "label": "INSTANCE with matching:\nT26 -> T45\nX48 -> X83" }, { "from": 397, "to": 364, "label": "INSTANCE with matching:\nT26 -> T48\nX48 -> X84" }, { "from": 435, "to": 437, "label": "EVAL with clause\np(d(e(*(X109, X110))), +(*(X109, X111), *(X110, X112))) :- ','(p(d(e(X109)), X112), p(d(e(X110)), X111)).\nand substitutionX109 -> T59,\nX110 -> T60,\nT26 -> *(T59, T60),\nX111 -> X113,\nX112 -> X114,\nX48 -> +(*(T59, X113), *(T60, X114)),\nT57 -> T59,\nT58 -> T60" }, { "from": 435, "to": 438, "label": "EVAL-BACKTRACK" }, { "from": 436, "to": 439, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 437, "to": 392, "label": "INSTANCE with matching:\nT45 -> T59\nX83 -> X114\nT46 -> T60\nX84 -> X113" }, { "from": 440, "to": 364, "label": "INSTANCE with matching:\nT26 -> T29\nX48 -> X49" }, { "from": 441, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(T62, T61)))\nT2 -> const(1)" }, { "from": 442, "to": 444, "label": "EVAL with clause\np(d(e(*(X141, X142))), +(*(X141, X143), *(X142, X144))) :- ','(p(d(e(X141)), X144), p(d(e(X142)), X143)).\nand substitutionX141 -> T73,\nX142 -> T74,\nT10 -> e(*(T73, T74)),\nX143 -> X145,\nX144 -> X146,\nX14 -> +(*(T73, X145), *(T74, X146)),\nT71 -> T73,\nT72 -> T74" }, { "from": 442, "to": 445, "label": "EVAL-BACKTRACK" }, { "from": 443, "to": 480, "label": "EVAL with clause\np(d(d(X154)), X155) :- ','(p(d(X154), X156), p(d(e(X156)), X155)).\nand substitutionX154 -> T85,\nT10 -> d(T85),\nX14 -> X157,\nX155 -> X157,\nT84 -> T85" }, { "from": 443, "to": 481, "label": "EVAL-BACKTRACK" }, { "from": 444, "to": 451, "label": "SPLIT 1" }, { "from": 444, "to": 452, "label": "SPLIT 2\nreplacements:X146 -> T75,\nT74 -> T76,\nT73 -> T77" }, { "from": 451, "to": 364, "label": "INSTANCE with matching:\nT26 -> T73\nX48 -> X146" }, { "from": 452, "to": 470, "label": "SPLIT 1" }, { "from": 452, "to": 472, "label": "SPLIT 2\nreplacements:X145 -> T78,\nT77 -> T79,\nT76 -> T80,\nT75 -> T81" }, { "from": 470, "to": 364, "label": "INSTANCE with matching:\nT26 -> T76\nX48 -> X145" }, { "from": 472, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T79, T78), *(T80, T81))))\nT2 -> const(1)" }, { "from": 480, "to": 482, "label": "SPLIT 1" }, { "from": 480, "to": 483, "label": "SPLIT 2\nreplacements:X156 -> T86" }, { "from": 482, "to": 484, "label": "CASE" }, { "from": 483, "to": 207, "label": "INSTANCE with matching:\nT10 -> e(T86)\nX14 -> X157" }, { "from": 484, "to": 485, "label": "PARALLEL" }, { "from": 484, "to": 486, "label": "PARALLEL" }, { "from": 485, "to": 487, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT85 -> e(t),\nX156 -> const(1)" }, { "from": 485, "to": 488, "label": "EVAL-BACKTRACK" }, { "from": 486, "to": 490, "label": "PARALLEL" }, { "from": 486, "to": 491, "label": "PARALLEL" }, { "from": 487, "to": 489, "label": "SUCCESS" }, { "from": 490, "to": 492, "label": "EVAL with clause\np(d(e(const(X162))), const(0)).\nand substitutionX162 -> T91,\nT85 -> e(const(T91)),\nX156 -> const(0)" }, { "from": 490, "to": 493, "label": "EVAL-BACKTRACK" }, { "from": 491, "to": 495, "label": "PARALLEL" }, { "from": 491, "to": 496, "label": "PARALLEL" }, { "from": 492, "to": 494, "label": "SUCCESS" }, { "from": 495, "to": 497, "label": "EVAL with clause\np(d(e(+(X187, X188))), +(X189, X190)) :- ','(p(d(e(X187)), X189), p(d(e(X188)), X190)).\nand substitutionX187 -> T102,\nX188 -> T103,\nT85 -> e(+(T102, T103)),\nX189 -> X191,\nX190 -> X192,\nX156 -> +(X191, X192),\nT100 -> T102,\nT101 -> T103" }, { "from": 495, "to": 498, "label": "EVAL-BACKTRACK" }, { "from": 496, "to": 501, "label": "PARALLEL" }, { "from": 496, "to": 502, "label": "PARALLEL" }, { "from": 497, "to": 392, "label": "INSTANCE with matching:\nT45 -> T102\nX83 -> X191\nT46 -> T103\nX84 -> X192" }, { "from": 501, "to": 505, "label": "EVAL with clause\np(d(e(*(X217, X218))), +(*(X217, X219), *(X218, X220))) :- ','(p(d(e(X217)), X220), p(d(e(X218)), X219)).\nand substitutionX217 -> T114,\nX218 -> T115,\nT85 -> e(*(T114, T115)),\nX219 -> X221,\nX220 -> X222,\nX156 -> +(*(T114, X221), *(T115, X222)),\nT112 -> T114,\nT113 -> T115" }, { "from": 501, "to": 506, "label": "EVAL-BACKTRACK" }, { "from": 502, "to": 514, "label": "EVAL with clause\np(d(d(X230)), X231) :- ','(p(d(X230), X232), p(d(e(X232)), X231)).\nand substitutionX230 -> T119,\nT85 -> d(T119),\nX156 -> X233,\nX231 -> X233,\nT118 -> T119" }, { "from": 502, "to": 515, "label": "EVAL-BACKTRACK" }, { "from": 505, "to": 392, "label": "INSTANCE with matching:\nT45 -> T114\nX83 -> X222\nT46 -> T115\nX84 -> X221" }, { "from": 514, "to": 518, "label": "SPLIT 1" }, { "from": 514, "to": 519, "label": "SPLIT 2\nreplacements:X232 -> T120" }, { "from": 518, "to": 482, "label": "INSTANCE with matching:\nT85 -> T119\nX156 -> X232" }, { "from": 519, "to": 364, "label": "INSTANCE with matching:\nT26 -> T120\nX48 -> X233" }, { "from": 528, "to": 534, "label": "SUCCESS" }, { "from": 533, "to": 591, "label": "EVAL with clause\np(d(e(+(X329, X330))), +(X331, X332)) :- ','(p(d(e(X329)), X331), p(d(e(X330)), X332)).\nand substitutionX329 -> T183,\nX330 -> T184,\nT1 -> d(e(+(T183, T184))),\nX331 -> T181,\nX332 -> T182,\nT2 -> +(T181, T182),\nT179 -> T183,\nT180 -> T184" }, { "from": 533, "to": 592, "label": "EVAL-BACKTRACK" }, { "from": 534, "to": 539, "label": "BACKTRACK\nfor clause: p(d(e(+(X, Y))), +(DX, DY)) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY))because of non-unification" }, { "from": 539, "to": 540, "label": "BACKTRACK\nfor clause: p(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) :- ','(p(d(e(X)), DX), p(d(e(Y)), DY))because of non-unification" }, { "from": 540, "to": 546, "label": "EVAL with clause\np(d(d(X246)), X247) :- ','(p(d(X246), X248), p(d(e(X248)), X247)).\nand substitutionX246 -> T129,\nT1 -> d(d(T129)),\nX247 -> const(0),\nT128 -> T129" }, { "from": 540, "to": 547, "label": "EVAL-BACKTRACK" }, { "from": 546, "to": 549, "label": "CASE" }, { "from": 549, "to": 551, "label": "PARALLEL" }, { "from": 549, "to": 552, "label": "PARALLEL" }, { "from": 551, "to": 553, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT129 -> e(t),\nX248 -> const(1)" }, { "from": 551, "to": 554, "label": "EVAL-BACKTRACK" }, { "from": 552, "to": 557, "label": "PARALLEL" }, { "from": 552, "to": 558, "label": "PARALLEL" }, { "from": 553, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> const(0)" }, { "from": 557, "to": 559, "label": "EVAL with clause\np(d(e(const(X253))), const(0)).\nand substitutionX253 -> T134,\nT129 -> e(const(T134)),\nX248 -> const(0)" }, { "from": 557, "to": 560, "label": "EVAL-BACKTRACK" }, { "from": 558, "to": 563, "label": "PARALLEL" }, { "from": 558, "to": 564, "label": "PARALLEL" }, { "from": 559, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> const(0)" }, { "from": 563, "to": 565, "label": "EVAL with clause\np(d(e(+(X278, X279))), +(X280, X281)) :- ','(p(d(e(X278)), X280), p(d(e(X279)), X281)).\nand substitutionX278 -> T145,\nX279 -> T146,\nT129 -> e(+(T145, T146)),\nX280 -> X282,\nX281 -> X283,\nX248 -> +(X282, X283),\nT143 -> T145,\nT144 -> T146" }, { "from": 563, "to": 566, "label": "EVAL-BACKTRACK" }, { "from": 564, "to": 571, "label": "PARALLEL" }, { "from": 564, "to": 572, "label": "PARALLEL" }, { "from": 565, "to": 567, "label": "SPLIT 1" }, { "from": 565, "to": 568, "label": "SPLIT 2\nreplacements:X282 -> T147,\nT146 -> T148" }, { "from": 567, "to": 364, "label": "INSTANCE with matching:\nT26 -> T145\nX48 -> X282" }, { "from": 568, "to": 569, "label": "SPLIT 1" }, { "from": 568, "to": 570, "label": "SPLIT 2\nreplacements:X283 -> T149,\nT147 -> T150" }, { "from": 569, "to": 364, "label": "INSTANCE with matching:\nT26 -> T148\nX48 -> X283" }, { "from": 570, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(T150, T149)))\nT2 -> const(0)" }, { "from": 571, "to": 573, "label": "EVAL with clause\np(d(e(*(X308, X309))), +(*(X308, X310), *(X309, X311))) :- ','(p(d(e(X308)), X311), p(d(e(X309)), X310)).\nand substitutionX308 -> T161,\nX309 -> T162,\nT129 -> e(*(T161, T162)),\nX310 -> X312,\nX311 -> X313,\nX248 -> +(*(T161, X312), *(T162, X313)),\nT159 -> T161,\nT160 -> T162" }, { "from": 571, "to": 574, "label": "EVAL-BACKTRACK" }, { "from": 572, "to": 587, "label": "EVAL with clause\np(d(d(X321)), X322) :- ','(p(d(X321), X323), p(d(e(X323)), X322)).\nand substitutionX321 -> T173,\nT129 -> d(T173),\nX248 -> X324,\nX322 -> X324,\nT172 -> T173" }, { "from": 572, "to": 588, "label": "EVAL-BACKTRACK" }, { "from": 573, "to": 577, "label": "SPLIT 1" }, { "from": 573, "to": 578, "label": "SPLIT 2\nreplacements:X313 -> T163,\nT162 -> T164,\nT161 -> T165" }, { "from": 577, "to": 364, "label": "INSTANCE with matching:\nT26 -> T161\nX48 -> X313" }, { "from": 578, "to": 579, "label": "SPLIT 1" }, { "from": 578, "to": 580, "label": "SPLIT 2\nreplacements:X312 -> T166,\nT165 -> T167,\nT164 -> T168,\nT163 -> T169" }, { "from": 579, "to": 364, "label": "INSTANCE with matching:\nT26 -> T164\nX48 -> X312" }, { "from": 580, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T167, T166), *(T168, T169))))\nT2 -> const(0)" }, { "from": 587, "to": 589, "label": "SPLIT 1" }, { "from": 587, "to": 590, "label": "SPLIT 2\nreplacements:X323 -> T174" }, { "from": 589, "to": 482, "label": "INSTANCE with matching:\nT85 -> T173\nX156 -> X323" }, { "from": 590, "to": 546, "label": "INSTANCE with matching:\nT129 -> e(T174)\nX248 -> X324" }, { "from": 591, "to": 593, "label": "CASE" }, { "from": 592, "to": 672, "label": "EVAL with clause\np(d(e(*(X412, X413))), +(*(X412, X414), *(X413, X415))) :- ','(p(d(e(X412)), X415), p(d(e(X413)), X414)).\nand substitutionX412 -> T274,\nX413 -> T275,\nT1 -> d(e(*(T274, T275))),\nX414 -> T276,\nX415 -> T277,\nT2 -> +(*(T274, T276), *(T275, T277))" }, { "from": 592, "to": 673, "label": "EVAL-BACKTRACK" }, { "from": 593, "to": 594, "label": "PARALLEL" }, { "from": 593, "to": 595, "label": "PARALLEL" }, { "from": 594, "to": 596, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT183 -> t,\nT181 -> const(1),\nT184 -> T185" }, { "from": 594, "to": 597, "label": "EVAL-BACKTRACK" }, { "from": 595, "to": 598, "label": "PARALLEL" }, { "from": 595, "to": 599, "label": "PARALLEL" }, { "from": 596, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T185))\nT2 -> T182" }, { "from": 598, "to": 600, "label": "EVAL with clause\np(d(e(const(X337))), const(0)).\nand substitutionX337 -> T190,\nT183 -> const(T190),\nT181 -> const(0),\nT184 -> T191" }, { "from": 598, "to": 601, "label": "EVAL-BACKTRACK" }, { "from": 599, "to": 602, "label": "PARALLEL" }, { "from": 599, "to": 603, "label": "PARALLEL" }, { "from": 600, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T191))\nT2 -> T182" }, { "from": 602, "to": 604, "label": "EVAL with clause\np(d(e(+(X354, X355))), +(X356, X357)) :- ','(p(d(e(X354)), X356), p(d(e(X355)), X357)).\nand substitutionX354 -> T212,\nX355 -> T213,\nT183 -> +(T212, T213),\nX356 -> T210,\nX357 -> T211,\nT181 -> +(T210, T211),\nT208 -> T212,\nT209 -> T213,\nT184 -> T214" }, { "from": 602, "to": 605, "label": "EVAL-BACKTRACK" }, { "from": 603, "to": 610, "label": "PARALLEL" }, { "from": 603, "to": 611, "label": "PARALLEL" }, { "from": 604, "to": 606, "label": "SPLIT 1" }, { "from": 604, "to": 607, "label": "SPLIT 2\nnew knowledge:\nT210 is ground\nreplacements:T213 -> T215,\nT214 -> T216" }, { "from": 606, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T212))\nT2 -> T210" }, { "from": 607, "to": 608, "label": "SPLIT 1" }, { "from": 607, "to": 609, "label": "SPLIT 2\nnew knowledge:\nT211 is ground\nreplacements:T216 -> T217" }, { "from": 608, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T215))\nT2 -> T211" }, { "from": 609, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T217))\nT2 -> T182" }, { "from": 610, "to": 612, "label": "EVAL with clause\np(d(e(*(X374, X375))), +(*(X374, X376), *(X375, X377))) :- ','(p(d(e(X374)), X377), p(d(e(X375)), X376)).\nand substitutionX374 -> T234,\nX375 -> T235,\nT183 -> *(T234, T235),\nX376 -> T236,\nX377 -> T237,\nT181 -> +(*(T234, T236), *(T235, T237)),\nT184 -> T238" }, { "from": 610, "to": 613, "label": "EVAL-BACKTRACK" }, { "from": 611, "to": 614, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 612, "to": 604, "label": "INSTANCE with matching:\nT212 -> T234\nT210 -> T237\nT213 -> T235\nT211 -> T236\nT214 -> T238" }, { "from": 614, "to": 615, "label": "FAILURE" }, { "from": 615, "to": 616, "label": "PARALLEL" }, { "from": 615, "to": 617, "label": "PARALLEL" }, { "from": 616, "to": 666, "label": "EVAL with clause\np(d(e(*(X396, X397))), +(*(X396, X398), *(X397, X399))) :- ','(p(d(e(X396)), X399), p(d(e(X397)), X398)).\nand substitutionX396 -> T255,\nX397 -> T256,\nT1 -> d(e(*(T255, T256))),\nX398 -> T257,\nT181 -> *(T255, T257),\nX399 -> T258,\nT182 -> *(T256, T258)" }, { "from": 616, "to": 667, "label": "EVAL-BACKTRACK" }, { "from": 617, "to": 668, "label": "EVAL with clause\np(d(d(X405)), X406) :- ','(p(d(X405), X407), p(d(e(X407)), X406)).\nand substitutionX405 -> T268,\nT1 -> d(d(T268)),\nT181 -> T266,\nT182 -> T267,\nX406 -> +(T266, T267),\nT265 -> T268" }, { "from": 617, "to": 669, "label": "EVAL-BACKTRACK" }, { "from": 666, "to": 607, "label": "INSTANCE with matching:\nT215 -> T255\nT211 -> T258\nT216 -> T256\nT182 -> T257" }, { "from": 668, "to": 670, "label": "SPLIT 1" }, { "from": 668, "to": 671, "label": "SPLIT 2\nreplacements:X407 -> T269" }, { "from": 670, "to": 482, "label": "INSTANCE with matching:\nT85 -> T268\nX156 -> X407" }, { "from": 671, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T269))\nT2 -> +(T266, T267)" }, { "from": 672, "to": 674, "label": "CASE" }, { "from": 673, "to": 743, "label": "EVAL with clause\np(d(d(X473)), X474) :- ','(p(d(X473), X475), p(d(e(X475)), X474)).\nand substitutionX473 -> T343,\nT1 -> d(d(T343)),\nT2 -> T342,\nX474 -> T342,\nT341 -> T343" }, { "from": 673, "to": 744, "label": "EVAL-BACKTRACK" }, { "from": 674, "to": 675, "label": "PARALLEL" }, { "from": 674, "to": 676, "label": "PARALLEL" }, { "from": 675, "to": 692, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT274 -> t,\nT277 -> const(1)" }, { "from": 675, "to": 726, "label": "EVAL-BACKTRACK" }, { "from": 676, "to": 727, "label": "PARALLEL" }, { "from": 676, "to": 728, "label": "PARALLEL" }, { "from": 692, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T275))\nT2 -> T276" }, { "from": 727, "to": 729, "label": "EVAL with clause\np(d(e(const(X420))), const(0)).\nand substitutionX420 -> T282,\nT274 -> const(T282),\nT277 -> const(0)" }, { "from": 727, "to": 730, "label": "EVAL-BACKTRACK" }, { "from": 728, "to": 731, "label": "PARALLEL" }, { "from": 728, "to": 732, "label": "PARALLEL" }, { "from": 729, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(T275))\nT2 -> T276" }, { "from": 731, "to": 733, "label": "EVAL with clause\np(d(e(+(X437, X438))), +(X439, X440)) :- ','(p(d(e(X437)), X439), p(d(e(X438)), X440)).\nand substitutionX437 -> T299,\nX438 -> T300,\nT274 -> +(T299, T300),\nX439 -> T301,\nX440 -> T302,\nT277 -> +(T301, T302)" }, { "from": 731, "to": 734, "label": "EVAL-BACKTRACK" }, { "from": 732, "to": 735, "label": "PARALLEL" }, { "from": 732, "to": 736, "label": "PARALLEL" }, { "from": 733, "to": 604, "label": "INSTANCE with matching:\nT212 -> T299\nT210 -> T301\nT213 -> T300\nT211 -> T302\nT214 -> T275\nT182 -> T276" }, { "from": 735, "to": 737, "label": "EVAL with clause\np(d(e(*(X457, X458))), +(*(X457, X459), *(X458, X460))) :- ','(p(d(e(X457)), X460), p(d(e(X458)), X459)).\nand substitutionX457 -> T319,\nX458 -> T320,\nT274 -> *(T319, T320),\nX459 -> T321,\nX460 -> T322,\nT277 -> +(*(T319, T321), *(T320, T322))" }, { "from": 735, "to": 738, "label": "EVAL-BACKTRACK" }, { "from": 736, "to": 739, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 737, "to": 604, "label": "INSTANCE with matching:\nT212 -> T319\nT210 -> T322\nT213 -> T320\nT211 -> T321\nT214 -> T275\nT182 -> T276" }, { "from": 739, "to": 740, "label": "FAILURE" }, { "from": 740, "to": 741, "label": "EVAL with clause\np(d(d(X468)), X469) :- ','(p(d(X468), X470), p(d(e(X470)), X469)).\nand substitutionX468 -> T338,\nT1 -> d(d(T338)),\nT274 -> T334,\nT276 -> T335,\nT275 -> T336,\nT277 -> T337,\nX469 -> +(*(T334, T335), *(T336, T337)),\nT333 -> T338" }, { "from": 740, "to": 742, "label": "EVAL-BACKTRACK" }, { "from": 741, "to": 668, "label": "INSTANCE with matching:\nT268 -> T338\nX407 -> X470\nT266 -> *(T334, T335)\nT267 -> *(T336, T337)" }, { "from": 743, "to": 745, "label": "CASE" }, { "from": 745, "to": 746, "label": "PARALLEL" }, { "from": 745, "to": 747, "label": "PARALLEL" }, { "from": 746, "to": 748, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT343 -> e(t),\nX475 -> const(1)" }, { "from": 746, "to": 749, "label": "EVAL-BACKTRACK" }, { "from": 747, "to": 750, "label": "PARALLEL" }, { "from": 747, "to": 751, "label": "PARALLEL" }, { "from": 748, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> T342" }, { "from": 750, "to": 752, "label": "EVAL with clause\np(d(e(const(X480))), const(0)).\nand substitutionX480 -> T348,\nT343 -> e(const(T348)),\nX475 -> const(0)" }, { "from": 750, "to": 753, "label": "EVAL-BACKTRACK" }, { "from": 751, "to": 754, "label": "PARALLEL" }, { "from": 751, "to": 755, "label": "PARALLEL" }, { "from": 752, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> T342" }, { "from": 754, "to": 756, "label": "EVAL with clause\np(d(e(+(X505, X506))), +(X507, X508)) :- ','(p(d(e(X505)), X507), p(d(e(X506)), X508)).\nand substitutionX505 -> T359,\nX506 -> T360,\nT343 -> e(+(T359, T360)),\nX507 -> X509,\nX508 -> X510,\nX475 -> +(X509, X510),\nT357 -> T359,\nT358 -> T360" }, { "from": 754, "to": 757, "label": "EVAL-BACKTRACK" }, { "from": 755, "to": 762, "label": "PARALLEL" }, { "from": 755, "to": 763, "label": "PARALLEL" }, { "from": 756, "to": 758, "label": "SPLIT 1" }, { "from": 756, "to": 759, "label": "SPLIT 2\nreplacements:X509 -> T361,\nT360 -> T362,\nT1 -> T363" }, { "from": 758, "to": 364, "label": "INSTANCE with matching:\nT26 -> T359\nX48 -> X509" }, { "from": 759, "to": 760, "label": "SPLIT 1" }, { "from": 759, "to": 761, "label": "SPLIT 2\nreplacements:X510 -> T364,\nT361 -> T365,\nT363 -> T366" }, { "from": 760, "to": 364, "label": "INSTANCE with matching:\nT26 -> T362\nX48 -> X510" }, { "from": 761, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(T365, T364)))\nT2 -> T342" }, { "from": 762, "to": 764, "label": "EVAL with clause\np(d(e(*(X535, X536))), +(*(X535, X537), *(X536, X538))) :- ','(p(d(e(X535)), X538), p(d(e(X536)), X537)).\nand substitutionX535 -> T377,\nX536 -> T378,\nT343 -> e(*(T377, T378)),\nX537 -> X539,\nX538 -> X540,\nX475 -> +(*(T377, X539), *(T378, X540)),\nT375 -> T377,\nT376 -> T378" }, { "from": 762, "to": 765, "label": "EVAL-BACKTRACK" }, { "from": 763, "to": 770, "label": "EVAL with clause\np(d(d(X548)), X549) :- ','(p(d(X548), X550), p(d(e(X550)), X549)).\nand substitutionX548 -> T391,\nT343 -> d(T391),\nX475 -> X551,\nX549 -> X551,\nT390 -> T391" }, { "from": 763, "to": 771, "label": "EVAL-BACKTRACK" }, { "from": 764, "to": 766, "label": "SPLIT 1" }, { "from": 764, "to": 767, "label": "SPLIT 2\nreplacements:X540 -> T379,\nT378 -> T380,\nT377 -> T381,\nT1 -> T382" }, { "from": 766, "to": 364, "label": "INSTANCE with matching:\nT26 -> T377\nX48 -> X540" }, { "from": 767, "to": 768, "label": "SPLIT 1" }, { "from": 767, "to": 769, "label": "SPLIT 2\nreplacements:X539 -> T383,\nT381 -> T384,\nT380 -> T385,\nT379 -> T386,\nT382 -> T387" }, { "from": 768, "to": 364, "label": "INSTANCE with matching:\nT26 -> T380\nX48 -> X539" }, { "from": 769, "to": 5, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T384, T383), *(T385, T386))))\nT2 -> T342" }, { "from": 770, "to": 772, "label": "SPLIT 1" }, { "from": 770, "to": 773, "label": "SPLIT 2\nreplacements:X550 -> T392,\nT1 -> T393" }, { "from": 772, "to": 482, "label": "INSTANCE with matching:\nT85 -> T391\nX156 -> X550" }, { "from": 773, "to": 743, "label": "INSTANCE with matching:\nT343 -> e(T392)\nX475 -> X551\nT1 -> T393" } ], "type": "Graph" } } ---------------------------------------- (120) Obligation: Triples: pH(+(X1, X2), +(X3, X4)) :- pI(X1, X3, X2, X4). pH(*(X1, X2), +(*(X1, X3), *(X2, X4))) :- pI(X1, X4, X2, X3). pI(X1, X2, X3, X4) :- pH(X1, X2). pI(X1, X2, X3, X4) :- ','(pcH(X1, X2), pH(X3, X4)). pB(e(t), const(1)) :- pA(d(e(const(1))), const(1)). pB(e(const(X1)), const(0)) :- pA(d(e(const(0))), const(1)). pB(e(+(X1, X2)), +(X3, X4)) :- pH(X1, X3). pB(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), pH(X2, X4)). pB(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pA(d(e(+(X3, X4))), const(1)))). pB(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- pH(X1, X4). pB(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), pH(X2, X3)). pB(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pA(d(e(+(*(X1, X3), *(X2, X4)))), const(1)))). pB(d(X1), X2) :- pJ(X1, X3). pB(d(X1), X2) :- ','(pcJ(X1, X3), pB(e(X3), X2)). pJ(e(+(X1, X2)), +(X3, X4)) :- pI(X1, X3, X2, X4). pJ(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- pI(X1, X4, X2, X3). pJ(d(X1), X2) :- pJ(X1, X3). pJ(d(X1), X2) :- ','(pcJ(X1, X3), pH(X3, X2)). pC(e(t), const(1)) :- pA(d(e(const(1))), const(0)). pC(e(const(X1)), const(0)) :- pA(d(e(const(0))), const(0)). pC(e(+(X1, X2)), +(X3, X4)) :- pH(X1, X3). pC(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), pH(X2, X4)). pC(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pA(d(e(+(X3, X4))), const(0)))). pC(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- pH(X1, X4). pC(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), pH(X2, X3)). pC(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pA(d(e(+(*(X1, X3), *(X2, X4)))), const(0)))). pC(d(X1), X2) :- pJ(X1, X3). pC(d(X1), X2) :- ','(pcJ(X1, X3), pC(e(X3), X2)). pD(X1, X2, X3, X4, X5, X6) :- pA(d(e(X1)), X2). pD(X1, X2, X3, X4, X5, X6) :- ','(pcA(d(e(X1)), X2), pE(X3, X4, X5, X6)). pE(X1, X2, X3, X4) :- pA(d(e(X1)), X2). pE(X1, X2, X3, X4) :- ','(pcA(d(e(X1)), X2), pA(d(e(X3)), X4)). pF(X1, X2, X3, X4) :- pJ(X1, X2). pF(X1, X2, X3, X4) :- ','(pcJ(X1, X2), pA(d(e(X2)), +(X3, X4))). pG(e(t), const(1), X1) :- pA(d(e(const(1))), X1). pG(e(const(X1)), const(0), X2) :- pA(d(e(const(0))), X2). pG(e(+(X1, X2)), +(X3, X4), X5) :- pH(X1, X3). pG(e(+(X1, X2)), +(X3, X4), X5) :- ','(pcH(X1, X3), pH(X2, X4)). pG(e(+(X1, X2)), +(X3, X4), X5) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pA(d(e(+(X3, X4))), X5))). pG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) :- pH(X1, X4). pG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) :- ','(pcH(X1, X4), pH(X2, X3)). pG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pA(d(e(+(*(X1, X3), *(X2, X4)))), X5))). pG(d(X1), X2, X3) :- pJ(X1, X4). pG(d(X1), X2, X3) :- ','(pcJ(X1, X4), pG(e(X4), X2, X3)). pA(d(d(X1)), const(1)) :- pB(X1, X2). pA(d(d(X1)), const(0)) :- pC(X1, X2). pA(d(e(+(t, X1))), +(const(1), X2)) :- pA(d(e(X1)), X2). pA(d(e(+(const(X1), X2))), +(const(0), X3)) :- pA(d(e(X2)), X3). pA(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) :- pD(X1, X4, X2, X5, X3, X6). pA(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) :- pD(X1, X5, X2, X4, X3, X6). pA(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) :- pE(X1, X4, X2, X3). pA(d(d(X1)), +(X2, X3)) :- pF(X1, X4, X2, X3). pA(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) :- pA(d(e(X1)), X2). pA(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) :- pA(d(e(X2)), X3). pA(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) :- pD(X1, X5, X2, X6, X3, X4). pA(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) :- pD(X1, X6, X2, X5, X3, X4). pA(d(d(X1)), +(*(X2, X3), *(X4, X5))) :- pF(X1, X6, *(X2, X3), *(X4, X5)). pA(d(d(X1)), X2) :- pG(X1, X3, X2). Clauses: pcA(d(e(t)), const(1)). pcA(d(d(X1)), const(1)) :- qcB(X1, X2). pcA(d(e(const(X1))), const(0)). pcA(d(d(X1)), const(0)) :- qcC(X1, X2). pcA(d(e(+(t, X1))), +(const(1), X2)) :- pcA(d(e(X1)), X2). pcA(d(e(+(const(X1), X2))), +(const(0), X3)) :- pcA(d(e(X2)), X3). pcA(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) :- qcD(X1, X4, X2, X5, X3, X6). pcA(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) :- qcD(X1, X5, X2, X4, X3, X6). pcA(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) :- qcE(X1, X4, X2, X3). pcA(d(d(X1)), +(X2, X3)) :- qcF(X1, X4, X2, X3). pcA(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) :- pcA(d(e(X1)), X2). pcA(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) :- pcA(d(e(X2)), X3). pcA(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) :- qcD(X1, X5, X2, X6, X3, X4). pcA(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) :- qcD(X1, X6, X2, X5, X3, X4). pcA(d(d(X1)), +(*(X2, X3), *(X4, X5))) :- qcF(X1, X6, *(X2, X3), *(X4, X5)). pcA(d(d(X1)), X2) :- qcG(X1, X3, X2). pcH(t, const(1)). pcH(const(X1), const(0)). pcH(+(X1, X2), +(X3, X4)) :- qcI(X1, X3, X2, X4). pcH(*(X1, X2), +(*(X1, X3), *(X2, X4))) :- qcI(X1, X4, X2, X3). qcI(X1, X2, X3, X4) :- ','(pcH(X1, X2), pcH(X3, X4)). qcB(e(t), const(1)) :- pcA(d(e(const(1))), const(1)). qcB(e(const(X1)), const(0)) :- pcA(d(e(const(0))), const(1)). qcB(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pcA(d(e(+(X3, X4))), const(1)))). qcB(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pcA(d(e(+(*(X1, X3), *(X2, X4)))), const(1)))). qcB(d(X1), X2) :- ','(pcJ(X1, X3), qcB(e(X3), X2)). pcJ(e(t), const(1)). pcJ(e(const(X1)), const(0)). pcJ(e(+(X1, X2)), +(X3, X4)) :- qcI(X1, X3, X2, X4). pcJ(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- qcI(X1, X4, X2, X3). pcJ(d(X1), X2) :- ','(pcJ(X1, X3), pcH(X3, X2)). qcC(e(t), const(1)) :- pcA(d(e(const(1))), const(0)). qcC(e(const(X1)), const(0)) :- pcA(d(e(const(0))), const(0)). qcC(e(+(X1, X2)), +(X3, X4)) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pcA(d(e(+(X3, X4))), const(0)))). qcC(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pcA(d(e(+(*(X1, X3), *(X2, X4)))), const(0)))). qcC(d(X1), X2) :- ','(pcJ(X1, X3), qcC(e(X3), X2)). qcD(X1, X2, X3, X4, X5, X6) :- ','(pcA(d(e(X1)), X2), qcE(X3, X4, X5, X6)). qcE(X1, X2, X3, X4) :- ','(pcA(d(e(X1)), X2), pcA(d(e(X3)), X4)). qcF(X1, X2, X3, X4) :- ','(pcJ(X1, X2), pcA(d(e(X2)), +(X3, X4))). qcG(e(t), const(1), X1) :- pcA(d(e(const(1))), X1). qcG(e(const(X1)), const(0), X2) :- pcA(d(e(const(0))), X2). qcG(e(+(X1, X2)), +(X3, X4), X5) :- ','(pcH(X1, X3), ','(pcH(X2, X4), pcA(d(e(+(X3, X4))), X5))). qcG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) :- ','(pcH(X1, X4), ','(pcH(X2, X3), pcA(d(e(+(*(X1, X3), *(X2, X4)))), X5))). qcG(d(X1), X2, X3) :- ','(pcJ(X1, X4), qcG(e(X4), X2, X3)). Afs: pA(x1, x2) = pA(x2) ---------------------------------------- (121) TriplesToPiDPProof (SOUND) We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes: pA_in_2: (f,b) (b,b) pB_in_2: (f,f) (b,f) pC_in_2: (b,f) (f,f) pD_in_6: (b,b,b,b,b,b) (f,b,f,b,f,b) (b,b,b,b,f,b) pcA_in_2: (b,b) (f,b) qcB_in_2: (b,f) (f,f) qcC_in_2: (b,f) (f,f) qcD_in_6: (b,b,b,b,b,b) (f,b,f,b,f,b) (b,b,b,b,f,b) qcE_in_4: (b,b,b,b) (b,b,f,b) (f,b,f,b) qcF_in_4: (b,f,b,b) (f,f,b,b) pcJ_in_2: (b,f) (f,f) qcI_in_4: (b,f,b,f) (f,f,f,f) pcH_in_2: (b,f) (f,f) qcG_in_3: (b,f,b) (f,f,b) pE_in_4: (b,b,b,b) (b,b,f,b) (f,b,f,b) pF_in_4: (b,f,b,b) (f,f,b,b) pJ_in_2: (b,f) (f,f) pI_in_4: (b,f,b,f) (f,f,f,f) pH_in_2: (b,f) (f,f) pG_in_3: (b,f,b) (f,f,b) Transforming TRIPLES into the following Term Rewriting System: Pi DP problem: The TRS P consists of the following rules: PA_IN_AG(d(d(X1)), const(1)) -> U65_AG(X1, pB_in_aa(X1, X2)) PA_IN_AG(d(d(X1)), const(1)) -> PB_IN_AA(X1, X2) PB_IN_AA(e(t), const(1)) -> U6_AA(pA_in_gg(d(e(const(1))), const(1))) PB_IN_AA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(1)) PA_IN_GG(d(d(X1)), const(1)) -> U65_GG(X1, pB_in_ga(X1, X2)) PA_IN_GG(d(d(X1)), const(1)) -> PB_IN_GA(X1, X2) PB_IN_GA(e(t), const(1)) -> U6_GA(pA_in_gg(d(e(const(1))), const(1))) PB_IN_GA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(1)) PA_IN_GG(d(d(X1)), const(0)) -> U66_GG(X1, pC_in_ga(X1, X2)) PA_IN_GG(d(d(X1)), const(0)) -> PC_IN_GA(X1, X2) PC_IN_GA(e(t), const(1)) -> U26_GA(pA_in_gg(d(e(const(1))), const(0))) PC_IN_GA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(0)) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> U67_GG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> U68_GG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U69_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X4, X2, X5, X3, X6)) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_GGGGGG(X1, X4, X2, X5, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U41_GGGGGG(X1, X2, X3, X4, X5, X6, pA_in_gg(d(e(X1)), X2)) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U70_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X4, X3, X6)) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGGG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U43_GGGGGG(X1, X2, X3, X4, X5, X6, pE_in_gggg(X3, X4, X5, X6)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGGG(X3, X4, X5, X6) PE_IN_GGGG(X1, X2, X3, X4) -> U44_GGGG(X1, X2, X3, X4, pA_in_gg(d(e(X1)), X2)) PE_IN_GGGG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U71_GG(X1, X2, X3, X4, pE_in_gggg(X1, X4, X2, X3)) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PE_IN_GGGG(X1, X2, X3, X4) -> U45_GGGG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U46_GGGG(X1, X2, X3, X4, pA_in_gg(d(e(X3)), X4)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_GG(d(e(X3)), X4) PA_IN_GG(d(d(X1)), +(X2, X3)) -> U72_GG(X1, X2, X3, pF_in_gagg(X1, X4, X2, X3)) PA_IN_GG(d(d(X1)), +(X2, X3)) -> PF_IN_GAGG(X1, X4, X2, X3) PF_IN_GAGG(X1, X2, X3, X4) -> U47_GAGG(X1, X2, X3, X4, pJ_in_ga(X1, X2)) PF_IN_GAGG(X1, X2, X3, X4) -> PJ_IN_GA(X1, X2) PJ_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U21_GA(X1, X2, X3, X4, pI_in_gaga(X1, X3, X2, X4)) PJ_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U3_GAGA(X1, X2, X3, X4, pH_in_ga(X1, X2)) PI_IN_GAGA(X1, X2, X3, X4) -> PH_IN_GA(X1, X2) PH_IN_GA(+(X1, X2), +(X3, X4)) -> U1_GA(X1, X2, X3, X4, pI_in_gaga(X1, X3, X2, X4)) PH_IN_GA(+(X1, X2), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U4_GAGA(X1, X2, X3, X4, pcH_in_ga(X1, X2)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U5_GAGA(X1, X2, X3, X4, pH_in_ga(X3, X4)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> PH_IN_GA(X3, X4) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U2_GA(X1, X2, X3, X4, pI_in_gaga(X1, X4, X2, X3)) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) PJ_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U22_GA(X1, X2, X3, X4, pI_in_gaga(X1, X4, X2, X3)) PJ_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) PJ_IN_GA(d(X1), X2) -> U23_GA(X1, X2, pJ_in_ga(X1, X3)) PJ_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PJ_IN_GA(d(X1), X2) -> U24_GA(X1, X2, pcJ_in_ga(X1, X3)) U24_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U25_GA(X1, X2, pH_in_ga(X3, X2)) U24_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PH_IN_GA(X3, X2) PF_IN_GAGG(X1, X2, X3, X4) -> U48_GAGG(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) U48_GAGG(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U49_GAGG(X1, X2, X3, X4, pA_in_gg(d(e(X2)), +(X3, X4))) U48_GAGG(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> PA_IN_GG(d(e(X2)), +(X3, X4)) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U73_GG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U74_GG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U75_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X6, X3, X4)) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U76_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X6, X2, X5, X3, X4)) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) PA_IN_GG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U77_GG(X1, X2, X3, X4, X5, pF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) PA_IN_GG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> PF_IN_GAGG(X1, X6, *(X2, X3), *(X4, X5)) PA_IN_GG(d(d(X1)), X2) -> U78_GG(X1, X2, pG_in_gag(X1, X3, X2)) PA_IN_GG(d(d(X1)), X2) -> PG_IN_GAG(X1, X3, X2) PG_IN_GAG(e(t), const(1), X1) -> U50_GAG(X1, pA_in_gg(d(e(const(1))), X1)) PG_IN_GAG(e(t), const(1), X1) -> PA_IN_GG(d(e(const(1))), X1) PG_IN_GAG(e(const(X1)), const(0), X2) -> U51_GAG(X1, X2, pA_in_gg(d(e(const(0))), X2)) PG_IN_GAG(e(const(X1)), const(0), X2) -> PA_IN_GG(d(e(const(0))), X2) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> U52_GAG(X1, X2, X3, X4, X5, pH_in_ga(X1, X3)) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> PH_IN_GA(X1, X3) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> U53_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U54_GAG(X1, X2, X3, X4, X5, pH_in_ga(X2, X4)) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U55_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U55_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U56_GAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(X3, X4))), X5)) U55_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), X5) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U57_GAG(X1, X2, X3, X4, X5, pH_in_ga(X1, X4)) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> PH_IN_GA(X1, X4) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U58_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U59_GAG(X1, X2, X3, X4, X5, pH_in_ga(X2, X3)) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U60_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U60_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U61_GAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U60_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), X5) PG_IN_GAG(d(X1), X2, X3) -> U62_GAG(X1, X2, X3, pJ_in_ga(X1, X4)) PG_IN_GAG(d(X1), X2, X3) -> PJ_IN_GA(X1, X4) PG_IN_GAG(d(X1), X2, X3) -> U63_GAG(X1, X2, X3, pcJ_in_ga(X1, X4)) U63_GAG(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U64_GAG(X1, X2, X3, pG_in_gag(e(X4), X2, X3)) U63_GAG(X1, X2, X3, pcJ_out_ga(X1, X4)) -> PG_IN_GAG(e(X4), X2, X3) PC_IN_GA(e(const(X1)), const(0)) -> U27_GA(X1, pA_in_gg(d(e(const(0))), const(0))) PC_IN_GA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(0)) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U28_GA(X1, X2, X3, X4, pH_in_ga(X1, X3)) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_GA(X1, X3) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U29_GA(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U30_GA(X1, X2, X3, X4, pH_in_ga(X2, X4)) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U31_GA(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U31_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U32_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(0))) U31_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(0)) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U33_GA(X1, X2, X3, X4, pH_in_ga(X1, X4)) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_GA(X1, X4) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U34_GA(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U35_GA(X1, X2, X3, X4, pH_in_ga(X2, X3)) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U36_GA(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U36_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U37_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U36_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(0)) PC_IN_GA(d(X1), X2) -> U38_GA(X1, X2, pJ_in_ga(X1, X3)) PC_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PC_IN_GA(d(X1), X2) -> U39_GA(X1, X2, pcJ_in_ga(X1, X3)) U39_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U40_GA(X1, X2, pC_in_ga(e(X3), X2)) U39_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PC_IN_GA(e(X3), X2) PB_IN_GA(e(const(X1)), const(0)) -> U7_GA(X1, pA_in_gg(d(e(const(0))), const(1))) PB_IN_GA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(1)) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U8_GA(X1, X2, X3, X4, pH_in_ga(X1, X3)) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_GA(X1, X3) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U9_GA(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U10_GA(X1, X2, X3, X4, pH_in_ga(X2, X4)) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U11_GA(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U11_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U12_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(1))) U11_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(1)) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U13_GA(X1, X2, X3, X4, pH_in_ga(X1, X4)) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_GA(X1, X4) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U14_GA(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U15_GA(X1, X2, X3, X4, pH_in_ga(X2, X3)) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U16_GA(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U16_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U17_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U16_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(1)) PB_IN_GA(d(X1), X2) -> U18_GA(X1, X2, pJ_in_ga(X1, X3)) PB_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PB_IN_GA(d(X1), X2) -> U19_GA(X1, X2, pcJ_in_ga(X1, X3)) U19_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U20_GA(X1, X2, pB_in_ga(e(X3), X2)) U19_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PB_IN_GA(e(X3), X2) PB_IN_AA(e(const(X1)), const(0)) -> U7_AA(X1, pA_in_gg(d(e(const(0))), const(1))) PB_IN_AA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(1)) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U8_AA(X1, X2, X3, X4, pH_in_aa(X1, X3)) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_AA(X1, X3) PH_IN_AA(+(X1, X2), +(X3, X4)) -> U1_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X3, X2, X4)) PH_IN_AA(+(X1, X2), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PI_IN_AAAA(X1, X2, X3, X4) -> U3_AAAA(X1, X2, X3, X4, pH_in_aa(X1, X2)) PI_IN_AAAA(X1, X2, X3, X4) -> PH_IN_AA(X1, X2) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U2_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X4, X2, X3)) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PI_IN_AAAA(X1, X2, X3, X4) -> U4_AAAA(X1, X2, X3, X4, pcH_in_aa(X1, X2)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U5_AAAA(X1, X2, X3, X4, pH_in_aa(X3, X4)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> PH_IN_AA(X3, X4) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U9_AA(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U10_AA(X1, X2, X3, X4, pH_in_aa(X2, X4)) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U11_AA(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U11_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U12_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(1))) U11_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(1)) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U13_AA(X1, X2, X3, X4, pH_in_aa(X1, X4)) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_AA(X1, X4) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U14_AA(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U15_AA(X1, X2, X3, X4, pH_in_aa(X2, X3)) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U16_AA(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U16_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U17_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U16_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(1)) PB_IN_AA(d(X1), X2) -> U18_AA(X1, X2, pJ_in_aa(X1, X3)) PB_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PJ_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U21_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X3, X2, X4)) PJ_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PJ_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U22_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X4, X2, X3)) PJ_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PJ_IN_AA(d(X1), X2) -> U23_AA(X1, X2, pJ_in_aa(X1, X3)) PJ_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PJ_IN_AA(d(X1), X2) -> U24_AA(X1, X2, pcJ_in_aa(X1, X3)) U24_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U25_AA(X1, X2, pH_in_ga(X3, X2)) U24_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PH_IN_GA(X3, X2) PB_IN_AA(d(X1), X2) -> U19_AA(X1, X2, pcJ_in_aa(X1, X3)) U19_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U20_AA(X1, X2, pB_in_ga(e(X3), X2)) U19_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PB_IN_GA(e(X3), X2) PA_IN_AG(d(d(X1)), const(0)) -> U66_AG(X1, pC_in_aa(X1, X2)) PA_IN_AG(d(d(X1)), const(0)) -> PC_IN_AA(X1, X2) PC_IN_AA(e(t), const(1)) -> U26_AA(pA_in_gg(d(e(const(1))), const(0))) PC_IN_AA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(0)) PC_IN_AA(e(const(X1)), const(0)) -> U27_AA(X1, pA_in_gg(d(e(const(0))), const(0))) PC_IN_AA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(0)) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U28_AA(X1, X2, X3, X4, pH_in_aa(X1, X3)) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_AA(X1, X3) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U29_AA(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U30_AA(X1, X2, X3, X4, pH_in_aa(X2, X4)) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U31_AA(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U31_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U32_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(0))) U31_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(0)) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U33_AA(X1, X2, X3, X4, pH_in_aa(X1, X4)) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_AA(X1, X4) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U34_AA(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U35_AA(X1, X2, X3, X4, pH_in_aa(X2, X3)) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U36_AA(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U36_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U37_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U36_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(0)) PC_IN_AA(d(X1), X2) -> U38_AA(X1, X2, pJ_in_aa(X1, X3)) PC_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PC_IN_AA(d(X1), X2) -> U39_AA(X1, X2, pcJ_in_aa(X1, X3)) U39_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U40_AA(X1, X2, pC_in_ga(e(X3), X2)) U39_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PC_IN_GA(e(X3), X2) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> U67_AG(X1, X2, pA_in_ag(d(e(X1)), X2)) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> U68_AG(X1, X2, X3, pA_in_ag(d(e(X2)), X3)) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_AG(d(e(X2)), X3) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U69_AG(X1, X2, X3, X4, X5, X6, pD_in_agagag(X1, X4, X2, X5, X3, X6)) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_AGAGAG(X1, X4, X2, X5, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U41_AGAGAG(X1, X2, X3, X4, X5, X6, pA_in_ag(d(e(X1)), X2)) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U70_AG(X1, X2, X3, X4, X5, X6, pD_in_ggggag(X1, X5, X2, X4, X3, X6)) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGAG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U41_GGGGAG(X1, X2, X3, X4, X5, X6, pA_in_gg(d(e(X1)), X2)) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U43_GGGGAG(X1, X2, X3, X4, X5, X6, pE_in_ggag(X3, X4, X5, X6)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGAG(X3, X4, X5, X6) PE_IN_GGAG(X1, X2, X3, X4) -> U44_GGAG(X1, X2, X3, X4, pA_in_gg(d(e(X1)), X2)) PE_IN_GGAG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PE_IN_GGAG(X1, X2, X3, X4) -> U45_GGAG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U46_GGAG(X1, X2, X3, X4, pA_in_ag(d(e(X3)), X4)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) PA_IN_AG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U71_AG(X1, X2, X3, X4, pE_in_gggg(X1, X4, X2, X3)) PA_IN_AG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PA_IN_AG(d(d(X1)), +(X2, X3)) -> U72_AG(X1, X2, X3, pF_in_aagg(X1, X4, X2, X3)) PA_IN_AG(d(d(X1)), +(X2, X3)) -> PF_IN_AAGG(X1, X4, X2, X3) PF_IN_AAGG(X1, X2, X3, X4) -> U47_AAGG(X1, X2, X3, X4, pJ_in_aa(X1, X2)) PF_IN_AAGG(X1, X2, X3, X4) -> PJ_IN_AA(X1, X2) PF_IN_AAGG(X1, X2, X3, X4) -> U48_AAGG(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U48_AAGG(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U49_AAGG(X1, X2, X3, X4, pA_in_gg(d(e(X2)), +(X3, X4))) U48_AAGG(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> PA_IN_GG(d(e(X2)), +(X3, X4)) PA_IN_AG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U73_AG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_AG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_AG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U74_AG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_AG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_AG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U75_AG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X6, X3, X4)) PA_IN_AG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_AG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U76_AG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X6, X2, X5, X3, X4)) PA_IN_AG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) PA_IN_AG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U77_AG(X1, X2, X3, X4, X5, pF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) PA_IN_AG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> PF_IN_AAGG(X1, X6, *(X2, X3), *(X4, X5)) PA_IN_AG(d(d(X1)), X2) -> U78_AG(X1, X2, pG_in_aag(X1, X3, X2)) PA_IN_AG(d(d(X1)), X2) -> PG_IN_AAG(X1, X3, X2) PG_IN_AAG(e(t), const(1), X1) -> U50_AAG(X1, pA_in_gg(d(e(const(1))), X1)) PG_IN_AAG(e(t), const(1), X1) -> PA_IN_GG(d(e(const(1))), X1) PG_IN_AAG(e(const(X1)), const(0), X2) -> U51_AAG(X1, X2, pA_in_gg(d(e(const(0))), X2)) PG_IN_AAG(e(const(X1)), const(0), X2) -> PA_IN_GG(d(e(const(0))), X2) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> U52_AAG(X1, X2, X3, X4, X5, pH_in_aa(X1, X3)) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> PH_IN_AA(X1, X3) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> U53_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U54_AAG(X1, X2, X3, X4, X5, pH_in_aa(X2, X4)) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U55_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U55_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U56_AAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(X3, X4))), X5)) U55_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), X5) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U57_AAG(X1, X2, X3, X4, X5, pH_in_aa(X1, X4)) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> PH_IN_AA(X1, X4) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U58_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U59_AAG(X1, X2, X3, X4, X5, pH_in_aa(X2, X3)) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U60_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U60_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U61_AAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U60_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), X5) PG_IN_AAG(d(X1), X2, X3) -> U62_AAG(X1, X2, X3, pJ_in_aa(X1, X4)) PG_IN_AAG(d(X1), X2, X3) -> PJ_IN_AA(X1, X4) PG_IN_AAG(d(X1), X2, X3) -> U63_AAG(X1, X2, X3, pcJ_in_aa(X1, X4)) U63_AAG(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U64_AAG(X1, X2, X3, pG_in_gag(e(X4), X2, X3)) U63_AAG(X1, X2, X3, pcJ_out_aa(X1, X4)) -> PG_IN_GAG(e(X4), X2, X3) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U43_AGAGAG(X1, X2, X3, X4, X5, X6, pE_in_agag(X3, X4, X5, X6)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> PE_IN_AGAG(X3, X4, X5, X6) PE_IN_AGAG(X1, X2, X3, X4) -> U44_AGAG(X1, X2, X3, X4, pA_in_ag(d(e(X1)), X2)) PE_IN_AGAG(X1, X2, X3, X4) -> PA_IN_AG(d(e(X1)), X2) PE_IN_AGAG(X1, X2, X3, X4) -> U45_AGAG(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U46_AGAG(X1, X2, X3, X4, pA_in_ag(d(e(X3)), X4)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: pA_in_ag(x1, x2) = pA_in_ag(x2) const(x1) = const 1 = 1 pB_in_aa(x1, x2) = pB_in_aa pA_in_gg(x1, x2) = pA_in_gg(x1, x2) d(x1) = d(x1) pB_in_ga(x1, x2) = pB_in_ga(x1) e(x1) = e(x1) t = t 0 = 0 pC_in_ga(x1, x2) = pC_in_ga(x1) +(x1, x2) = +(x1, x2) pD_in_gggggg(x1, x2, x3, x4, x5, x6) = pD_in_gggggg(x1, x2, x3, x4, x5, x6) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pE_in_gggg(x1, x2, x3, x4) = pE_in_gggg(x1, x2, x3, x4) pF_in_gagg(x1, x2, x3, x4) = pF_in_gagg(x1, x3, x4) pJ_in_ga(x1, x2) = pJ_in_ga(x1) pI_in_gaga(x1, x2, x3, x4) = pI_in_gaga(x1, x3) pH_in_ga(x1, x2) = pH_in_ga(x1) pG_in_gag(x1, x2, x3) = pG_in_gag(x1, x3) pH_in_aa(x1, x2) = pH_in_aa pI_in_aaaa(x1, x2, x3, x4) = pI_in_aaaa pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pJ_in_aa(x1, x2) = pJ_in_aa pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pC_in_aa(x1, x2) = pC_in_aa pD_in_agagag(x1, x2, x3, x4, x5, x6) = pD_in_agagag(x2, x4, x6) pD_in_ggggag(x1, x2, x3, x4, x5, x6) = pD_in_ggggag(x1, x2, x3, x4, x6) pE_in_ggag(x1, x2, x3, x4) = pE_in_ggag(x1, x2, x4) pF_in_aagg(x1, x2, x3, x4) = pF_in_aagg(x3, x4) pG_in_aag(x1, x2, x3) = pG_in_aag(x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) pE_in_agag(x1, x2, x3, x4) = pE_in_agag(x2, x4) PA_IN_AG(x1, x2) = PA_IN_AG(x2) U65_AG(x1, x2) = U65_AG(x2) PB_IN_AA(x1, x2) = PB_IN_AA U6_AA(x1) = U6_AA(x1) PA_IN_GG(x1, x2) = PA_IN_GG(x1, x2) U65_GG(x1, x2) = U65_GG(x1, x2) PB_IN_GA(x1, x2) = PB_IN_GA(x1) U6_GA(x1) = U6_GA(x1) U66_GG(x1, x2) = U66_GG(x1, x2) PC_IN_GA(x1, x2) = PC_IN_GA(x1) U26_GA(x1) = U26_GA(x1) U67_GG(x1, x2, x3) = U67_GG(x1, x2, x3) U68_GG(x1, x2, x3, x4) = U68_GG(x2, x3, x4) U69_GG(x1, x2, x3, x4, x5, x6, x7) = U69_GG(x1, x2, x3, x4, x5, x6, x7) PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) U41_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U41_GGGGGG(x1, x2, x3, x4, x5, x6, x7) U70_GG(x1, x2, x3, x4, x5, x6, x7) = U70_GG(x1, x2, x3, x4, x5, x6, x7) U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) U43_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U43_GGGGGG(x1, x2, x3, x4, x5, x6, x7) PE_IN_GGGG(x1, x2, x3, x4) = PE_IN_GGGG(x1, x2, x3, x4) U44_GGGG(x1, x2, x3, x4, x5) = U44_GGGG(x1, x2, x3, x4, x5) U71_GG(x1, x2, x3, x4, x5) = U71_GG(x1, x2, x3, x4, x5) U45_GGGG(x1, x2, x3, x4, x5) = U45_GGGG(x1, x2, x3, x4, x5) U46_GGGG(x1, x2, x3, x4, x5) = U46_GGGG(x1, x2, x3, x4, x5) U72_GG(x1, x2, x3, x4) = U72_GG(x1, x2, x3, x4) PF_IN_GAGG(x1, x2, x3, x4) = PF_IN_GAGG(x1, x3, x4) U47_GAGG(x1, x2, x3, x4, x5) = U47_GAGG(x1, x3, x4, x5) PJ_IN_GA(x1, x2) = PJ_IN_GA(x1) U21_GA(x1, x2, x3, x4, x5) = U21_GA(x1, x2, x5) PI_IN_GAGA(x1, x2, x3, x4) = PI_IN_GAGA(x1, x3) U3_GAGA(x1, x2, x3, x4, x5) = U3_GAGA(x1, x3, x5) PH_IN_GA(x1, x2) = PH_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U4_GAGA(x1, x2, x3, x4, x5) = U4_GAGA(x1, x3, x5) U5_GAGA(x1, x2, x3, x4, x5) = U5_GAGA(x1, x3, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x5) U23_GA(x1, x2, x3) = U23_GA(x1, x3) U24_GA(x1, x2, x3) = U24_GA(x1, x3) U25_GA(x1, x2, x3) = U25_GA(x1, x3) U48_GAGG(x1, x2, x3, x4, x5) = U48_GAGG(x1, x3, x4, x5) U49_GAGG(x1, x2, x3, x4, x5) = U49_GAGG(x1, x3, x4, x5) U73_GG(x1, x2, x3) = U73_GG(x1, x2, x3) U74_GG(x1, x2, x3, x4) = U74_GG(x2, x3, x4) U75_GG(x1, x2, x3, x4, x5, x6, x7) = U75_GG(x1, x2, x3, x4, x5, x6, x7) U76_GG(x1, x2, x3, x4, x5, x6, x7) = U76_GG(x1, x2, x3, x4, x5, x6, x7) U77_GG(x1, x2, x3, x4, x5, x6) = U77_GG(x1, x2, x3, x4, x5, x6) U78_GG(x1, x2, x3) = U78_GG(x1, x2, x3) PG_IN_GAG(x1, x2, x3) = PG_IN_GAG(x1, x3) U50_GAG(x1, x2) = U50_GAG(x1, x2) U51_GAG(x1, x2, x3) = U51_GAG(x2, x3) U52_GAG(x1, x2, x3, x4, x5, x6) = U52_GAG(x1, x2, x5, x6) U53_GAG(x1, x2, x3, x4, x5, x6) = U53_GAG(x1, x2, x5, x6) U54_GAG(x1, x2, x3, x4, x5, x6) = U54_GAG(x1, x2, x5, x6) U55_GAG(x1, x2, x3, x4, x5, x6) = U55_GAG(x1, x2, x3, x5, x6) U56_GAG(x1, x2, x3, x4, x5, x6) = U56_GAG(x1, x2, x5, x6) U57_GAG(x1, x2, x3, x4, x5, x6) = U57_GAG(x1, x2, x5, x6) U58_GAG(x1, x2, x3, x4, x5, x6) = U58_GAG(x1, x2, x5, x6) U59_GAG(x1, x2, x3, x4, x5, x6) = U59_GAG(x1, x2, x5, x6) U60_GAG(x1, x2, x3, x4, x5, x6) = U60_GAG(x1, x2, x4, x5, x6) U61_GAG(x1, x2, x3, x4, x5, x6) = U61_GAG(x1, x2, x5, x6) U62_GAG(x1, x2, x3, x4) = U62_GAG(x1, x3, x4) U63_GAG(x1, x2, x3, x4) = U63_GAG(x1, x3, x4) U64_GAG(x1, x2, x3, x4) = U64_GAG(x1, x3, x4) U27_GA(x1, x2) = U27_GA(x2) U28_GA(x1, x2, x3, x4, x5) = U28_GA(x1, x2, x5) U29_GA(x1, x2, x3, x4, x5) = U29_GA(x1, x2, x5) U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x5) U31_GA(x1, x2, x3, x4, x5) = U31_GA(x1, x2, x3, x5) U32_GA(x1, x2, x3, x4, x5) = U32_GA(x1, x2, x5) U33_GA(x1, x2, x3, x4, x5) = U33_GA(x1, x2, x5) U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x2, x5) U35_GA(x1, x2, x3, x4, x5) = U35_GA(x1, x2, x5) U36_GA(x1, x2, x3, x4, x5) = U36_GA(x1, x2, x4, x5) U37_GA(x1, x2, x3, x4, x5) = U37_GA(x1, x2, x5) U38_GA(x1, x2, x3) = U38_GA(x1, x3) U39_GA(x1, x2, x3) = U39_GA(x1, x3) U40_GA(x1, x2, x3) = U40_GA(x1, x3) U7_GA(x1, x2) = U7_GA(x2) U8_GA(x1, x2, x3, x4, x5) = U8_GA(x1, x2, x5) U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x5) U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x2, x5) U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x3, x5) U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x5) U13_GA(x1, x2, x3, x4, x5) = U13_GA(x1, x2, x5) U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x5) U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x2, x5) U16_GA(x1, x2, x3, x4, x5) = U16_GA(x1, x2, x4, x5) U17_GA(x1, x2, x3, x4, x5) = U17_GA(x1, x2, x5) U18_GA(x1, x2, x3) = U18_GA(x1, x3) U19_GA(x1, x2, x3) = U19_GA(x1, x3) U20_GA(x1, x2, x3) = U20_GA(x1, x3) U7_AA(x1, x2) = U7_AA(x2) U8_AA(x1, x2, x3, x4, x5) = U8_AA(x5) PH_IN_AA(x1, x2) = PH_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) PI_IN_AAAA(x1, x2, x3, x4) = PI_IN_AAAA U3_AAAA(x1, x2, x3, x4, x5) = U3_AAAA(x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U4_AAAA(x1, x2, x3, x4, x5) = U4_AAAA(x5) U5_AAAA(x1, x2, x3, x4, x5) = U5_AAAA(x5) U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5) U10_AA(x1, x2, x3, x4, x5) = U10_AA(x5) U11_AA(x1, x2, x3, x4, x5) = U11_AA(x3, x5) U12_AA(x1, x2, x3, x4, x5) = U12_AA(x5) U13_AA(x1, x2, x3, x4, x5) = U13_AA(x5) U14_AA(x1, x2, x3, x4, x5) = U14_AA(x5) U15_AA(x1, x2, x3, x4, x5) = U15_AA(x5) U16_AA(x1, x2, x3, x4, x5) = U16_AA(x1, x4, x5) U17_AA(x1, x2, x3, x4, x5) = U17_AA(x5) U18_AA(x1, x2, x3) = U18_AA(x3) PJ_IN_AA(x1, x2) = PJ_IN_AA U21_AA(x1, x2, x3, x4, x5) = U21_AA(x5) U22_AA(x1, x2, x3, x4, x5) = U22_AA(x5) U23_AA(x1, x2, x3) = U23_AA(x3) U24_AA(x1, x2, x3) = U24_AA(x3) U25_AA(x1, x2, x3) = U25_AA(x3) U19_AA(x1, x2, x3) = U19_AA(x3) U20_AA(x1, x2, x3) = U20_AA(x3) U66_AG(x1, x2) = U66_AG(x2) PC_IN_AA(x1, x2) = PC_IN_AA U26_AA(x1) = U26_AA(x1) U27_AA(x1, x2) = U27_AA(x2) U28_AA(x1, x2, x3, x4, x5) = U28_AA(x5) U29_AA(x1, x2, x3, x4, x5) = U29_AA(x5) U30_AA(x1, x2, x3, x4, x5) = U30_AA(x5) U31_AA(x1, x2, x3, x4, x5) = U31_AA(x3, x5) U32_AA(x1, x2, x3, x4, x5) = U32_AA(x5) U33_AA(x1, x2, x3, x4, x5) = U33_AA(x5) U34_AA(x1, x2, x3, x4, x5) = U34_AA(x5) U35_AA(x1, x2, x3, x4, x5) = U35_AA(x5) U36_AA(x1, x2, x3, x4, x5) = U36_AA(x1, x4, x5) U37_AA(x1, x2, x3, x4, x5) = U37_AA(x5) U38_AA(x1, x2, x3) = U38_AA(x3) U39_AA(x1, x2, x3) = U39_AA(x3) U40_AA(x1, x2, x3) = U40_AA(x3) U67_AG(x1, x2, x3) = U67_AG(x2, x3) U68_AG(x1, x2, x3, x4) = U68_AG(x3, x4) U69_AG(x1, x2, x3, x4, x5, x6, x7) = U69_AG(x4, x5, x6, x7) PD_IN_AGAGAG(x1, x2, x3, x4, x5, x6) = PD_IN_AGAGAG(x2, x4, x6) U41_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U41_AGAGAG(x2, x4, x6, x7) U70_AG(x1, x2, x3, x4, x5, x6, x7) = U70_AG(x1, x2, x4, x5, x6, x7) PD_IN_GGGGAG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGAG(x1, x2, x3, x4, x6) U41_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U41_GGGGAG(x1, x2, x3, x4, x6, x7) U42_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGAG(x1, x2, x3, x4, x6, x7) U43_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U43_GGGGAG(x1, x2, x3, x4, x6, x7) PE_IN_GGAG(x1, x2, x3, x4) = PE_IN_GGAG(x1, x2, x4) U44_GGAG(x1, x2, x3, x4, x5) = U44_GGAG(x1, x2, x4, x5) U45_GGAG(x1, x2, x3, x4, x5) = U45_GGAG(x1, x2, x4, x5) U46_GGAG(x1, x2, x3, x4, x5) = U46_GGAG(x1, x2, x4, x5) U71_AG(x1, x2, x3, x4, x5) = U71_AG(x1, x2, x3, x4, x5) U72_AG(x1, x2, x3, x4) = U72_AG(x2, x3, x4) PF_IN_AAGG(x1, x2, x3, x4) = PF_IN_AAGG(x3, x4) U47_AAGG(x1, x2, x3, x4, x5) = U47_AAGG(x3, x4, x5) U48_AAGG(x1, x2, x3, x4, x5) = U48_AAGG(x3, x4, x5) U49_AAGG(x1, x2, x3, x4, x5) = U49_AAGG(x3, x4, x5) U73_AG(x1, x2, x3) = U73_AG(x1, x2, x3) U74_AG(x1, x2, x3, x4) = U74_AG(x2, x3, x4) U75_AG(x1, x2, x3, x4, x5, x6, x7) = U75_AG(x1, x2, x3, x4, x5, x6, x7) U76_AG(x1, x2, x3, x4, x5, x6, x7) = U76_AG(x1, x2, x3, x4, x5, x6, x7) U77_AG(x1, x2, x3, x4, x5, x6) = U77_AG(x2, x3, x4, x5, x6) U78_AG(x1, x2, x3) = U78_AG(x2, x3) PG_IN_AAG(x1, x2, x3) = PG_IN_AAG(x3) U50_AAG(x1, x2) = U50_AAG(x1, x2) U51_AAG(x1, x2, x3) = U51_AAG(x2, x3) U52_AAG(x1, x2, x3, x4, x5, x6) = U52_AAG(x5, x6) U53_AAG(x1, x2, x3, x4, x5, x6) = U53_AAG(x5, x6) U54_AAG(x1, x2, x3, x4, x5, x6) = U54_AAG(x5, x6) U55_AAG(x1, x2, x3, x4, x5, x6) = U55_AAG(x3, x5, x6) U56_AAG(x1, x2, x3, x4, x5, x6) = U56_AAG(x5, x6) U57_AAG(x1, x2, x3, x4, x5, x6) = U57_AAG(x5, x6) U58_AAG(x1, x2, x3, x4, x5, x6) = U58_AAG(x5, x6) U59_AAG(x1, x2, x3, x4, x5, x6) = U59_AAG(x5, x6) U60_AAG(x1, x2, x3, x4, x5, x6) = U60_AAG(x1, x4, x5, x6) U61_AAG(x1, x2, x3, x4, x5, x6) = U61_AAG(x5, x6) U62_AAG(x1, x2, x3, x4) = U62_AAG(x3, x4) U63_AAG(x1, x2, x3, x4) = U63_AAG(x3, x4) U64_AAG(x1, x2, x3, x4) = U64_AAG(x3, x4) U42_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U42_AGAGAG(x2, x4, x6, x7) U43_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U43_AGAGAG(x2, x4, x6, x7) PE_IN_AGAG(x1, x2, x3, x4) = PE_IN_AGAG(x2, x4) U44_AGAG(x1, x2, x3, x4, x5) = U44_AGAG(x2, x4, x5) U45_AGAG(x1, x2, x3, x4, x5) = U45_AGAG(x2, x4, x5) U46_AGAG(x1, x2, x3, x4, x5) = U46_AGAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES ---------------------------------------- (122) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_AG(d(d(X1)), const(1)) -> U65_AG(X1, pB_in_aa(X1, X2)) PA_IN_AG(d(d(X1)), const(1)) -> PB_IN_AA(X1, X2) PB_IN_AA(e(t), const(1)) -> U6_AA(pA_in_gg(d(e(const(1))), const(1))) PB_IN_AA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(1)) PA_IN_GG(d(d(X1)), const(1)) -> U65_GG(X1, pB_in_ga(X1, X2)) PA_IN_GG(d(d(X1)), const(1)) -> PB_IN_GA(X1, X2) PB_IN_GA(e(t), const(1)) -> U6_GA(pA_in_gg(d(e(const(1))), const(1))) PB_IN_GA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(1)) PA_IN_GG(d(d(X1)), const(0)) -> U66_GG(X1, pC_in_ga(X1, X2)) PA_IN_GG(d(d(X1)), const(0)) -> PC_IN_GA(X1, X2) PC_IN_GA(e(t), const(1)) -> U26_GA(pA_in_gg(d(e(const(1))), const(0))) PC_IN_GA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(0)) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> U67_GG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> U68_GG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U69_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X4, X2, X5, X3, X6)) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_GGGGGG(X1, X4, X2, X5, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U41_GGGGGG(X1, X2, X3, X4, X5, X6, pA_in_gg(d(e(X1)), X2)) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U70_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X4, X3, X6)) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGGG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U43_GGGGGG(X1, X2, X3, X4, X5, X6, pE_in_gggg(X3, X4, X5, X6)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGGG(X3, X4, X5, X6) PE_IN_GGGG(X1, X2, X3, X4) -> U44_GGGG(X1, X2, X3, X4, pA_in_gg(d(e(X1)), X2)) PE_IN_GGGG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U71_GG(X1, X2, X3, X4, pE_in_gggg(X1, X4, X2, X3)) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PE_IN_GGGG(X1, X2, X3, X4) -> U45_GGGG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U46_GGGG(X1, X2, X3, X4, pA_in_gg(d(e(X3)), X4)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_GG(d(e(X3)), X4) PA_IN_GG(d(d(X1)), +(X2, X3)) -> U72_GG(X1, X2, X3, pF_in_gagg(X1, X4, X2, X3)) PA_IN_GG(d(d(X1)), +(X2, X3)) -> PF_IN_GAGG(X1, X4, X2, X3) PF_IN_GAGG(X1, X2, X3, X4) -> U47_GAGG(X1, X2, X3, X4, pJ_in_ga(X1, X2)) PF_IN_GAGG(X1, X2, X3, X4) -> PJ_IN_GA(X1, X2) PJ_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U21_GA(X1, X2, X3, X4, pI_in_gaga(X1, X3, X2, X4)) PJ_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U3_GAGA(X1, X2, X3, X4, pH_in_ga(X1, X2)) PI_IN_GAGA(X1, X2, X3, X4) -> PH_IN_GA(X1, X2) PH_IN_GA(+(X1, X2), +(X3, X4)) -> U1_GA(X1, X2, X3, X4, pI_in_gaga(X1, X3, X2, X4)) PH_IN_GA(+(X1, X2), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U4_GAGA(X1, X2, X3, X4, pcH_in_ga(X1, X2)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U5_GAGA(X1, X2, X3, X4, pH_in_ga(X3, X4)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> PH_IN_GA(X3, X4) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U2_GA(X1, X2, X3, X4, pI_in_gaga(X1, X4, X2, X3)) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) PJ_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U22_GA(X1, X2, X3, X4, pI_in_gaga(X1, X4, X2, X3)) PJ_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) PJ_IN_GA(d(X1), X2) -> U23_GA(X1, X2, pJ_in_ga(X1, X3)) PJ_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PJ_IN_GA(d(X1), X2) -> U24_GA(X1, X2, pcJ_in_ga(X1, X3)) U24_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U25_GA(X1, X2, pH_in_ga(X3, X2)) U24_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PH_IN_GA(X3, X2) PF_IN_GAGG(X1, X2, X3, X4) -> U48_GAGG(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) U48_GAGG(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U49_GAGG(X1, X2, X3, X4, pA_in_gg(d(e(X2)), +(X3, X4))) U48_GAGG(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> PA_IN_GG(d(e(X2)), +(X3, X4)) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U73_GG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U74_GG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U75_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X6, X3, X4)) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U76_GG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X6, X2, X5, X3, X4)) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) PA_IN_GG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U77_GG(X1, X2, X3, X4, X5, pF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) PA_IN_GG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> PF_IN_GAGG(X1, X6, *(X2, X3), *(X4, X5)) PA_IN_GG(d(d(X1)), X2) -> U78_GG(X1, X2, pG_in_gag(X1, X3, X2)) PA_IN_GG(d(d(X1)), X2) -> PG_IN_GAG(X1, X3, X2) PG_IN_GAG(e(t), const(1), X1) -> U50_GAG(X1, pA_in_gg(d(e(const(1))), X1)) PG_IN_GAG(e(t), const(1), X1) -> PA_IN_GG(d(e(const(1))), X1) PG_IN_GAG(e(const(X1)), const(0), X2) -> U51_GAG(X1, X2, pA_in_gg(d(e(const(0))), X2)) PG_IN_GAG(e(const(X1)), const(0), X2) -> PA_IN_GG(d(e(const(0))), X2) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> U52_GAG(X1, X2, X3, X4, X5, pH_in_ga(X1, X3)) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> PH_IN_GA(X1, X3) PG_IN_GAG(e(+(X1, X2)), +(X3, X4), X5) -> U53_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U54_GAG(X1, X2, X3, X4, X5, pH_in_ga(X2, X4)) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U53_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U55_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U55_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U56_GAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(X3, X4))), X5)) U55_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), X5) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U57_GAG(X1, X2, X3, X4, X5, pH_in_ga(X1, X4)) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> PH_IN_GA(X1, X4) PG_IN_GAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U58_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U59_GAG(X1, X2, X3, X4, X5, pH_in_ga(X2, X3)) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U58_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U60_GAG(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U60_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U61_GAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U60_GAG(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), X5) PG_IN_GAG(d(X1), X2, X3) -> U62_GAG(X1, X2, X3, pJ_in_ga(X1, X4)) PG_IN_GAG(d(X1), X2, X3) -> PJ_IN_GA(X1, X4) PG_IN_GAG(d(X1), X2, X3) -> U63_GAG(X1, X2, X3, pcJ_in_ga(X1, X4)) U63_GAG(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U64_GAG(X1, X2, X3, pG_in_gag(e(X4), X2, X3)) U63_GAG(X1, X2, X3, pcJ_out_ga(X1, X4)) -> PG_IN_GAG(e(X4), X2, X3) PC_IN_GA(e(const(X1)), const(0)) -> U27_GA(X1, pA_in_gg(d(e(const(0))), const(0))) PC_IN_GA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(0)) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U28_GA(X1, X2, X3, X4, pH_in_ga(X1, X3)) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_GA(X1, X3) PC_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U29_GA(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U30_GA(X1, X2, X3, X4, pH_in_ga(X2, X4)) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U29_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U31_GA(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U31_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U32_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(0))) U31_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(0)) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U33_GA(X1, X2, X3, X4, pH_in_ga(X1, X4)) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_GA(X1, X4) PC_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U34_GA(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U35_GA(X1, X2, X3, X4, pH_in_ga(X2, X3)) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U34_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U36_GA(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U36_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U37_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U36_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(0)) PC_IN_GA(d(X1), X2) -> U38_GA(X1, X2, pJ_in_ga(X1, X3)) PC_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PC_IN_GA(d(X1), X2) -> U39_GA(X1, X2, pcJ_in_ga(X1, X3)) U39_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U40_GA(X1, X2, pC_in_ga(e(X3), X2)) U39_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PC_IN_GA(e(X3), X2) PB_IN_GA(e(const(X1)), const(0)) -> U7_GA(X1, pA_in_gg(d(e(const(0))), const(1))) PB_IN_GA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(1)) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U8_GA(X1, X2, X3, X4, pH_in_ga(X1, X3)) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_GA(X1, X3) PB_IN_GA(e(+(X1, X2)), +(X3, X4)) -> U9_GA(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U10_GA(X1, X2, X3, X4, pH_in_ga(X2, X4)) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> PH_IN_GA(X2, X4) U9_GA(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U11_GA(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U11_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U12_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(1))) U11_GA(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(1)) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U13_GA(X1, X2, X3, X4, pH_in_ga(X1, X4)) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_GA(X1, X4) PB_IN_GA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U14_GA(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U15_GA(X1, X2, X3, X4, pH_in_ga(X2, X3)) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> PH_IN_GA(X2, X3) U14_GA(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U16_GA(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U16_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U17_GA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U16_GA(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(1)) PB_IN_GA(d(X1), X2) -> U18_GA(X1, X2, pJ_in_ga(X1, X3)) PB_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) PB_IN_GA(d(X1), X2) -> U19_GA(X1, X2, pcJ_in_ga(X1, X3)) U19_GA(X1, X2, pcJ_out_ga(X1, X3)) -> U20_GA(X1, X2, pB_in_ga(e(X3), X2)) U19_GA(X1, X2, pcJ_out_ga(X1, X3)) -> PB_IN_GA(e(X3), X2) PB_IN_AA(e(const(X1)), const(0)) -> U7_AA(X1, pA_in_gg(d(e(const(0))), const(1))) PB_IN_AA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(1)) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U8_AA(X1, X2, X3, X4, pH_in_aa(X1, X3)) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_AA(X1, X3) PH_IN_AA(+(X1, X2), +(X3, X4)) -> U1_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X3, X2, X4)) PH_IN_AA(+(X1, X2), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PI_IN_AAAA(X1, X2, X3, X4) -> U3_AAAA(X1, X2, X3, X4, pH_in_aa(X1, X2)) PI_IN_AAAA(X1, X2, X3, X4) -> PH_IN_AA(X1, X2) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U2_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X4, X2, X3)) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PI_IN_AAAA(X1, X2, X3, X4) -> U4_AAAA(X1, X2, X3, X4, pcH_in_aa(X1, X2)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U5_AAAA(X1, X2, X3, X4, pH_in_aa(X3, X4)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> PH_IN_AA(X3, X4) PB_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U9_AA(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U10_AA(X1, X2, X3, X4, pH_in_aa(X2, X4)) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U9_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U11_AA(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U11_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U12_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(1))) U11_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(1)) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U13_AA(X1, X2, X3, X4, pH_in_aa(X1, X4)) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_AA(X1, X4) PB_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U14_AA(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U15_AA(X1, X2, X3, X4, pH_in_aa(X2, X3)) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U14_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U16_AA(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U16_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U17_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U16_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(1)) PB_IN_AA(d(X1), X2) -> U18_AA(X1, X2, pJ_in_aa(X1, X3)) PB_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PJ_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U21_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X3, X2, X4)) PJ_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PJ_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U22_AA(X1, X2, X3, X4, pI_in_aaaa(X1, X4, X2, X3)) PJ_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PJ_IN_AA(d(X1), X2) -> U23_AA(X1, X2, pJ_in_aa(X1, X3)) PJ_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PJ_IN_AA(d(X1), X2) -> U24_AA(X1, X2, pcJ_in_aa(X1, X3)) U24_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U25_AA(X1, X2, pH_in_ga(X3, X2)) U24_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PH_IN_GA(X3, X2) PB_IN_AA(d(X1), X2) -> U19_AA(X1, X2, pcJ_in_aa(X1, X3)) U19_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U20_AA(X1, X2, pB_in_ga(e(X3), X2)) U19_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PB_IN_GA(e(X3), X2) PA_IN_AG(d(d(X1)), const(0)) -> U66_AG(X1, pC_in_aa(X1, X2)) PA_IN_AG(d(d(X1)), const(0)) -> PC_IN_AA(X1, X2) PC_IN_AA(e(t), const(1)) -> U26_AA(pA_in_gg(d(e(const(1))), const(0))) PC_IN_AA(e(t), const(1)) -> PA_IN_GG(d(e(const(1))), const(0)) PC_IN_AA(e(const(X1)), const(0)) -> U27_AA(X1, pA_in_gg(d(e(const(0))), const(0))) PC_IN_AA(e(const(X1)), const(0)) -> PA_IN_GG(d(e(const(0))), const(0)) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U28_AA(X1, X2, X3, X4, pH_in_aa(X1, X3)) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> PH_IN_AA(X1, X3) PC_IN_AA(e(+(X1, X2)), +(X3, X4)) -> U29_AA(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U30_AA(X1, X2, X3, X4, pH_in_aa(X2, X4)) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U29_AA(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U31_AA(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U31_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U32_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(X3, X4))), const(0))) U31_AA(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), const(0)) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U33_AA(X1, X2, X3, X4, pH_in_aa(X1, X4)) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> PH_IN_AA(X1, X4) PC_IN_AA(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U34_AA(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U35_AA(X1, X2, X3, X4, pH_in_aa(X2, X3)) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U34_AA(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U36_AA(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U36_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U37_AA(X1, X2, X3, X4, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U36_AA(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), const(0)) PC_IN_AA(d(X1), X2) -> U38_AA(X1, X2, pJ_in_aa(X1, X3)) PC_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) PC_IN_AA(d(X1), X2) -> U39_AA(X1, X2, pcJ_in_aa(X1, X3)) U39_AA(X1, X2, pcJ_out_aa(X1, X3)) -> U40_AA(X1, X2, pC_in_ga(e(X3), X2)) U39_AA(X1, X2, pcJ_out_aa(X1, X3)) -> PC_IN_GA(e(X3), X2) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> U67_AG(X1, X2, pA_in_ag(d(e(X1)), X2)) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> U68_AG(X1, X2, X3, pA_in_ag(d(e(X2)), X3)) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_AG(d(e(X2)), X3) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U69_AG(X1, X2, X3, X4, X5, X6, pD_in_agagag(X1, X4, X2, X5, X3, X6)) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_AGAGAG(X1, X4, X2, X5, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U41_AGAGAG(X1, X2, X3, X4, X5, X6, pA_in_ag(d(e(X1)), X2)) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U70_AG(X1, X2, X3, X4, X5, X6, pD_in_ggggag(X1, X5, X2, X4, X3, X6)) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGAG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U41_GGGGAG(X1, X2, X3, X4, X5, X6, pA_in_gg(d(e(X1)), X2)) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U43_GGGGAG(X1, X2, X3, X4, X5, X6, pE_in_ggag(X3, X4, X5, X6)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGAG(X3, X4, X5, X6) PE_IN_GGAG(X1, X2, X3, X4) -> U44_GGAG(X1, X2, X3, X4, pA_in_gg(d(e(X1)), X2)) PE_IN_GGAG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PE_IN_GGAG(X1, X2, X3, X4) -> U45_GGAG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U46_GGAG(X1, X2, X3, X4, pA_in_ag(d(e(X3)), X4)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) PA_IN_AG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U71_AG(X1, X2, X3, X4, pE_in_gggg(X1, X4, X2, X3)) PA_IN_AG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PA_IN_AG(d(d(X1)), +(X2, X3)) -> U72_AG(X1, X2, X3, pF_in_aagg(X1, X4, X2, X3)) PA_IN_AG(d(d(X1)), +(X2, X3)) -> PF_IN_AAGG(X1, X4, X2, X3) PF_IN_AAGG(X1, X2, X3, X4) -> U47_AAGG(X1, X2, X3, X4, pJ_in_aa(X1, X2)) PF_IN_AAGG(X1, X2, X3, X4) -> PJ_IN_AA(X1, X2) PF_IN_AAGG(X1, X2, X3, X4) -> U48_AAGG(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U48_AAGG(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U49_AAGG(X1, X2, X3, X4, pA_in_gg(d(e(X2)), +(X3, X4))) U48_AAGG(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> PA_IN_GG(d(e(X2)), +(X3, X4)) PA_IN_AG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U73_AG(X1, X2, pA_in_gg(d(e(X1)), X2)) PA_IN_AG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_AG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U74_AG(X1, X2, X3, pA_in_gg(d(e(X2)), X3)) PA_IN_AG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_AG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U75_AG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X5, X2, X6, X3, X4)) PA_IN_AG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_AG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U76_AG(X1, X2, X3, X4, X5, X6, pD_in_gggggg(X1, X6, X2, X5, X3, X4)) PA_IN_AG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) PA_IN_AG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U77_AG(X1, X2, X3, X4, X5, pF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) PA_IN_AG(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> PF_IN_AAGG(X1, X6, *(X2, X3), *(X4, X5)) PA_IN_AG(d(d(X1)), X2) -> U78_AG(X1, X2, pG_in_aag(X1, X3, X2)) PA_IN_AG(d(d(X1)), X2) -> PG_IN_AAG(X1, X3, X2) PG_IN_AAG(e(t), const(1), X1) -> U50_AAG(X1, pA_in_gg(d(e(const(1))), X1)) PG_IN_AAG(e(t), const(1), X1) -> PA_IN_GG(d(e(const(1))), X1) PG_IN_AAG(e(const(X1)), const(0), X2) -> U51_AAG(X1, X2, pA_in_gg(d(e(const(0))), X2)) PG_IN_AAG(e(const(X1)), const(0), X2) -> PA_IN_GG(d(e(const(0))), X2) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> U52_AAG(X1, X2, X3, X4, X5, pH_in_aa(X1, X3)) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> PH_IN_AA(X1, X3) PG_IN_AAG(e(+(X1, X2)), +(X3, X4), X5) -> U53_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U54_AAG(X1, X2, X3, X4, X5, pH_in_aa(X2, X4)) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> PH_IN_AA(X2, X4) U53_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U55_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U55_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U56_AAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(X3, X4))), X5)) U55_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> PA_IN_GG(d(e(+(X3, X4))), X5) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U57_AAG(X1, X2, X3, X4, X5, pH_in_aa(X1, X4)) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> PH_IN_AA(X1, X4) PG_IN_AAG(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U58_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U59_AAG(X1, X2, X3, X4, X5, pH_in_aa(X2, X3)) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> PH_IN_AA(X2, X3) U58_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U60_AAG(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U60_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U61_AAG(X1, X2, X3, X4, X5, pA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U60_AAG(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> PA_IN_GG(d(e(+(*(X1, X3), *(X2, X4)))), X5) PG_IN_AAG(d(X1), X2, X3) -> U62_AAG(X1, X2, X3, pJ_in_aa(X1, X4)) PG_IN_AAG(d(X1), X2, X3) -> PJ_IN_AA(X1, X4) PG_IN_AAG(d(X1), X2, X3) -> U63_AAG(X1, X2, X3, pcJ_in_aa(X1, X4)) U63_AAG(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U64_AAG(X1, X2, X3, pG_in_gag(e(X4), X2, X3)) U63_AAG(X1, X2, X3, pcJ_out_aa(X1, X4)) -> PG_IN_GAG(e(X4), X2, X3) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U43_AGAGAG(X1, X2, X3, X4, X5, X6, pE_in_agag(X3, X4, X5, X6)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> PE_IN_AGAG(X3, X4, X5, X6) PE_IN_AGAG(X1, X2, X3, X4) -> U44_AGAG(X1, X2, X3, X4, pA_in_ag(d(e(X1)), X2)) PE_IN_AGAG(X1, X2, X3, X4) -> PA_IN_AG(d(e(X1)), X2) PE_IN_AGAG(X1, X2, X3, X4) -> U45_AGAG(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U46_AGAG(X1, X2, X3, X4, pA_in_ag(d(e(X3)), X4)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: pA_in_ag(x1, x2) = pA_in_ag(x2) const(x1) = const 1 = 1 pB_in_aa(x1, x2) = pB_in_aa pA_in_gg(x1, x2) = pA_in_gg(x1, x2) d(x1) = d(x1) pB_in_ga(x1, x2) = pB_in_ga(x1) e(x1) = e(x1) t = t 0 = 0 pC_in_ga(x1, x2) = pC_in_ga(x1) +(x1, x2) = +(x1, x2) pD_in_gggggg(x1, x2, x3, x4, x5, x6) = pD_in_gggggg(x1, x2, x3, x4, x5, x6) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pE_in_gggg(x1, x2, x3, x4) = pE_in_gggg(x1, x2, x3, x4) pF_in_gagg(x1, x2, x3, x4) = pF_in_gagg(x1, x3, x4) pJ_in_ga(x1, x2) = pJ_in_ga(x1) pI_in_gaga(x1, x2, x3, x4) = pI_in_gaga(x1, x3) pH_in_ga(x1, x2) = pH_in_ga(x1) pG_in_gag(x1, x2, x3) = pG_in_gag(x1, x3) pH_in_aa(x1, x2) = pH_in_aa pI_in_aaaa(x1, x2, x3, x4) = pI_in_aaaa pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pJ_in_aa(x1, x2) = pJ_in_aa pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pC_in_aa(x1, x2) = pC_in_aa pD_in_agagag(x1, x2, x3, x4, x5, x6) = pD_in_agagag(x2, x4, x6) pD_in_ggggag(x1, x2, x3, x4, x5, x6) = pD_in_ggggag(x1, x2, x3, x4, x6) pE_in_ggag(x1, x2, x3, x4) = pE_in_ggag(x1, x2, x4) pF_in_aagg(x1, x2, x3, x4) = pF_in_aagg(x3, x4) pG_in_aag(x1, x2, x3) = pG_in_aag(x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) pE_in_agag(x1, x2, x3, x4) = pE_in_agag(x2, x4) PA_IN_AG(x1, x2) = PA_IN_AG(x2) U65_AG(x1, x2) = U65_AG(x2) PB_IN_AA(x1, x2) = PB_IN_AA U6_AA(x1) = U6_AA(x1) PA_IN_GG(x1, x2) = PA_IN_GG(x1, x2) U65_GG(x1, x2) = U65_GG(x1, x2) PB_IN_GA(x1, x2) = PB_IN_GA(x1) U6_GA(x1) = U6_GA(x1) U66_GG(x1, x2) = U66_GG(x1, x2) PC_IN_GA(x1, x2) = PC_IN_GA(x1) U26_GA(x1) = U26_GA(x1) U67_GG(x1, x2, x3) = U67_GG(x1, x2, x3) U68_GG(x1, x2, x3, x4) = U68_GG(x2, x3, x4) U69_GG(x1, x2, x3, x4, x5, x6, x7) = U69_GG(x1, x2, x3, x4, x5, x6, x7) PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) U41_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U41_GGGGGG(x1, x2, x3, x4, x5, x6, x7) U70_GG(x1, x2, x3, x4, x5, x6, x7) = U70_GG(x1, x2, x3, x4, x5, x6, x7) U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) U43_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U43_GGGGGG(x1, x2, x3, x4, x5, x6, x7) PE_IN_GGGG(x1, x2, x3, x4) = PE_IN_GGGG(x1, x2, x3, x4) U44_GGGG(x1, x2, x3, x4, x5) = U44_GGGG(x1, x2, x3, x4, x5) U71_GG(x1, x2, x3, x4, x5) = U71_GG(x1, x2, x3, x4, x5) U45_GGGG(x1, x2, x3, x4, x5) = U45_GGGG(x1, x2, x3, x4, x5) U46_GGGG(x1, x2, x3, x4, x5) = U46_GGGG(x1, x2, x3, x4, x5) U72_GG(x1, x2, x3, x4) = U72_GG(x1, x2, x3, x4) PF_IN_GAGG(x1, x2, x3, x4) = PF_IN_GAGG(x1, x3, x4) U47_GAGG(x1, x2, x3, x4, x5) = U47_GAGG(x1, x3, x4, x5) PJ_IN_GA(x1, x2) = PJ_IN_GA(x1) U21_GA(x1, x2, x3, x4, x5) = U21_GA(x1, x2, x5) PI_IN_GAGA(x1, x2, x3, x4) = PI_IN_GAGA(x1, x3) U3_GAGA(x1, x2, x3, x4, x5) = U3_GAGA(x1, x3, x5) PH_IN_GA(x1, x2) = PH_IN_GA(x1) U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5) U4_GAGA(x1, x2, x3, x4, x5) = U4_GAGA(x1, x3, x5) U5_GAGA(x1, x2, x3, x4, x5) = U5_GAGA(x1, x3, x5) U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x5) U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x5) U23_GA(x1, x2, x3) = U23_GA(x1, x3) U24_GA(x1, x2, x3) = U24_GA(x1, x3) U25_GA(x1, x2, x3) = U25_GA(x1, x3) U48_GAGG(x1, x2, x3, x4, x5) = U48_GAGG(x1, x3, x4, x5) U49_GAGG(x1, x2, x3, x4, x5) = U49_GAGG(x1, x3, x4, x5) U73_GG(x1, x2, x3) = U73_GG(x1, x2, x3) U74_GG(x1, x2, x3, x4) = U74_GG(x2, x3, x4) U75_GG(x1, x2, x3, x4, x5, x6, x7) = U75_GG(x1, x2, x3, x4, x5, x6, x7) U76_GG(x1, x2, x3, x4, x5, x6, x7) = U76_GG(x1, x2, x3, x4, x5, x6, x7) U77_GG(x1, x2, x3, x4, x5, x6) = U77_GG(x1, x2, x3, x4, x5, x6) U78_GG(x1, x2, x3) = U78_GG(x1, x2, x3) PG_IN_GAG(x1, x2, x3) = PG_IN_GAG(x1, x3) U50_GAG(x1, x2) = U50_GAG(x1, x2) U51_GAG(x1, x2, x3) = U51_GAG(x2, x3) U52_GAG(x1, x2, x3, x4, x5, x6) = U52_GAG(x1, x2, x5, x6) U53_GAG(x1, x2, x3, x4, x5, x6) = U53_GAG(x1, x2, x5, x6) U54_GAG(x1, x2, x3, x4, x5, x6) = U54_GAG(x1, x2, x5, x6) U55_GAG(x1, x2, x3, x4, x5, x6) = U55_GAG(x1, x2, x3, x5, x6) U56_GAG(x1, x2, x3, x4, x5, x6) = U56_GAG(x1, x2, x5, x6) U57_GAG(x1, x2, x3, x4, x5, x6) = U57_GAG(x1, x2, x5, x6) U58_GAG(x1, x2, x3, x4, x5, x6) = U58_GAG(x1, x2, x5, x6) U59_GAG(x1, x2, x3, x4, x5, x6) = U59_GAG(x1, x2, x5, x6) U60_GAG(x1, x2, x3, x4, x5, x6) = U60_GAG(x1, x2, x4, x5, x6) U61_GAG(x1, x2, x3, x4, x5, x6) = U61_GAG(x1, x2, x5, x6) U62_GAG(x1, x2, x3, x4) = U62_GAG(x1, x3, x4) U63_GAG(x1, x2, x3, x4) = U63_GAG(x1, x3, x4) U64_GAG(x1, x2, x3, x4) = U64_GAG(x1, x3, x4) U27_GA(x1, x2) = U27_GA(x2) U28_GA(x1, x2, x3, x4, x5) = U28_GA(x1, x2, x5) U29_GA(x1, x2, x3, x4, x5) = U29_GA(x1, x2, x5) U30_GA(x1, x2, x3, x4, x5) = U30_GA(x1, x2, x5) U31_GA(x1, x2, x3, x4, x5) = U31_GA(x1, x2, x3, x5) U32_GA(x1, x2, x3, x4, x5) = U32_GA(x1, x2, x5) U33_GA(x1, x2, x3, x4, x5) = U33_GA(x1, x2, x5) U34_GA(x1, x2, x3, x4, x5) = U34_GA(x1, x2, x5) U35_GA(x1, x2, x3, x4, x5) = U35_GA(x1, x2, x5) U36_GA(x1, x2, x3, x4, x5) = U36_GA(x1, x2, x4, x5) U37_GA(x1, x2, x3, x4, x5) = U37_GA(x1, x2, x5) U38_GA(x1, x2, x3) = U38_GA(x1, x3) U39_GA(x1, x2, x3) = U39_GA(x1, x3) U40_GA(x1, x2, x3) = U40_GA(x1, x3) U7_GA(x1, x2) = U7_GA(x2) U8_GA(x1, x2, x3, x4, x5) = U8_GA(x1, x2, x5) U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x5) U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x2, x5) U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x3, x5) U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x5) U13_GA(x1, x2, x3, x4, x5) = U13_GA(x1, x2, x5) U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x5) U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x2, x5) U16_GA(x1, x2, x3, x4, x5) = U16_GA(x1, x2, x4, x5) U17_GA(x1, x2, x3, x4, x5) = U17_GA(x1, x2, x5) U18_GA(x1, x2, x3) = U18_GA(x1, x3) U19_GA(x1, x2, x3) = U19_GA(x1, x3) U20_GA(x1, x2, x3) = U20_GA(x1, x3) U7_AA(x1, x2) = U7_AA(x2) U8_AA(x1, x2, x3, x4, x5) = U8_AA(x5) PH_IN_AA(x1, x2) = PH_IN_AA U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5) PI_IN_AAAA(x1, x2, x3, x4) = PI_IN_AAAA U3_AAAA(x1, x2, x3, x4, x5) = U3_AAAA(x5) U2_AA(x1, x2, x3, x4, x5) = U2_AA(x5) U4_AAAA(x1, x2, x3, x4, x5) = U4_AAAA(x5) U5_AAAA(x1, x2, x3, x4, x5) = U5_AAAA(x5) U9_AA(x1, x2, x3, x4, x5) = U9_AA(x5) U10_AA(x1, x2, x3, x4, x5) = U10_AA(x5) U11_AA(x1, x2, x3, x4, x5) = U11_AA(x3, x5) U12_AA(x1, x2, x3, x4, x5) = U12_AA(x5) U13_AA(x1, x2, x3, x4, x5) = U13_AA(x5) U14_AA(x1, x2, x3, x4, x5) = U14_AA(x5) U15_AA(x1, x2, x3, x4, x5) = U15_AA(x5) U16_AA(x1, x2, x3, x4, x5) = U16_AA(x1, x4, x5) U17_AA(x1, x2, x3, x4, x5) = U17_AA(x5) U18_AA(x1, x2, x3) = U18_AA(x3) PJ_IN_AA(x1, x2) = PJ_IN_AA U21_AA(x1, x2, x3, x4, x5) = U21_AA(x5) U22_AA(x1, x2, x3, x4, x5) = U22_AA(x5) U23_AA(x1, x2, x3) = U23_AA(x3) U24_AA(x1, x2, x3) = U24_AA(x3) U25_AA(x1, x2, x3) = U25_AA(x3) U19_AA(x1, x2, x3) = U19_AA(x3) U20_AA(x1, x2, x3) = U20_AA(x3) U66_AG(x1, x2) = U66_AG(x2) PC_IN_AA(x1, x2) = PC_IN_AA U26_AA(x1) = U26_AA(x1) U27_AA(x1, x2) = U27_AA(x2) U28_AA(x1, x2, x3, x4, x5) = U28_AA(x5) U29_AA(x1, x2, x3, x4, x5) = U29_AA(x5) U30_AA(x1, x2, x3, x4, x5) = U30_AA(x5) U31_AA(x1, x2, x3, x4, x5) = U31_AA(x3, x5) U32_AA(x1, x2, x3, x4, x5) = U32_AA(x5) U33_AA(x1, x2, x3, x4, x5) = U33_AA(x5) U34_AA(x1, x2, x3, x4, x5) = U34_AA(x5) U35_AA(x1, x2, x3, x4, x5) = U35_AA(x5) U36_AA(x1, x2, x3, x4, x5) = U36_AA(x1, x4, x5) U37_AA(x1, x2, x3, x4, x5) = U37_AA(x5) U38_AA(x1, x2, x3) = U38_AA(x3) U39_AA(x1, x2, x3) = U39_AA(x3) U40_AA(x1, x2, x3) = U40_AA(x3) U67_AG(x1, x2, x3) = U67_AG(x2, x3) U68_AG(x1, x2, x3, x4) = U68_AG(x3, x4) U69_AG(x1, x2, x3, x4, x5, x6, x7) = U69_AG(x4, x5, x6, x7) PD_IN_AGAGAG(x1, x2, x3, x4, x5, x6) = PD_IN_AGAGAG(x2, x4, x6) U41_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U41_AGAGAG(x2, x4, x6, x7) U70_AG(x1, x2, x3, x4, x5, x6, x7) = U70_AG(x1, x2, x4, x5, x6, x7) PD_IN_GGGGAG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGAG(x1, x2, x3, x4, x6) U41_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U41_GGGGAG(x1, x2, x3, x4, x6, x7) U42_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGAG(x1, x2, x3, x4, x6, x7) U43_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U43_GGGGAG(x1, x2, x3, x4, x6, x7) PE_IN_GGAG(x1, x2, x3, x4) = PE_IN_GGAG(x1, x2, x4) U44_GGAG(x1, x2, x3, x4, x5) = U44_GGAG(x1, x2, x4, x5) U45_GGAG(x1, x2, x3, x4, x5) = U45_GGAG(x1, x2, x4, x5) U46_GGAG(x1, x2, x3, x4, x5) = U46_GGAG(x1, x2, x4, x5) U71_AG(x1, x2, x3, x4, x5) = U71_AG(x1, x2, x3, x4, x5) U72_AG(x1, x2, x3, x4) = U72_AG(x2, x3, x4) PF_IN_AAGG(x1, x2, x3, x4) = PF_IN_AAGG(x3, x4) U47_AAGG(x1, x2, x3, x4, x5) = U47_AAGG(x3, x4, x5) U48_AAGG(x1, x2, x3, x4, x5) = U48_AAGG(x3, x4, x5) U49_AAGG(x1, x2, x3, x4, x5) = U49_AAGG(x3, x4, x5) U73_AG(x1, x2, x3) = U73_AG(x1, x2, x3) U74_AG(x1, x2, x3, x4) = U74_AG(x2, x3, x4) U75_AG(x1, x2, x3, x4, x5, x6, x7) = U75_AG(x1, x2, x3, x4, x5, x6, x7) U76_AG(x1, x2, x3, x4, x5, x6, x7) = U76_AG(x1, x2, x3, x4, x5, x6, x7) U77_AG(x1, x2, x3, x4, x5, x6) = U77_AG(x2, x3, x4, x5, x6) U78_AG(x1, x2, x3) = U78_AG(x2, x3) PG_IN_AAG(x1, x2, x3) = PG_IN_AAG(x3) U50_AAG(x1, x2) = U50_AAG(x1, x2) U51_AAG(x1, x2, x3) = U51_AAG(x2, x3) U52_AAG(x1, x2, x3, x4, x5, x6) = U52_AAG(x5, x6) U53_AAG(x1, x2, x3, x4, x5, x6) = U53_AAG(x5, x6) U54_AAG(x1, x2, x3, x4, x5, x6) = U54_AAG(x5, x6) U55_AAG(x1, x2, x3, x4, x5, x6) = U55_AAG(x3, x5, x6) U56_AAG(x1, x2, x3, x4, x5, x6) = U56_AAG(x5, x6) U57_AAG(x1, x2, x3, x4, x5, x6) = U57_AAG(x5, x6) U58_AAG(x1, x2, x3, x4, x5, x6) = U58_AAG(x5, x6) U59_AAG(x1, x2, x3, x4, x5, x6) = U59_AAG(x5, x6) U60_AAG(x1, x2, x3, x4, x5, x6) = U60_AAG(x1, x4, x5, x6) U61_AAG(x1, x2, x3, x4, x5, x6) = U61_AAG(x5, x6) U62_AAG(x1, x2, x3, x4) = U62_AAG(x3, x4) U63_AAG(x1, x2, x3, x4) = U63_AAG(x3, x4) U64_AAG(x1, x2, x3, x4) = U64_AAG(x3, x4) U42_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U42_AGAGAG(x2, x4, x6, x7) U43_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U43_AGAGAG(x2, x4, x6, x7) PE_IN_AGAG(x1, x2, x3, x4) = PE_IN_AGAG(x2, x4) U44_AGAG(x1, x2, x3, x4, x5) = U44_AGAG(x2, x4, x5) U45_AGAG(x1, x2, x3, x4, x5) = U45_AGAG(x2, x4, x5) U46_AGAG(x1, x2, x3, x4, x5) = U46_AGAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (123) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 241 less nodes. ---------------------------------------- (124) Complex Obligation (AND) ---------------------------------------- (125) Obligation: Pi DP problem: The TRS P consists of the following rules: PH_IN_AA(+(X1, X2), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PI_IN_AAAA(X1, X2, X3, X4) -> PH_IN_AA(X1, X2) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PI_IN_AAAA(X1, X2, X3, X4) -> U4_AAAA(X1, X2, X3, X4, pcH_in_aa(X1, X2)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> PH_IN_AA(X3, X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PH_IN_AA(x1, x2) = PH_IN_AA PI_IN_AAAA(x1, x2, x3, x4) = PI_IN_AAAA U4_AAAA(x1, x2, x3, x4, x5) = U4_AAAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (126) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (127) Obligation: Pi DP problem: The TRS P consists of the following rules: PH_IN_AA(+(X1, X2), +(X3, X4)) -> PI_IN_AAAA(X1, X3, X2, X4) PI_IN_AAAA(X1, X2, X3, X4) -> PH_IN_AA(X1, X2) PH_IN_AA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_AAAA(X1, X4, X2, X3) PI_IN_AAAA(X1, X2, X3, X4) -> U4_AAAA(X1, X2, X3, X4, pcH_in_aa(X1, X2)) U4_AAAA(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> PH_IN_AA(X3, X4) The TRS R consists of the following rules: pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) PH_IN_AA(x1, x2) = PH_IN_AA PI_IN_AAAA(x1, x2, x3, x4) = PI_IN_AAAA U4_AAAA(x1, x2, x3, x4, x5) = U4_AAAA(x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (128) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: PH_IN_AA -> PI_IN_AAAA PI_IN_AAAA -> PH_IN_AA PI_IN_AAAA -> U4_AAAA(pcH_in_aa) U4_AAAA(pcH_out_aa(X1, X2)) -> PH_IN_AA The TRS R consists of the following rules: pcH_in_aa -> pcH_out_aa(t, const) pcH_in_aa -> pcH_out_aa(const, const) pcH_in_aa -> U94_aa(qcI_in_aaaa) pcH_in_aa -> U95_aa(qcI_in_aaaa) U94_aa(qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) U95_aa(qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) qcI_in_aaaa -> U96_aaaa(pcH_in_aa) U96_aaaa(pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, pcH_in_aa) U97_aaaa(X1, X2, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) The set Q consists of the following terms: pcH_in_aa U94_aa(x0) U95_aa(x0) qcI_in_aaaa U96_aaaa(x0) U97_aaaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (130) TransformationProof (SOUND) By narrowing [LPAR04] the rule PI_IN_AAAA -> U4_AAAA(pcH_in_aa) at position [0] we obtained the following new rules [LPAR04]: (PI_IN_AAAA -> U4_AAAA(pcH_out_aa(t, const)),PI_IN_AAAA -> U4_AAAA(pcH_out_aa(t, const))) (PI_IN_AAAA -> U4_AAAA(pcH_out_aa(const, const)),PI_IN_AAAA -> U4_AAAA(pcH_out_aa(const, const))) (PI_IN_AAAA -> U4_AAAA(U94_aa(qcI_in_aaaa)),PI_IN_AAAA -> U4_AAAA(U94_aa(qcI_in_aaaa))) (PI_IN_AAAA -> U4_AAAA(U95_aa(qcI_in_aaaa)),PI_IN_AAAA -> U4_AAAA(U95_aa(qcI_in_aaaa))) ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: PH_IN_AA -> PI_IN_AAAA PI_IN_AAAA -> PH_IN_AA U4_AAAA(pcH_out_aa(X1, X2)) -> PH_IN_AA PI_IN_AAAA -> U4_AAAA(pcH_out_aa(t, const)) PI_IN_AAAA -> U4_AAAA(pcH_out_aa(const, const)) PI_IN_AAAA -> U4_AAAA(U94_aa(qcI_in_aaaa)) PI_IN_AAAA -> U4_AAAA(U95_aa(qcI_in_aaaa)) The TRS R consists of the following rules: pcH_in_aa -> pcH_out_aa(t, const) pcH_in_aa -> pcH_out_aa(const, const) pcH_in_aa -> U94_aa(qcI_in_aaaa) pcH_in_aa -> U95_aa(qcI_in_aaaa) U94_aa(qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) U95_aa(qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) qcI_in_aaaa -> U96_aaaa(pcH_in_aa) U96_aaaa(pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, pcH_in_aa) U97_aaaa(X1, X2, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) The set Q consists of the following terms: pcH_in_aa U94_aa(x0) U95_aa(x0) qcI_in_aaaa U96_aaaa(x0) U97_aaaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (132) Obligation: Pi DP problem: The TRS P consists of the following rules: PI_IN_GAGA(X1, X2, X3, X4) -> PH_IN_GA(X1, X2) PH_IN_GA(+(X1, X2), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U4_GAGA(X1, X2, X3, X4, pcH_in_ga(X1, X2)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> PH_IN_GA(X3, X4) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PI_IN_GAGA(x1, x2, x3, x4) = PI_IN_GAGA(x1, x3) PH_IN_GA(x1, x2) = PH_IN_GA(x1) U4_GAGA(x1, x2, x3, x4, x5) = U4_GAGA(x1, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (133) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (134) Obligation: Pi DP problem: The TRS P consists of the following rules: PI_IN_GAGA(X1, X2, X3, X4) -> PH_IN_GA(X1, X2) PH_IN_GA(+(X1, X2), +(X3, X4)) -> PI_IN_GAGA(X1, X3, X2, X4) PI_IN_GAGA(X1, X2, X3, X4) -> U4_GAGA(X1, X2, X3, X4, pcH_in_ga(X1, X2)) U4_GAGA(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> PH_IN_GA(X3, X4) PH_IN_GA(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> PI_IN_GAGA(X1, X4, X2, X3) The TRS R consists of the following rules: pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) PI_IN_GAGA(x1, x2, x3, x4) = PI_IN_GAGA(x1, x3) PH_IN_GA(x1, x2) = PH_IN_GA(x1) U4_GAGA(x1, x2, x3, x4, x5) = U4_GAGA(x1, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (135) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PJ_IN_AA(x1, x2) = PJ_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (136) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (137) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_AA(d(X1), X2) -> PJ_IN_AA(X1, X3) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) PJ_IN_AA(x1, x2) = PJ_IN_AA We have to consider all (P,R,Pi)-chains ---------------------------------------- (138) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PJ_IN_GA(x1, x2) = PJ_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (139) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (140) Obligation: Pi DP problem: The TRS P consists of the following rules: PJ_IN_GA(d(X1), X2) -> PJ_IN_GA(X1, X3) R is empty. The argument filtering Pi contains the following mapping: d(x1) = d(x1) PJ_IN_GA(x1, x2) = PJ_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (141) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_GGGGGG(X1, X4, X2, X5, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGGG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGGG(X3, X4, X5, X6) PE_IN_GGGG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PE_IN_GGGG(X1, X2, X3, X4) -> U45_GGGG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_GG(d(e(X3)), X4) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PA_IN_GG(x1, x2) = PA_IN_GG(x1, x2) PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) PE_IN_GGGG(x1, x2, x3, x4) = PE_IN_GGGG(x1, x2, x3, x4) U45_GGGG(x1, x2, x3, x4, x5) = U45_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (142) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (143) Obligation: Pi DP problem: The TRS P consists of the following rules: PA_IN_GG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_GGGGGG(X1, X4, X2, X5, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGGG(X1, X5, X2, X4, X3, X6) PD_IN_GGGGGG(X1, X2, X3, X4, X5, X6) -> U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGGG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGGG(X3, X4, X5, X6) PE_IN_GGGG(X1, X2, X3, X4) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> PE_IN_GGGG(X1, X4, X2, X3) PE_IN_GGGG(X1, X2, X3, X4) -> U45_GGGG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGGG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_GG(d(e(X3)), X4) PA_IN_GG(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> PA_IN_GG(d(e(X1)), X2) PA_IN_GG(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> PA_IN_GG(d(e(X2)), X3) PA_IN_GG(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> PD_IN_GGGGGG(X1, X5, X2, X6, X3, X4) PA_IN_GG(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> PD_IN_GGGGGG(X1, X6, X2, X5, X3, X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) PA_IN_GG(x1, x2) = PA_IN_GG(x1, x2) PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGGG(x1, x2, x3, x4, x5, x6) U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGGG(x1, x2, x3, x4, x5, x6, x7) PE_IN_GGGG(x1, x2, x3, x4) = PE_IN_GGGG(x1, x2, x3, x4) U45_GGGG(x1, x2, x3, x4, x5) = U45_GGGG(x1, x2, x3, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (144) Obligation: Pi DP problem: The TRS P consists of the following rules: PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGAG(X3, X4, X5, X6) PE_IN_GGAG(X1, X2, X3, X4) -> U45_GGAG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_AG(d(e(X2)), X3) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_AGAGAG(X1, X4, X2, X5, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGAG(X1, X5, X2, X4, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> PE_IN_AGAG(X3, X4, X5, X6) PE_IN_AGAG(X1, X2, X3, X4) -> PA_IN_AG(d(e(X1)), X2) PE_IN_AGAG(X1, X2, X3, X4) -> U45_AGAG(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(d(X1)), const(1)) -> U80_gg(X1, qcB_in_ga(X1, X2)) qcB_in_ga(e(t), const(1)) -> U98_ga(pcA_in_gg(d(e(const(1))), const(1))) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(d(X1)), const(0)) -> U81_gg(X1, qcC_in_ga(X1, X2)) qcC_in_ga(e(t), const(1)) -> U112_ga(pcA_in_gg(d(e(const(1))), const(0))) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(d(X1)), +(X2, X3)) -> U87_gg(X1, X2, X3, qcF_in_gagg(X1, X4, X2, X3)) qcF_in_gagg(X1, X2, X3, X4) -> U126_gagg(X1, X2, X3, X4, pcJ_in_ga(X1, X2)) pcJ_in_ga(e(t), const(1)) -> pcJ_out_ga(e(t), const(1)) pcJ_in_ga(e(const(X1)), const(0)) -> pcJ_out_ga(e(const(X1)), const(0)) pcJ_in_ga(e(+(X1, X2)), +(X3, X4)) -> U108_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) qcI_in_gaga(X1, X2, X3, X4) -> U96_gaga(X1, X2, X3, X4, pcH_in_ga(X1, X2)) pcH_in_ga(t, const(1)) -> pcH_out_ga(t, const(1)) pcH_in_ga(const(X1), const(0)) -> pcH_out_ga(const(X1), const(0)) pcH_in_ga(+(X1, X2), +(X3, X4)) -> U94_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X3, X2, X4)) U94_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcH_out_ga(+(X1, X2), +(X3, X4)) pcH_in_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U95_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcH_out_ga(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_gaga(X1, X2, X3, X4, pcH_out_ga(X1, X2)) -> U97_gaga(X1, X2, X3, X4, pcH_in_ga(X3, X4)) U97_gaga(X1, X2, X3, X4, pcH_out_ga(X3, X4)) -> qcI_out_gaga(X1, X2, X3, X4) U108_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X3, X2, X4)) -> pcJ_out_ga(e(+(X1, X2)), +(X3, X4)) pcJ_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_ga(X1, X2, X3, X4, qcI_in_gaga(X1, X4, X2, X3)) U109_ga(X1, X2, X3, X4, qcI_out_gaga(X1, X4, X2, X3)) -> pcJ_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_ga(d(X1), X2) -> U110_ga(X1, X2, pcJ_in_ga(X1, X3)) U110_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U111_ga(X1, X2, pcH_in_ga(X3, X2)) U111_ga(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_ga(d(X1), X2) U126_gagg(X1, X2, X3, X4, pcJ_out_ga(X1, X2)) -> U127_gagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_gg(X1, X2, X3, X4, X5, qcF_in_gagg(X1, X6, *(X2, X3), *(X4, X5))) U92_gg(X1, X2, X3, X4, X5, qcF_out_gagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_gg(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_gg(d(d(X1)), X2) -> U93_gg(X1, X2, qcG_in_gag(X1, X3, X2)) qcG_in_gag(e(t), const(1), X1) -> U128_gag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_gag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_gag(e(t), const(1), X1) qcG_in_gag(e(const(X1)), const(0), X2) -> U129_gag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_gag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_gag(e(const(X1)), const(0), X2) qcG_in_gag(e(+(X1, X2)), +(X3, X4), X5) -> U130_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X3)) U130_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X3)) -> U131_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X4)) U131_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X4)) -> U132_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_gag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_gag(X1, X2, X3, X4, X5, pcH_in_ga(X1, X4)) U133_gag(X1, X2, X3, X4, X5, pcH_out_ga(X1, X4)) -> U134_gag(X1, X2, X3, X4, X5, pcH_in_ga(X2, X3)) U134_gag(X1, X2, X3, X4, X5, pcH_out_ga(X2, X3)) -> U135_gag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_gag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_gag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_gag(d(X1), X2, X3) -> U136_gag(X1, X2, X3, pcJ_in_ga(X1, X4)) U136_gag(X1, X2, X3, pcJ_out_ga(X1, X4)) -> U137_gag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_gag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_gag(d(X1), X2, X3) U93_gg(X1, X2, qcG_out_gag(X1, X3, X2)) -> pcA_out_gg(d(d(X1)), X2) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U127_gagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_gagg(X1, X2, X3, X4) U87_gg(X1, X2, X3, qcF_out_gagg(X1, X4, X2, X3)) -> pcA_out_gg(d(d(X1)), +(X2, X3)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U112_ga(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_ga(e(t), const(1)) qcC_in_ga(e(const(X1)), const(0)) -> U113_ga(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_ga(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_ga(e(const(X1)), const(0)) qcC_in_ga(e(+(X1, X2)), +(X3, X4)) -> U114_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U114_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U115_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U115_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U116_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_ga(e(+(X1, X2)), +(X3, X4)) qcC_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U117_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U118_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U118_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U119_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_ga(d(X1), X2) -> U120_ga(X1, X2, pcJ_in_ga(X1, X3)) U120_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U121_ga(X1, X2, qcC_in_ga(e(X3), X2)) U121_ga(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_ga(d(X1), X2) U81_gg(X1, qcC_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(0)) U98_ga(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_ga(e(t), const(1)) qcB_in_ga(e(const(X1)), const(0)) -> U99_ga(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_ga(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_ga(e(const(X1)), const(0)) qcB_in_ga(e(+(X1, X2)), +(X3, X4)) -> U100_ga(X1, X2, X3, X4, pcH_in_ga(X1, X3)) U100_ga(X1, X2, X3, X4, pcH_out_ga(X1, X3)) -> U101_ga(X1, X2, X3, X4, pcH_in_ga(X2, X4)) U101_ga(X1, X2, X3, X4, pcH_out_ga(X2, X4)) -> U102_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_ga(e(+(X1, X2)), +(X3, X4)) qcB_in_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_ga(X1, X2, X3, X4, pcH_in_ga(X1, X4)) U103_ga(X1, X2, X3, X4, pcH_out_ga(X1, X4)) -> U104_ga(X1, X2, X3, X4, pcH_in_ga(X2, X3)) U104_ga(X1, X2, X3, X4, pcH_out_ga(X2, X3)) -> U105_ga(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_ga(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_ga(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_ga(d(X1), X2) -> U106_ga(X1, X2, pcJ_in_ga(X1, X3)) U106_ga(X1, X2, pcJ_out_ga(X1, X3)) -> U107_ga(X1, X2, qcB_in_ga(e(X3), X2)) U107_ga(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_ga(d(X1), X2) U80_gg(X1, qcB_out_ga(X1, X2)) -> pcA_out_gg(d(d(X1)), const(1)) pcH_in_aa(t, const(1)) -> pcH_out_aa(t, const(1)) pcH_in_aa(const(X1), const(0)) -> pcH_out_aa(const(X1), const(0)) pcH_in_aa(+(X1, X2), +(X3, X4)) -> U94_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) qcI_in_aaaa(X1, X2, X3, X4) -> U96_aaaa(X1, X2, X3, X4, pcH_in_aa(X1, X2)) pcH_in_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) -> U95_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U95_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcH_out_aa(*(X1, X2), +(*(X1, X3), *(X2, X4))) U96_aaaa(X1, X2, X3, X4, pcH_out_aa(X1, X2)) -> U97_aaaa(X1, X2, X3, X4, pcH_in_aa(X3, X4)) U97_aaaa(X1, X2, X3, X4, pcH_out_aa(X3, X4)) -> qcI_out_aaaa(X1, X2, X3, X4) U94_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcH_out_aa(+(X1, X2), +(X3, X4)) pcJ_in_aa(e(t), const(1)) -> pcJ_out_aa(e(t), const(1)) pcJ_in_aa(e(const(X1)), const(0)) -> pcJ_out_aa(e(const(X1)), const(0)) pcJ_in_aa(e(+(X1, X2)), +(X3, X4)) -> U108_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X3, X2, X4)) U108_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X3, X2, X4)) -> pcJ_out_aa(e(+(X1, X2)), +(X3, X4)) pcJ_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U109_aa(X1, X2, X3, X4, qcI_in_aaaa(X1, X4, X2, X3)) U109_aa(X1, X2, X3, X4, qcI_out_aaaa(X1, X4, X2, X3)) -> pcJ_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) pcJ_in_aa(d(X1), X2) -> U110_aa(X1, X2, pcJ_in_aa(X1, X3)) U110_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U111_aa(X1, X2, pcH_in_ga(X3, X2)) U111_aa(X1, X2, pcH_out_ga(X3, X2)) -> pcJ_out_aa(d(X1), X2) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(d(X1)), const(1)) -> U80_ag(X1, qcB_in_aa(X1, X2)) qcB_in_aa(e(t), const(1)) -> U98_aa(pcA_in_gg(d(e(const(1))), const(1))) U98_aa(pcA_out_gg(d(e(const(1))), const(1))) -> qcB_out_aa(e(t), const(1)) qcB_in_aa(e(const(X1)), const(0)) -> U99_aa(X1, pcA_in_gg(d(e(const(0))), const(1))) U99_aa(X1, pcA_out_gg(d(e(const(0))), const(1))) -> qcB_out_aa(e(const(X1)), const(0)) qcB_in_aa(e(+(X1, X2)), +(X3, X4)) -> U100_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U100_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U101_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U101_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U102_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(1))) U102_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(1))) -> qcB_out_aa(e(+(X1, X2)), +(X3, X4)) qcB_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U103_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U103_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U104_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U104_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U105_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) U105_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(1))) -> qcB_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcB_in_aa(d(X1), X2) -> U106_aa(X1, X2, pcJ_in_aa(X1, X3)) U106_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U107_aa(X1, X2, qcB_in_ga(e(X3), X2)) U107_aa(X1, X2, qcB_out_ga(e(X3), X2)) -> qcB_out_aa(d(X1), X2) U80_ag(X1, qcB_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(d(X1)), const(0)) -> U81_ag(X1, qcC_in_aa(X1, X2)) qcC_in_aa(e(t), const(1)) -> U112_aa(pcA_in_gg(d(e(const(1))), const(0))) U112_aa(pcA_out_gg(d(e(const(1))), const(0))) -> qcC_out_aa(e(t), const(1)) qcC_in_aa(e(const(X1)), const(0)) -> U113_aa(X1, pcA_in_gg(d(e(const(0))), const(0))) U113_aa(X1, pcA_out_gg(d(e(const(0))), const(0))) -> qcC_out_aa(e(const(X1)), const(0)) qcC_in_aa(e(+(X1, X2)), +(X3, X4)) -> U114_aa(X1, X2, X3, X4, pcH_in_aa(X1, X3)) U114_aa(X1, X2, X3, X4, pcH_out_aa(X1, X3)) -> U115_aa(X1, X2, X3, X4, pcH_in_aa(X2, X4)) U115_aa(X1, X2, X3, X4, pcH_out_aa(X2, X4)) -> U116_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(X3, X4))), const(0))) U116_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(X3, X4))), const(0))) -> qcC_out_aa(e(+(X1, X2)), +(X3, X4)) qcC_in_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) -> U117_aa(X1, X2, X3, X4, pcH_in_aa(X1, X4)) U117_aa(X1, X2, X3, X4, pcH_out_aa(X1, X4)) -> U118_aa(X1, X2, X3, X4, pcH_in_aa(X2, X3)) U118_aa(X1, X2, X3, X4, pcH_out_aa(X2, X3)) -> U119_aa(X1, X2, X3, X4, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) U119_aa(X1, X2, X3, X4, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), const(0))) -> qcC_out_aa(e(*(X1, X2)), +(*(X1, X3), *(X2, X4))) qcC_in_aa(d(X1), X2) -> U120_aa(X1, X2, pcJ_in_aa(X1, X3)) U120_aa(X1, X2, pcJ_out_aa(X1, X3)) -> U121_aa(X1, X2, qcC_in_ga(e(X3), X2)) U121_aa(X1, X2, qcC_out_ga(e(X3), X2)) -> qcC_out_aa(d(X1), X2) U81_ag(X1, qcC_out_aa(X1, X2)) -> pcA_out_ag(d(d(X1)), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) pcA_in_ag(d(d(X1)), +(X2, X3)) -> U87_ag(X1, X2, X3, qcF_in_aagg(X1, X4, X2, X3)) qcF_in_aagg(X1, X2, X3, X4) -> U126_aagg(X1, X2, X3, X4, pcJ_in_aa(X1, X2)) U126_aagg(X1, X2, X3, X4, pcJ_out_aa(X1, X2)) -> U127_aagg(X1, X2, X3, X4, pcA_in_gg(d(e(X2)), +(X3, X4))) U127_aagg(X1, X2, X3, X4, pcA_out_gg(d(e(X2)), +(X3, X4))) -> qcF_out_aagg(X1, X2, X3, X4) U87_ag(X1, X2, X3, qcF_out_aagg(X1, X4, X2, X3)) -> pcA_out_ag(d(d(X1)), +(X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) pcA_in_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) -> U92_ag(X1, X2, X3, X4, X5, qcF_in_aagg(X1, X6, *(X2, X3), *(X4, X5))) U92_ag(X1, X2, X3, X4, X5, qcF_out_aagg(X1, X6, *(X2, X3), *(X4, X5))) -> pcA_out_ag(d(d(X1)), +(*(X2, X3), *(X4, X5))) pcA_in_ag(d(d(X1)), X2) -> U93_ag(X1, X2, qcG_in_aag(X1, X3, X2)) qcG_in_aag(e(t), const(1), X1) -> U128_aag(X1, pcA_in_gg(d(e(const(1))), X1)) U128_aag(X1, pcA_out_gg(d(e(const(1))), X1)) -> qcG_out_aag(e(t), const(1), X1) qcG_in_aag(e(const(X1)), const(0), X2) -> U129_aag(X1, X2, pcA_in_gg(d(e(const(0))), X2)) U129_aag(X1, X2, pcA_out_gg(d(e(const(0))), X2)) -> qcG_out_aag(e(const(X1)), const(0), X2) qcG_in_aag(e(+(X1, X2)), +(X3, X4), X5) -> U130_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X3)) U130_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X3)) -> U131_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X4)) U131_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X4)) -> U132_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(X3, X4))), X5)) U132_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(X3, X4))), X5)) -> qcG_out_aag(e(+(X1, X2)), +(X3, X4), X5) qcG_in_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) -> U133_aag(X1, X2, X3, X4, X5, pcH_in_aa(X1, X4)) U133_aag(X1, X2, X3, X4, X5, pcH_out_aa(X1, X4)) -> U134_aag(X1, X2, X3, X4, X5, pcH_in_aa(X2, X3)) U134_aag(X1, X2, X3, X4, X5, pcH_out_aa(X2, X3)) -> U135_aag(X1, X2, X3, X4, X5, pcA_in_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) U135_aag(X1, X2, X3, X4, X5, pcA_out_gg(d(e(+(*(X1, X3), *(X2, X4)))), X5)) -> qcG_out_aag(e(*(X1, X2)), +(*(X1, X3), *(X2, X4)), X5) qcG_in_aag(d(X1), X2, X3) -> U136_aag(X1, X2, X3, pcJ_in_aa(X1, X4)) U136_aag(X1, X2, X3, pcJ_out_aa(X1, X4)) -> U137_aag(X1, X2, X3, qcG_in_gag(e(X4), X2, X3)) U137_aag(X1, X2, X3, qcG_out_gag(e(X4), X2, X3)) -> qcG_out_aag(d(X1), X2, X3) U93_ag(X1, X2, qcG_out_aag(X1, X3, X2)) -> pcA_out_ag(d(d(X1)), X2) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U80_gg(x1, x2) = U80_gg(x1, x2) qcB_in_ga(x1, x2) = qcB_in_ga(x1) U98_ga(x1) = U98_ga(x1) U81_gg(x1, x2) = U81_gg(x1, x2) qcC_in_ga(x1, x2) = qcC_in_ga(x1) U112_ga(x1) = U112_ga(x1) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U87_gg(x1, x2, x3, x4) = U87_gg(x1, x2, x3, x4) qcF_in_gagg(x1, x2, x3, x4) = qcF_in_gagg(x1, x3, x4) U126_gagg(x1, x2, x3, x4, x5) = U126_gagg(x1, x3, x4, x5) pcJ_in_ga(x1, x2) = pcJ_in_ga(x1) pcJ_out_ga(x1, x2) = pcJ_out_ga(x1, x2) U108_ga(x1, x2, x3, x4, x5) = U108_ga(x1, x2, x5) qcI_in_gaga(x1, x2, x3, x4) = qcI_in_gaga(x1, x3) U96_gaga(x1, x2, x3, x4, x5) = U96_gaga(x1, x3, x5) pcH_in_ga(x1, x2) = pcH_in_ga(x1) pcH_out_ga(x1, x2) = pcH_out_ga(x1, x2) U94_ga(x1, x2, x3, x4, x5) = U94_ga(x1, x2, x5) qcI_out_gaga(x1, x2, x3, x4) = qcI_out_gaga(x1, x2, x3, x4) U95_ga(x1, x2, x3, x4, x5) = U95_ga(x1, x2, x5) U97_gaga(x1, x2, x3, x4, x5) = U97_gaga(x1, x2, x3, x5) U109_ga(x1, x2, x3, x4, x5) = U109_ga(x1, x2, x5) U110_ga(x1, x2, x3) = U110_ga(x1, x3) U111_ga(x1, x2, x3) = U111_ga(x1, x3) U127_gagg(x1, x2, x3, x4, x5) = U127_gagg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U92_gg(x1, x2, x3, x4, x5, x6) = U92_gg(x1, x2, x3, x4, x5, x6) qcF_out_gagg(x1, x2, x3, x4) = qcF_out_gagg(x1, x2, x3, x4) U93_gg(x1, x2, x3) = U93_gg(x1, x2, x3) qcG_in_gag(x1, x2, x3) = qcG_in_gag(x1, x3) U128_gag(x1, x2) = U128_gag(x1, x2) qcG_out_gag(x1, x2, x3) = qcG_out_gag(x1, x2, x3) U129_gag(x1, x2, x3) = U129_gag(x2, x3) U130_gag(x1, x2, x3, x4, x5, x6) = U130_gag(x1, x2, x5, x6) U131_gag(x1, x2, x3, x4, x5, x6) = U131_gag(x1, x2, x3, x5, x6) U132_gag(x1, x2, x3, x4, x5, x6) = U132_gag(x1, x2, x3, x4, x5, x6) U133_gag(x1, x2, x3, x4, x5, x6) = U133_gag(x1, x2, x5, x6) U134_gag(x1, x2, x3, x4, x5, x6) = U134_gag(x1, x2, x4, x5, x6) U135_gag(x1, x2, x3, x4, x5, x6) = U135_gag(x1, x2, x3, x4, x5, x6) U136_gag(x1, x2, x3, x4) = U136_gag(x1, x3, x4) U137_gag(x1, x2, x3, x4) = U137_gag(x1, x3, x4) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) qcC_out_ga(x1, x2) = qcC_out_ga(x1, x2) U113_ga(x1, x2) = U113_ga(x2) U114_ga(x1, x2, x3, x4, x5) = U114_ga(x1, x2, x5) U115_ga(x1, x2, x3, x4, x5) = U115_ga(x1, x2, x3, x5) U116_ga(x1, x2, x3, x4, x5) = U116_ga(x1, x2, x3, x4, x5) U117_ga(x1, x2, x3, x4, x5) = U117_ga(x1, x2, x5) U118_ga(x1, x2, x3, x4, x5) = U118_ga(x1, x2, x4, x5) U119_ga(x1, x2, x3, x4, x5) = U119_ga(x1, x2, x3, x4, x5) U120_ga(x1, x2, x3) = U120_ga(x1, x3) U121_ga(x1, x2, x3) = U121_ga(x1, x3) qcB_out_ga(x1, x2) = qcB_out_ga(x1, x2) U99_ga(x1, x2) = U99_ga(x2) U100_ga(x1, x2, x3, x4, x5) = U100_ga(x1, x2, x5) U101_ga(x1, x2, x3, x4, x5) = U101_ga(x1, x2, x3, x5) U102_ga(x1, x2, x3, x4, x5) = U102_ga(x1, x2, x3, x4, x5) U103_ga(x1, x2, x3, x4, x5) = U103_ga(x1, x2, x5) U104_ga(x1, x2, x3, x4, x5) = U104_ga(x1, x2, x4, x5) U105_ga(x1, x2, x3, x4, x5) = U105_ga(x1, x2, x3, x4, x5) U106_ga(x1, x2, x3) = U106_ga(x1, x3) U107_ga(x1, x2, x3) = U107_ga(x1, x3) pcH_in_aa(x1, x2) = pcH_in_aa pcH_out_aa(x1, x2) = pcH_out_aa(x1, x2) U94_aa(x1, x2, x3, x4, x5) = U94_aa(x5) qcI_in_aaaa(x1, x2, x3, x4) = qcI_in_aaaa U96_aaaa(x1, x2, x3, x4, x5) = U96_aaaa(x5) U95_aa(x1, x2, x3, x4, x5) = U95_aa(x5) qcI_out_aaaa(x1, x2, x3, x4) = qcI_out_aaaa(x1, x2, x3, x4) U97_aaaa(x1, x2, x3, x4, x5) = U97_aaaa(x1, x2, x5) pcJ_in_aa(x1, x2) = pcJ_in_aa pcJ_out_aa(x1, x2) = pcJ_out_aa(x1, x2) U108_aa(x1, x2, x3, x4, x5) = U108_aa(x5) U109_aa(x1, x2, x3, x4, x5) = U109_aa(x5) U110_aa(x1, x2, x3) = U110_aa(x3) U111_aa(x1, x2, x3) = U111_aa(x1, x3) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U80_ag(x1, x2) = U80_ag(x2) qcB_in_aa(x1, x2) = qcB_in_aa U98_aa(x1) = U98_aa(x1) qcB_out_aa(x1, x2) = qcB_out_aa(x1, x2) U99_aa(x1, x2) = U99_aa(x2) U100_aa(x1, x2, x3, x4, x5) = U100_aa(x5) U101_aa(x1, x2, x3, x4, x5) = U101_aa(x1, x3, x5) U102_aa(x1, x2, x3, x4, x5) = U102_aa(x1, x2, x3, x4, x5) U103_aa(x1, x2, x3, x4, x5) = U103_aa(x5) U104_aa(x1, x2, x3, x4, x5) = U104_aa(x1, x4, x5) U105_aa(x1, x2, x3, x4, x5) = U105_aa(x1, x2, x3, x4, x5) U106_aa(x1, x2, x3) = U106_aa(x3) U107_aa(x1, x2, x3) = U107_aa(x1, x3) U81_ag(x1, x2) = U81_ag(x2) qcC_in_aa(x1, x2) = qcC_in_aa U112_aa(x1) = U112_aa(x1) qcC_out_aa(x1, x2) = qcC_out_aa(x1, x2) U113_aa(x1, x2) = U113_aa(x2) U114_aa(x1, x2, x3, x4, x5) = U114_aa(x5) U115_aa(x1, x2, x3, x4, x5) = U115_aa(x1, x3, x5) U116_aa(x1, x2, x3, x4, x5) = U116_aa(x1, x2, x3, x4, x5) U117_aa(x1, x2, x3, x4, x5) = U117_aa(x5) U118_aa(x1, x2, x3, x4, x5) = U118_aa(x1, x4, x5) U119_aa(x1, x2, x3, x4, x5) = U119_aa(x1, x2, x3, x4, x5) U120_aa(x1, x2, x3) = U120_aa(x3) U121_aa(x1, x2, x3) = U121_aa(x1, x3) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U87_ag(x1, x2, x3, x4) = U87_ag(x2, x3, x4) qcF_in_aagg(x1, x2, x3, x4) = qcF_in_aagg(x3, x4) U126_aagg(x1, x2, x3, x4, x5) = U126_aagg(x3, x4, x5) U127_aagg(x1, x2, x3, x4, x5) = U127_aagg(x1, x2, x3, x4, x5) qcF_out_aagg(x1, x2, x3, x4) = qcF_out_aagg(x1, x2, x3, x4) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) U92_ag(x1, x2, x3, x4, x5, x6) = U92_ag(x2, x3, x4, x5, x6) U93_ag(x1, x2, x3) = U93_ag(x2, x3) qcG_in_aag(x1, x2, x3) = qcG_in_aag(x3) U128_aag(x1, x2) = U128_aag(x1, x2) qcG_out_aag(x1, x2, x3) = qcG_out_aag(x1, x2, x3) U129_aag(x1, x2, x3) = U129_aag(x2, x3) U130_aag(x1, x2, x3, x4, x5, x6) = U130_aag(x5, x6) U131_aag(x1, x2, x3, x4, x5, x6) = U131_aag(x1, x3, x5, x6) U132_aag(x1, x2, x3, x4, x5, x6) = U132_aag(x1, x2, x3, x4, x5, x6) U133_aag(x1, x2, x3, x4, x5, x6) = U133_aag(x5, x6) U134_aag(x1, x2, x3, x4, x5, x6) = U134_aag(x1, x4, x5, x6) U135_aag(x1, x2, x3, x4, x5, x6) = U135_aag(x1, x2, x3, x4, x5, x6) U136_aag(x1, x2, x3, x4) = U136_aag(x3, x4) U137_aag(x1, x2, x3, x4) = U137_aag(x1, x3, x4) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PA_IN_AG(x1, x2) = PA_IN_AG(x2) PD_IN_AGAGAG(x1, x2, x3, x4, x5, x6) = PD_IN_AGAGAG(x2, x4, x6) PD_IN_GGGGAG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGAG(x1, x2, x3, x4, x6) U42_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGAG(x1, x2, x3, x4, x6, x7) PE_IN_GGAG(x1, x2, x3, x4) = PE_IN_GGAG(x1, x2, x4) U45_GGAG(x1, x2, x3, x4, x5) = U45_GGAG(x1, x2, x4, x5) U42_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U42_AGAGAG(x2, x4, x6, x7) PE_IN_AGAG(x1, x2, x3, x4) = PE_IN_AGAG(x2, x4) U45_AGAG(x1, x2, x3, x4, x5) = U45_AGAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (145) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (146) Obligation: Pi DP problem: The TRS P consists of the following rules: PD_IN_GGGGAG(X1, X2, X3, X4, X5, X6) -> U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U42_GGGGAG(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> PE_IN_GGAG(X3, X4, X5, X6) PE_IN_GGAG(X1, X2, X3, X4) -> U45_GGAG(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U45_GGAG(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) PA_IN_AG(d(e(+(t, X1))), +(const(1), X2)) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(const(X1), X2))), +(const(0), X3)) -> PA_IN_AG(d(e(X2)), X3) PA_IN_AG(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> PD_IN_AGAGAG(X1, X4, X2, X5, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> PA_IN_AG(d(e(X1)), X2) PA_IN_AG(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> PD_IN_GGGGAG(X1, X5, X2, X4, X3, X6) PD_IN_AGAGAG(X1, X2, X3, X4, X5, X6) -> U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) U42_AGAGAG(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> PE_IN_AGAG(X3, X4, X5, X6) PE_IN_AGAG(X1, X2, X3, X4) -> PA_IN_AG(d(e(X1)), X2) PE_IN_AGAG(X1, X2, X3, X4) -> U45_AGAG(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) U45_AGAG(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> PA_IN_AG(d(e(X3)), X4) The TRS R consists of the following rules: pcA_in_gg(d(e(t)), const(1)) -> pcA_out_gg(d(e(t)), const(1)) pcA_in_gg(d(e(const(X1))), const(0)) -> pcA_out_gg(d(e(const(X1))), const(0)) pcA_in_gg(d(e(+(t, X1))), +(const(1), X2)) -> U82_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X4, X2, X5, X3, X6)) pcA_in_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X4, X3, X6)) pcA_in_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_gg(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) pcA_in_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_gg(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_gg(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) pcA_in_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_gg(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) pcA_in_ag(d(e(t)), const(1)) -> pcA_out_ag(d(e(t)), const(1)) pcA_in_ag(d(e(const(X1))), const(0)) -> pcA_out_ag(d(e(const(X1))), const(0)) pcA_in_ag(d(e(+(t, X1))), +(const(1), X2)) -> U82_ag(X1, X2, pcA_in_ag(d(e(X1)), X2)) pcA_in_ag(d(e(+(const(X1), X2))), +(const(0), X3)) -> U83_ag(X1, X2, X3, pcA_in_ag(d(e(X2)), X3)) pcA_in_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) -> U84_ag(X1, X2, X3, X4, X5, X6, qcD_in_agagag(X1, X4, X2, X5, X3, X6)) pcA_in_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) -> U85_ag(X1, X2, X3, X4, X5, X6, qcD_in_ggggag(X1, X5, X2, X4, X3, X6)) pcA_in_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) -> U86_ag(X1, X2, X3, X4, qcE_in_gggg(X1, X4, X2, X3)) pcA_in_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) -> U88_ag(X1, X2, pcA_in_gg(d(e(X1)), X2)) pcA_in_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) -> U89_ag(X1, X2, X3, pcA_in_gg(d(e(X2)), X3)) pcA_in_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) -> U90_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X5, X2, X6, X3, X4)) pcA_in_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) -> U91_ag(X1, X2, X3, X4, X5, X6, qcD_in_gggggg(X1, X6, X2, X5, X3, X4)) U82_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(+(t, X1))), +(const(1), X2)) U83_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(+(const(X1), X2))), +(const(0), X3)) U84_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X4, X2, X5, X3, X6)) -> pcA_out_gg(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U85_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X4, X3, X6)) -> pcA_out_gg(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U86_gg(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_gg(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U88_gg(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_gg(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U89_gg(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_gg(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U90_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_gg(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) U91_gg(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_gg(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) U82_ag(X1, X2, pcA_out_ag(d(e(X1)), X2)) -> pcA_out_ag(d(e(+(t, X1))), +(const(1), X2)) U83_ag(X1, X2, X3, pcA_out_ag(d(e(X2)), X3)) -> pcA_out_ag(d(e(+(const(X1), X2))), +(const(0), X3)) U84_ag(X1, X2, X3, X4, X5, X6, qcD_out_agagag(X1, X4, X2, X5, X3, X6)) -> pcA_out_ag(d(e(+(+(X1, X2), X3))), +(+(X4, X5), X6)) U85_ag(X1, X2, X3, X4, X5, X6, qcD_out_ggggag(X1, X5, X2, X4, X3, X6)) -> pcA_out_ag(d(e(+(*(X1, X2), X3))), +(+(*(X1, X4), *(X2, X5)), X6)) U86_ag(X1, X2, X3, X4, qcE_out_gggg(X1, X4, X2, X3)) -> pcA_out_ag(d(e(*(X1, X2))), +(*(X1, X3), *(X2, X4))) U88_ag(X1, X2, pcA_out_gg(d(e(X1)), X2)) -> pcA_out_ag(d(e(*(t, X1))), +(*(t, X2), *(X1, const(1)))) U89_ag(X1, X2, X3, pcA_out_gg(d(e(X2)), X3)) -> pcA_out_ag(d(e(*(const(X1), X2))), +(*(const(X1), X3), *(X2, const(0)))) U90_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X5, X2, X6, X3, X4)) -> pcA_out_ag(d(e(*(+(X1, X2), X3))), +(*(+(X1, X2), X4), *(X3, +(X5, X6)))) U91_ag(X1, X2, X3, X4, X5, X6, qcD_out_gggggg(X1, X6, X2, X5, X3, X4)) -> pcA_out_ag(d(e(*(*(X1, X2), X3))), +(*(*(X1, X2), X4), *(X3, +(*(X1, X5), *(X2, X6))))) qcD_in_gggggg(X1, X2, X3, X4, X5, X6) -> U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) qcE_in_gggg(X1, X2, X3, X4) -> U124_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) qcD_in_agagag(X1, X2, X3, X4, X5, X6) -> U122_agagag(X1, X2, X3, X4, X5, X6, pcA_in_ag(d(e(X1)), X2)) qcD_in_ggggag(X1, X2, X3, X4, X5, X6) -> U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_in_gg(d(e(X1)), X2)) U122_gggggg(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_in_gggg(X3, X4, X5, X6)) U124_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_gggg(X1, X2, X3, X4, pcA_in_gg(d(e(X3)), X4)) U122_agagag(X1, X2, X3, X4, X5, X6, pcA_out_ag(d(e(X1)), X2)) -> U123_agagag(X1, X2, X3, X4, X5, X6, qcE_in_agag(X3, X4, X5, X6)) U122_ggggag(X1, X2, X3, X4, X5, X6, pcA_out_gg(d(e(X1)), X2)) -> U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_in_ggag(X3, X4, X5, X6)) U123_gggggg(X1, X2, X3, X4, X5, X6, qcE_out_gggg(X3, X4, X5, X6)) -> qcD_out_gggggg(X1, X2, X3, X4, X5, X6) U125_gggg(X1, X2, X3, X4, pcA_out_gg(d(e(X3)), X4)) -> qcE_out_gggg(X1, X2, X3, X4) U123_agagag(X1, X2, X3, X4, X5, X6, qcE_out_agag(X3, X4, X5, X6)) -> qcD_out_agagag(X1, X2, X3, X4, X5, X6) U123_ggggag(X1, X2, X3, X4, X5, X6, qcE_out_ggag(X3, X4, X5, X6)) -> qcD_out_ggggag(X1, X2, X3, X4, X5, X6) qcE_in_agag(X1, X2, X3, X4) -> U124_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X1)), X2)) qcE_in_ggag(X1, X2, X3, X4) -> U124_ggag(X1, X2, X3, X4, pcA_in_gg(d(e(X1)), X2)) U124_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X1)), X2)) -> U125_agag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U124_ggag(X1, X2, X3, X4, pcA_out_gg(d(e(X1)), X2)) -> U125_ggag(X1, X2, X3, X4, pcA_in_ag(d(e(X3)), X4)) U125_agag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_agag(X1, X2, X3, X4) U125_ggag(X1, X2, X3, X4, pcA_out_ag(d(e(X3)), X4)) -> qcE_out_ggag(X1, X2, X3, X4) The argument filtering Pi contains the following mapping: const(x1) = const 1 = 1 d(x1) = d(x1) e(x1) = e(x1) t = t 0 = 0 +(x1, x2) = +(x1, x2) *(x1, x2) = *(x1, x2) pcA_in_gg(x1, x2) = pcA_in_gg(x1, x2) pcA_out_gg(x1, x2) = pcA_out_gg(x1, x2) U82_gg(x1, x2, x3) = U82_gg(x1, x2, x3) U83_gg(x1, x2, x3, x4) = U83_gg(x2, x3, x4) U84_gg(x1, x2, x3, x4, x5, x6, x7) = U84_gg(x1, x2, x3, x4, x5, x6, x7) qcD_in_gggggg(x1, x2, x3, x4, x5, x6) = qcD_in_gggggg(x1, x2, x3, x4, x5, x6) U122_gggggg(x1, x2, x3, x4, x5, x6, x7) = U122_gggggg(x1, x2, x3, x4, x5, x6, x7) U85_gg(x1, x2, x3, x4, x5, x6, x7) = U85_gg(x1, x2, x3, x4, x5, x6, x7) qcD_out_gggggg(x1, x2, x3, x4, x5, x6) = qcD_out_gggggg(x1, x2, x3, x4, x5, x6) U86_gg(x1, x2, x3, x4, x5) = U86_gg(x1, x2, x3, x4, x5) qcE_in_gggg(x1, x2, x3, x4) = qcE_in_gggg(x1, x2, x3, x4) U124_gggg(x1, x2, x3, x4, x5) = U124_gggg(x1, x2, x3, x4, x5) U88_gg(x1, x2, x3) = U88_gg(x1, x2, x3) U89_gg(x1, x2, x3, x4) = U89_gg(x2, x3, x4) U90_gg(x1, x2, x3, x4, x5, x6, x7) = U90_gg(x1, x2, x3, x4, x5, x6, x7) U91_gg(x1, x2, x3, x4, x5, x6, x7) = U91_gg(x1, x2, x3, x4, x5, x6, x7) U125_gggg(x1, x2, x3, x4, x5) = U125_gggg(x1, x2, x3, x4, x5) qcE_out_gggg(x1, x2, x3, x4) = qcE_out_gggg(x1, x2, x3, x4) U123_gggggg(x1, x2, x3, x4, x5, x6, x7) = U123_gggggg(x1, x2, x3, x4, x5, x6, x7) pcA_in_ag(x1, x2) = pcA_in_ag(x2) pcA_out_ag(x1, x2) = pcA_out_ag(x1, x2) U82_ag(x1, x2, x3) = U82_ag(x2, x3) U83_ag(x1, x2, x3, x4) = U83_ag(x3, x4) U84_ag(x1, x2, x3, x4, x5, x6, x7) = U84_ag(x4, x5, x6, x7) qcD_in_agagag(x1, x2, x3, x4, x5, x6) = qcD_in_agagag(x2, x4, x6) U122_agagag(x1, x2, x3, x4, x5, x6, x7) = U122_agagag(x2, x4, x6, x7) U85_ag(x1, x2, x3, x4, x5, x6, x7) = U85_ag(x1, x2, x4, x5, x6, x7) qcD_in_ggggag(x1, x2, x3, x4, x5, x6) = qcD_in_ggggag(x1, x2, x3, x4, x6) U122_ggggag(x1, x2, x3, x4, x5, x6, x7) = U122_ggggag(x1, x2, x3, x4, x6, x7) U123_ggggag(x1, x2, x3, x4, x5, x6, x7) = U123_ggggag(x1, x2, x3, x4, x6, x7) qcE_in_ggag(x1, x2, x3, x4) = qcE_in_ggag(x1, x2, x4) U124_ggag(x1, x2, x3, x4, x5) = U124_ggag(x1, x2, x4, x5) U125_ggag(x1, x2, x3, x4, x5) = U125_ggag(x1, x2, x4, x5) U86_ag(x1, x2, x3, x4, x5) = U86_ag(x1, x2, x3, x4, x5) U88_ag(x1, x2, x3) = U88_ag(x1, x2, x3) U89_ag(x1, x2, x3, x4) = U89_ag(x2, x3, x4) U90_ag(x1, x2, x3, x4, x5, x6, x7) = U90_ag(x1, x2, x3, x4, x5, x6, x7) U91_ag(x1, x2, x3, x4, x5, x6, x7) = U91_ag(x1, x2, x3, x4, x5, x6, x7) qcE_out_ggag(x1, x2, x3, x4) = qcE_out_ggag(x1, x2, x3, x4) qcD_out_ggggag(x1, x2, x3, x4, x5, x6) = qcD_out_ggggag(x1, x2, x3, x4, x5, x6) U123_agagag(x1, x2, x3, x4, x5, x6, x7) = U123_agagag(x1, x2, x4, x6, x7) qcE_in_agag(x1, x2, x3, x4) = qcE_in_agag(x2, x4) U124_agag(x1, x2, x3, x4, x5) = U124_agag(x2, x4, x5) U125_agag(x1, x2, x3, x4, x5) = U125_agag(x1, x2, x4, x5) qcE_out_agag(x1, x2, x3, x4) = qcE_out_agag(x1, x2, x3, x4) qcD_out_agagag(x1, x2, x3, x4, x5, x6) = qcD_out_agagag(x1, x2, x3, x4, x5, x6) PA_IN_AG(x1, x2) = PA_IN_AG(x2) PD_IN_AGAGAG(x1, x2, x3, x4, x5, x6) = PD_IN_AGAGAG(x2, x4, x6) PD_IN_GGGGAG(x1, x2, x3, x4, x5, x6) = PD_IN_GGGGAG(x1, x2, x3, x4, x6) U42_GGGGAG(x1, x2, x3, x4, x5, x6, x7) = U42_GGGGAG(x1, x2, x3, x4, x6, x7) PE_IN_GGAG(x1, x2, x3, x4) = PE_IN_GGAG(x1, x2, x4) U45_GGAG(x1, x2, x3, x4, x5) = U45_GGAG(x1, x2, x4, x5) U42_AGAGAG(x1, x2, x3, x4, x5, x6, x7) = U42_AGAGAG(x2, x4, x6, x7) PE_IN_AGAG(x1, x2, x3, x4) = PE_IN_AGAG(x2, x4) U45_AGAG(x1, x2, x3, x4, x5) = U45_AGAG(x2, x4, x5) We have to consider all (P,R,Pi)-chains