MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern p(a,g) w.r.t. the given Prolog program could not be shown: (0) Prolog (1) PrologToPiTRSProof [SOUND, 27 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 52 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 8 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, 30 ms] (32) PiTRS (33) DependencyPairsProof [EQUIVALENT, 39 ms] (34) PiDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) AND (37) PiDP (38) UsableRulesProof [EQUIVALENT, 0 ms] (39) PiDP (40) PiDPToQDPProof [SOUND, 0 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, 62 ms] (62) QTRS (63) DependencyPairsProof [EQUIVALENT, 0 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) AND (67) QDP (68) NonLoopProof [COMPLETE, 15 ms] (69) NO (70) QDP (71) QDPSizeChangeProof [EQUIVALENT, 0 ms] (72) YES (73) QDP (74) NonLoopProof [COMPLETE, 1185 ms] (75) NO (76) PrologToIRSwTTransformerProof [SOUND, 56 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, 6 ms] (86) IRSwT (87) FilterProof [EQUIVALENT, 0 ms] (88) IntTRS (89) IntTRSPeriodicNontermProof [COMPLETE, 5 ms] (90) NO (91) IRSwT (92) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (93) IRSwT (94) IntTRSCompressionProof [EQUIVALENT, 0 ms] (95) IRSwT (96) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (97) IRSwT (98) IRSwTTerminationDigraphProof [EQUIVALENT, 28 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, 7 ms] (107) NO (108) IRSwT (109) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms] (110) IRSwT (111) IntTRSCompressionProof [EQUIVALENT, 21 ms] (112) IRSwT (113) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (114) IRSwT (115) IRSwTTerminationDigraphProof [EQUIVALENT, 26 ms] (116) IRSwT (117) IntTRSCompressionProof [EQUIVALENT, 8 ms] (118) IRSwT (119) PrologToDTProblemTransformerProof [SOUND, 114 ms] (120) TRIPLES (121) TriplesToPiDPProof [SOUND, 1176 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) PiDP (131) UsableRulesProof [EQUIVALENT, 0 ms] (132) PiDP (133) PiDP (134) UsableRulesProof [EQUIVALENT, 0 ms] (135) PiDP (136) PiDP (137) UsableRulesProof [EQUIVALENT, 0 ms] (138) PiDP (139) PiDP (140) UsableRulesProof [EQUIVALENT, 0 ms] (141) PiDP (142) PiDP (143) UsableRulesProof [EQUIVALENT, 0 ms] (144) 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": 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": { "44": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "45": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "46": { "goal": [{ "clause": 1, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "47": { "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": [] } }, "48": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "49": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "type": "Nodes", "430": { "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": [] } }, "432": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "434": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "436": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "437": { "goal": [{ "clause": 2, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "514": { "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": [] } }, "438": { "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": [] } }, "515": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "516": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "517": { "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": [] } }, "518": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "519": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "1": { "goal": [{ "clause": -1, "scope": -1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "243": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "441": { "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": [] } }, "2": { "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": [] } }, "442": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "520": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T103)) T101)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T101"], "free": [], "exprvars": [] } }, "521": { "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": [] } }, "5": { "goal": [{ "clause": 0, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "445": { "goal": [{ "clause": 3, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "522": { "goal": [{ "clause": 0, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "446": { "goal": [{ "clause": 4, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "523": { "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": [] } }, "249": { "goal": [{ "clause": 2, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "524": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "404": { "goal": [{ "clause": 3, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "525": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "526": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "406": { "goal": [{ "clause": 4, "scope": 1, "term": "(p T1 T2)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T2"], "free": [], "exprvars": [] } }, "527": { "goal": [{ "clause": 1, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "528": { "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": [] } }, "408": { "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": [] } }, "529": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "250": { "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": [] } }, "530": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "410": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "531": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "378": { "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": [] } }, "532": { "goal": [{ "clause": 2, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "379": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "412": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T47)) T50)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T47", "T50" ], "free": [], "exprvars": [] } }, "533": { "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": [] } }, "534": { "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": [] } }, "414": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "535": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "415": { "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": [] } }, "536": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T119)) X134)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X134"], "exprvars": [] } }, "537": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T122)) X135)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X135"], "exprvars": [] } }, "417": { "goal": [{ "clause": 0, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "538": { "goal": [{ "clause": 3, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "539": { "goal": [{ "clause": 4, "scope": 3, "term": "(p (d T102) X100)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "380": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T28)) T26)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T26"], "free": [], "exprvars": [] } }, "381": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T30)) T27)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T27"], "free": [], "exprvars": [] } }, "540": { "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": [] } }, "420": { "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": [] } }, "541": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "542": { "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": [] } }, "422": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "543": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "544": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d T138) X175)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X175"], "exprvars": [] } }, "424": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "545": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T139)) X176)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X176"], "exprvars": [] } }, "426": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "429": { "goal": [{ "clause": 1, "scope": 2, "term": "(p (d (e T48)) T49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [ "T49", "T48" ], "free": [], "exprvars": [] } }, "42": { "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": [] } }, "43": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 1, "to": 2, "label": "CASE" }, { "from": 2, "to": 5, "label": "PARALLEL" }, { "from": 2, "to": 42, "label": "PARALLEL" }, { "from": 5, "to": 43, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 5, "to": 44, "label": "EVAL-BACKTRACK" }, { "from": 42, "to": 46, "label": "PARALLEL" }, { "from": 42, "to": 47, "label": "PARALLEL" }, { "from": 43, "to": 45, "label": "SUCCESS" }, { "from": 46, "to": 48, "label": "EVAL with clause\np(d(e(const(X5))), const(0)).\nand substitutionX5 -> T7,\nT1 -> d(e(const(T7))),\nT2 -> const(0)" }, { "from": 46, "to": 49, "label": "EVAL-BACKTRACK" }, { "from": 47, "to": 249, "label": "PARALLEL" }, { "from": 47, "to": 250, "label": "PARALLEL" }, { "from": 48, "to": 243, "label": "SUCCESS" }, { "from": 249, "to": 378, "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": 249, "to": 379, "label": "EVAL-BACKTRACK" }, { "from": 250, "to": 404, "label": "PARALLEL" }, { "from": 250, "to": 406, "label": "PARALLEL" }, { "from": 378, "to": 380, "label": "SPLIT 1" }, { "from": 378, "to": 381, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:T29 -> T30" }, { "from": 380, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T28))\nT2 -> T26" }, { "from": 381, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T30))\nT2 -> T27" }, { "from": 404, "to": 408, "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": 404, "to": 410, "label": "EVAL-BACKTRACK" }, { "from": 406, "to": 517, "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": 406, "to": 518, "label": "EVAL-BACKTRACK" }, { "from": 408, "to": 412, "label": "SPLIT 1" }, { "from": 408, "to": 414, "label": "SPLIT 2\nnew knowledge:\nT47 is ground\nT50 is ground" }, { "from": 412, "to": 414, "label": "INSTANCE with matching:\nT48 -> T47\nT49 -> T50" }, { "from": 414, "to": 415, "label": "CASE" }, { "from": 415, "to": 417, "label": "PARALLEL" }, { "from": 415, "to": 420, "label": "PARALLEL" }, { "from": 417, "to": 422, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT48 -> t,\nT49 -> const(1)" }, { "from": 417, "to": 424, "label": "EVAL-BACKTRACK" }, { "from": 420, "to": 429, "label": "PARALLEL" }, { "from": 420, "to": 430, "label": "PARALLEL" }, { "from": 422, "to": 426, "label": "SUCCESS" }, { "from": 429, "to": 432, "label": "EVAL with clause\np(d(e(const(X50))), const(0)).\nand substitutionX50 -> T55,\nT48 -> const(T55),\nT49 -> const(0)" }, { "from": 429, "to": 434, "label": "EVAL-BACKTRACK" }, { "from": 430, "to": 437, "label": "PARALLEL" }, { "from": 430, "to": 438, "label": "PARALLEL" }, { "from": 432, "to": 436, "label": "SUCCESS" }, { "from": 437, "to": 441, "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": 437, "to": 442, "label": "EVAL-BACKTRACK" }, { "from": 438, "to": 445, "label": "PARALLEL" }, { "from": 438, "to": 446, "label": "PARALLEL" }, { "from": 441, "to": 408, "label": "INSTANCE with matching:\nT47 -> T72\nT50 -> T74\nT48 -> T73\nT49 -> T75" }, { "from": 445, "to": 514, "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": 445, "to": 515, "label": "EVAL-BACKTRACK" }, { "from": 446, "to": 516, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 514, "to": 408, "label": "INSTANCE with matching:\nT47 -> T92\nT50 -> T95\nT48 -> T93\nT49 -> T94" }, { "from": 517, "to": 519, "label": "SPLIT 1" }, { "from": 517, "to": 520, "label": "SPLIT 2\nreplacements:X100 -> T103" }, { "from": 519, "to": 521, "label": "CASE" }, { "from": 520, "to": 1, "label": "INSTANCE with matching:\nT1 -> d(e(T103))\nT2 -> T101" }, { "from": 521, "to": 522, "label": "PARALLEL" }, { "from": 521, "to": 523, "label": "PARALLEL" }, { "from": 522, "to": 524, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT102 -> e(t),\nX100 -> const(1)" }, { "from": 522, "to": 525, "label": "EVAL-BACKTRACK" }, { "from": 523, "to": 527, "label": "PARALLEL" }, { "from": 523, "to": 528, "label": "PARALLEL" }, { "from": 524, "to": 526, "label": "SUCCESS" }, { "from": 527, "to": 529, "label": "EVAL with clause\np(d(e(const(X105))), const(0)).\nand substitutionX105 -> T108,\nT102 -> e(const(T108)),\nX100 -> const(0)" }, { "from": 527, "to": 530, "label": "EVAL-BACKTRACK" }, { "from": 528, "to": 532, "label": "PARALLEL" }, { "from": 528, "to": 533, "label": "PARALLEL" }, { "from": 529, "to": 531, "label": "SUCCESS" }, { "from": 532, "to": 534, "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": 532, "to": 535, "label": "EVAL-BACKTRACK" }, { "from": 533, "to": 538, "label": "PARALLEL" }, { "from": 533, "to": 539, "label": "PARALLEL" }, { "from": 534, "to": 536, "label": "SPLIT 1" }, { "from": 534, "to": 537, "label": "SPLIT 2\nreplacements:X134 -> T121,\nT120 -> T122" }, { "from": 536, "to": 519, "label": "INSTANCE with matching:\nT102 -> e(T119)\nX100 -> X134" }, { "from": 537, "to": 519, "label": "INSTANCE with matching:\nT102 -> e(T122)\nX100 -> X135" }, { "from": 538, "to": 540, "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": 538, "to": 541, "label": "EVAL-BACKTRACK" }, { "from": 539, "to": 542, "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": 539, "to": 543, "label": "EVAL-BACKTRACK" }, { "from": 540, "to": 534, "label": "INSTANCE with matching:\nT119 -> T133\nX134 -> X165\nT120 -> T134\nX135 -> X164" }, { "from": 542, "to": 544, "label": "SPLIT 1" }, { "from": 542, "to": 545, "label": "SPLIT 2\nreplacements:X175 -> T139" }, { "from": 544, "to": 519, "label": "INSTANCE with matching:\nT102 -> T138\nX100 -> X175" }, { "from": 545, "to": 519, "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: f1_in(const(1)) -> f1_out1 f1_in(const(0)) -> f1_out1 f1_in(+(T26, T27)) -> U1(f378_in(T26, T27), +(T26, T27)) U1(f378_out1, +(T26, T27)) -> f1_out1 f1_in(+(*(T47, T49), *(T48, T50))) -> U2(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f408_out1, +(*(T47, T49), *(T48, T50))) -> f1_out1 f1_in(T101) -> U3(f517_in(T101), T101) U3(f517_out1, T101) -> f1_out1 f414_in(t, const(1)) -> f414_out1 f414_in(const(T55), const(0)) -> f414_out1 f414_in(+(T72, T73), +(T74, T75)) -> U4(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f408_out1, +(T72, T73), +(T74, T75)) -> f414_out1 f414_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f408_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f414_out1 f519_in -> f519_out1 f519_in -> U6(f534_in) U6(f534_out1) -> f519_out1 f519_in -> U7(f534_in) U7(f534_out1) -> f519_out1 f519_in -> U8(f542_in) U8(f542_out1) -> f519_out1 f378_in(T26, T27) -> U9(f1_in(T26), T26, T27) U9(f1_out1, T26, T27) -> U10(f1_in(T27), T26, T27) U10(f1_out1, T26, T27) -> f378_out1 f408_in(T47, T50, T48, T49) -> U11(f414_in(T47, T50), T47, T50, T48, T49) U11(f414_out1, T47, T50, T48, T49) -> U12(f414_in(T48, T49), T47, T50, T48, T49) U12(f414_out1, T47, T50, T48, T49) -> f408_out1 f517_in(T101) -> U13(f519_in, T101) U13(f519_out1, T101) -> U14(f1_in(T101), T101) U14(f1_out1, T101) -> f517_out1 f534_in -> U15(f519_in) U15(f519_out1) -> U16(f519_in) U16(f519_out1) -> f534_out1 f542_in -> U17(f519_in) U17(f519_out1) -> U18(f519_in) U18(f519_out1) -> f542_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: F1_IN(+(T26, T27)) -> U1^1(f378_in(T26, T27), +(T26, T27)) F1_IN(+(T26, T27)) -> F378_IN(T26, T27) F1_IN(+(*(T47, T49), *(T48, T50))) -> U2^1(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) F1_IN(+(*(T47, T49), *(T48, T50))) -> F408_IN(T47, T50, T48, T49) F1_IN(T101) -> U3^1(f517_in(T101), T101) F1_IN(T101) -> F517_IN(T101) F414_IN(+(T72, T73), +(T74, T75)) -> U4^1(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) F414_IN(+(T72, T73), +(T74, T75)) -> F408_IN(T72, T74, T73, T75) F414_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5^1(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) F414_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F408_IN(T92, T95, T93, T94) F519_IN -> U6^1(f534_in) F519_IN -> F534_IN F519_IN -> U7^1(f534_in) F519_IN -> U8^1(f542_in) F519_IN -> F542_IN F378_IN(T26, T27) -> U9^1(f1_in(T26), T26, T27) F378_IN(T26, T27) -> F1_IN(T26) U9^1(f1_out1, T26, T27) -> U10^1(f1_in(T27), T26, T27) U9^1(f1_out1, T26, T27) -> F1_IN(T27) F408_IN(T47, T50, T48, T49) -> U11^1(f414_in(T47, T50), T47, T50, T48, T49) F408_IN(T47, T50, T48, T49) -> F414_IN(T47, T50) U11^1(f414_out1, T47, T50, T48, T49) -> U12^1(f414_in(T48, T49), T47, T50, T48, T49) U11^1(f414_out1, T47, T50, T48, T49) -> F414_IN(T48, T49) F517_IN(T101) -> U13^1(f519_in, T101) F517_IN(T101) -> F519_IN U13^1(f519_out1, T101) -> U14^1(f1_in(T101), T101) U13^1(f519_out1, T101) -> F1_IN(T101) F534_IN -> U15^1(f519_in) F534_IN -> F519_IN U15^1(f519_out1) -> U16^1(f519_in) U15^1(f519_out1) -> F519_IN F542_IN -> U17^1(f519_in) F542_IN -> F519_IN U17^1(f519_out1) -> U18^1(f519_in) U17^1(f519_out1) -> F519_IN The TRS R consists of the following rules: f1_in(const(1)) -> f1_out1 f1_in(const(0)) -> f1_out1 f1_in(+(T26, T27)) -> U1(f378_in(T26, T27), +(T26, T27)) U1(f378_out1, +(T26, T27)) -> f1_out1 f1_in(+(*(T47, T49), *(T48, T50))) -> U2(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f408_out1, +(*(T47, T49), *(T48, T50))) -> f1_out1 f1_in(T101) -> U3(f517_in(T101), T101) U3(f517_out1, T101) -> f1_out1 f414_in(t, const(1)) -> f414_out1 f414_in(const(T55), const(0)) -> f414_out1 f414_in(+(T72, T73), +(T74, T75)) -> U4(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f408_out1, +(T72, T73), +(T74, T75)) -> f414_out1 f414_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f408_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f414_out1 f519_in -> f519_out1 f519_in -> U6(f534_in) U6(f534_out1) -> f519_out1 f519_in -> U7(f534_in) U7(f534_out1) -> f519_out1 f519_in -> U8(f542_in) U8(f542_out1) -> f519_out1 f378_in(T26, T27) -> U9(f1_in(T26), T26, T27) U9(f1_out1, T26, T27) -> U10(f1_in(T27), T26, T27) U10(f1_out1, T26, T27) -> f378_out1 f408_in(T47, T50, T48, T49) -> U11(f414_in(T47, T50), T47, T50, T48, T49) U11(f414_out1, T47, T50, T48, T49) -> U12(f414_in(T48, T49), T47, T50, T48, T49) U12(f414_out1, T47, T50, T48, T49) -> f408_out1 f517_in(T101) -> U13(f519_in, T101) U13(f519_out1, T101) -> U14(f1_in(T101), T101) U14(f1_out1, T101) -> f517_out1 f534_in -> U15(f519_in) U15(f519_out1) -> U16(f519_in) U16(f519_out1) -> f534_out1 f542_in -> U17(f519_in) U17(f519_out1) -> U18(f519_in) U18(f519_out1) -> f542_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: F519_IN -> F534_IN F534_IN -> U15^1(f519_in) U15^1(f519_out1) -> F519_IN F519_IN -> F542_IN F542_IN -> U17^1(f519_in) U17^1(f519_out1) -> F519_IN F542_IN -> F519_IN F534_IN -> F519_IN The TRS R consists of the following rules: f1_in(const(1)) -> f1_out1 f1_in(const(0)) -> f1_out1 f1_in(+(T26, T27)) -> U1(f378_in(T26, T27), +(T26, T27)) U1(f378_out1, +(T26, T27)) -> f1_out1 f1_in(+(*(T47, T49), *(T48, T50))) -> U2(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f408_out1, +(*(T47, T49), *(T48, T50))) -> f1_out1 f1_in(T101) -> U3(f517_in(T101), T101) U3(f517_out1, T101) -> f1_out1 f414_in(t, const(1)) -> f414_out1 f414_in(const(T55), const(0)) -> f414_out1 f414_in(+(T72, T73), +(T74, T75)) -> U4(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f408_out1, +(T72, T73), +(T74, T75)) -> f414_out1 f414_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f408_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f414_out1 f519_in -> f519_out1 f519_in -> U6(f534_in) U6(f534_out1) -> f519_out1 f519_in -> U7(f534_in) U7(f534_out1) -> f519_out1 f519_in -> U8(f542_in) U8(f542_out1) -> f519_out1 f378_in(T26, T27) -> U9(f1_in(T26), T26, T27) U9(f1_out1, T26, T27) -> U10(f1_in(T27), T26, T27) U10(f1_out1, T26, T27) -> f378_out1 f408_in(T47, T50, T48, T49) -> U11(f414_in(T47, T50), T47, T50, T48, T49) U11(f414_out1, T47, T50, T48, T49) -> U12(f414_in(T48, T49), T47, T50, T48, T49) U12(f414_out1, T47, T50, T48, T49) -> f408_out1 f517_in(T101) -> U13(f519_in, T101) U13(f519_out1, T101) -> U14(f1_in(T101), T101) U14(f1_out1, T101) -> f517_out1 f534_in -> U15(f519_in) U15(f519_out1) -> U16(f519_in) U16(f519_out1) -> f534_out1 f542_in -> U17(f519_in) U17(f519_out1) -> U18(f519_in) U18(f519_out1) -> f542_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 F534_IN[ ]^n[ ] -> F534_IN[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: F534_IN[ ]^n[ ] -> F534_IN[ ]^n[ ] by Narrowing at position: [] F534_IN[ ]^n[ ] -> F519_IN[ ]^n[ ] by Rule from TRS P F519_IN[ ]^n[ ] -> F534_IN[ ]^n[ ] by Rule from TRS P ---------------------------------------- (69) NO ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: F414_IN(+(T72, T73), +(T74, T75)) -> F408_IN(T72, T74, T73, T75) F408_IN(T47, T50, T48, T49) -> U11^1(f414_in(T47, T50), T47, T50, T48, T49) U11^1(f414_out1, T47, T50, T48, T49) -> F414_IN(T48, T49) F414_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F408_IN(T92, T95, T93, T94) F408_IN(T47, T50, T48, T49) -> F414_IN(T47, T50) The TRS R consists of the following rules: f1_in(const(1)) -> f1_out1 f1_in(const(0)) -> f1_out1 f1_in(+(T26, T27)) -> U1(f378_in(T26, T27), +(T26, T27)) U1(f378_out1, +(T26, T27)) -> f1_out1 f1_in(+(*(T47, T49), *(T48, T50))) -> U2(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f408_out1, +(*(T47, T49), *(T48, T50))) -> f1_out1 f1_in(T101) -> U3(f517_in(T101), T101) U3(f517_out1, T101) -> f1_out1 f414_in(t, const(1)) -> f414_out1 f414_in(const(T55), const(0)) -> f414_out1 f414_in(+(T72, T73), +(T74, T75)) -> U4(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f408_out1, +(T72, T73), +(T74, T75)) -> f414_out1 f414_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f408_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f414_out1 f519_in -> f519_out1 f519_in -> U6(f534_in) U6(f534_out1) -> f519_out1 f519_in -> U7(f534_in) U7(f534_out1) -> f519_out1 f519_in -> U8(f542_in) U8(f542_out1) -> f519_out1 f378_in(T26, T27) -> U9(f1_in(T26), T26, T27) U9(f1_out1, T26, T27) -> U10(f1_in(T27), T26, T27) U10(f1_out1, T26, T27) -> f378_out1 f408_in(T47, T50, T48, T49) -> U11(f414_in(T47, T50), T47, T50, T48, T49) U11(f414_out1, T47, T50, T48, T49) -> U12(f414_in(T48, T49), T47, T50, T48, T49) U12(f414_out1, T47, T50, T48, T49) -> f408_out1 f517_in(T101) -> U13(f519_in, T101) U13(f519_out1, T101) -> U14(f1_in(T101), T101) U14(f1_out1, T101) -> f517_out1 f534_in -> U15(f519_in) U15(f519_out1) -> U16(f519_in) U16(f519_out1) -> f534_out1 f542_in -> U17(f519_in) U17(f519_out1) -> U18(f519_in) U18(f519_out1) -> f542_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: *F408_IN(T47, T50, T48, T49) -> F414_IN(T47, T50) The graph contains the following edges 1 >= 1, 2 >= 2 *F408_IN(T47, T50, T48, T49) -> U11^1(f414_in(T47, T50), T47, T50, T48, T49) The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5 *U11^1(f414_out1, T47, T50, T48, T49) -> F414_IN(T48, T49) The graph contains the following edges 4 >= 1, 5 >= 2 *F414_IN(+(T72, T73), +(T74, T75)) -> F408_IN(T72, T74, T73, T75) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *F414_IN(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> F408_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: F1_IN(+(T26, T27)) -> F378_IN(T26, T27) F378_IN(T26, T27) -> U9^1(f1_in(T26), T26, T27) U9^1(f1_out1, T26, T27) -> F1_IN(T27) F1_IN(T101) -> F517_IN(T101) F517_IN(T101) -> U13^1(f519_in, T101) U13^1(f519_out1, T101) -> F1_IN(T101) F378_IN(T26, T27) -> F1_IN(T26) The TRS R consists of the following rules: f1_in(const(1)) -> f1_out1 f1_in(const(0)) -> f1_out1 f1_in(+(T26, T27)) -> U1(f378_in(T26, T27), +(T26, T27)) U1(f378_out1, +(T26, T27)) -> f1_out1 f1_in(+(*(T47, T49), *(T48, T50))) -> U2(f408_in(T47, T50, T48, T49), +(*(T47, T49), *(T48, T50))) U2(f408_out1, +(*(T47, T49), *(T48, T50))) -> f1_out1 f1_in(T101) -> U3(f517_in(T101), T101) U3(f517_out1, T101) -> f1_out1 f414_in(t, const(1)) -> f414_out1 f414_in(const(T55), const(0)) -> f414_out1 f414_in(+(T72, T73), +(T74, T75)) -> U4(f408_in(T72, T74, T73, T75), +(T72, T73), +(T74, T75)) U4(f408_out1, +(T72, T73), +(T74, T75)) -> f414_out1 f414_in(*(T92, T93), +(*(T92, T94), *(T93, T95))) -> U5(f408_in(T92, T95, T93, T94), *(T92, T93), +(*(T92, T94), *(T93, T95))) U5(f408_out1, *(T92, T93), +(*(T92, T94), *(T93, T95))) -> f414_out1 f519_in -> f519_out1 f519_in -> U6(f534_in) U6(f534_out1) -> f519_out1 f519_in -> U7(f534_in) U7(f534_out1) -> f519_out1 f519_in -> U8(f542_in) U8(f542_out1) -> f519_out1 f378_in(T26, T27) -> U9(f1_in(T26), T26, T27) U9(f1_out1, T26, T27) -> U10(f1_in(T27), T26, T27) U10(f1_out1, T26, T27) -> f378_out1 f408_in(T47, T50, T48, T49) -> U11(f414_in(T47, T50), T47, T50, T48, T49) U11(f414_out1, T47, T50, T48, T49) -> U12(f414_in(T48, T49), T47, T50, T48, T49) U12(f414_out1, T47, T50, T48, T49) -> f408_out1 f517_in(T101) -> U13(f519_in, T101) U13(f519_out1, T101) -> U14(f1_in(T101), T101) U14(f1_out1, T101) -> f517_out1 f534_in -> U15(f519_in) U15(f519_out1) -> U16(f519_in) U16(f519_out1) -> f534_out1 f542_in -> U17(f519_in) U17(f519_out1) -> U18(f519_in) U18(f519_out1) -> f542_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(f519_out1, x0)[ ]^n[ ] -> U13^1(f519_out1, x0)[ ]^n[ ] This rule is correct for the QDP as the following derivation shows: U13^1(f519_out1, x0)[ ]^n[ ] -> U13^1(f519_out1, x0)[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation U13^1(f519_out1, T101)[ ]^n[ ] -> F1_IN(T101)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation F1_IN(x0)[ ]^n[ ] -> U13^1(f519_out1, x0)[ ]^n[ ] by Narrowing at position: [0] F1_IN(x0)[ ]^n[ ] -> U13^1(f519_in, x0)[ ]^n[ ] by Narrowing at position: [] intermediate steps: Instantiation F1_IN(T101)[ ]^n[ ] -> F517_IN(T101)[ ]^n[ ] by Rule from TRS P intermediate steps: Instantiation - Instantiation F517_IN(T101)[ ]^n[ ] -> U13^1(f519_in, T101)[ ]^n[ ] by Rule from TRS P f519_in[ ]^n[ ] -> f519_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": 3, "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", "431": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "630": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "433": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": 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}], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X100"], "exprvars": [] } }, "627": { "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": [] } }, "628": { "goal": [{ "clause": -1, "scope": -1, "term": "(true)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "629": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } } }, "edges": [ { "from": 3, "to": 4, "label": "CASE" }, { "from": 4, "to": 6, "label": "PARALLEL" }, { "from": 4, "to": 7, "label": "PARALLEL" }, { "from": 6, "to": 8, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 6, "to": 9, "label": "EVAL-BACKTRACK" }, { "from": 7, "to": 14, "label": "PARALLEL" }, { "from": 7, "to": 15, "label": "PARALLEL" }, { "from": 8, "to": 13, "label": "SUCCESS" }, { "from": 14, "to": 16, "label": "EVAL with clause\np(d(e(const(X5))), const(0)).\nand substitutionX5 -> T7,\nT1 -> d(e(const(T7))),\nT2 -> const(0)" }, { "from": 14, "to": 17, "label": "EVAL-BACKTRACK" }, { "from": 15, "to": 22, "label": "PARALLEL" }, { "from": 15, "to": 23, "label": "PARALLEL" }, { "from": 16, "to": 18, "label": "SUCCESS" }, { "from": 22, "to": 320, "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": 22, "to": 322, "label": "EVAL-BACKTRACK" }, { "from": 23, "to": 403, "label": "PARALLEL" }, { "from": 23, "to": 405, "label": "PARALLEL" }, { "from": 320, "to": 339, "label": "SPLIT 1" }, { "from": 320, "to": 340, "label": "SPLIT 2\nnew knowledge:\nT26 is ground\nreplacements:T29 -> T30" }, { "from": 339, "to": 3, "label": "INSTANCE with matching:\nT1 -> d(e(T28))\nT2 -> T26" }, { "from": 340, "to": 3, "label": "INSTANCE with matching:\nT1 -> d(e(T30))\nT2 -> T27" }, { "from": 403, "to": 407, "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": 403, "to": 409, "label": "EVAL-BACKTRACK" }, { "from": 405, "to": 616, "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": 405, "to": 617, "label": "EVAL-BACKTRACK" }, { "from": 407, "to": 411, "label": "SPLIT 1" }, { "from": 407, "to": 413, "label": "SPLIT 2\nnew knowledge:\nT47 is ground\nT50 is ground" }, { "from": 411, "to": 413, "label": "INSTANCE with matching:\nT48 -> T47\nT49 -> T50" }, { "from": 413, "to": 416, "label": "CASE" }, { "from": 416, "to": 418, "label": "PARALLEL" }, { "from": 416, "to": 419, "label": "PARALLEL" }, { "from": 418, "to": 421, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT48 -> t,\nT49 -> const(1)" }, { "from": 418, "to": 423, "label": "EVAL-BACKTRACK" }, { "from": 419, "to": 427, "label": "PARALLEL" }, { "from": 419, "to": 428, "label": "PARALLEL" }, { "from": 421, "to": 425, "label": "SUCCESS" }, { "from": 427, "to": 431, "label": "EVAL with clause\np(d(e(const(X50))), const(0)).\nand substitutionX50 -> T55,\nT48 -> const(T55),\nT49 -> const(0)" }, { "from": 427, "to": 433, "label": "EVAL-BACKTRACK" }, { "from": 428, "to": 439, "label": "PARALLEL" }, { "from": 428, "to": 440, "label": "PARALLEL" }, { "from": 431, "to": 435, "label": "SUCCESS" }, { "from": 439, "to": 443, "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": 439, "to": 444, "label": "EVAL-BACKTRACK" }, { "from": 440, "to": 449, "label": "PARALLEL" }, { "from": 440, "to": 450, "label": "PARALLEL" }, { "from": 443, "to": 407, "label": "INSTANCE with matching:\nT47 -> T72\nT50 -> T74\nT48 -> T73\nT49 -> T75" }, { "from": 449, "to": 613, "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": 449, "to": 614, "label": "EVAL-BACKTRACK" }, { "from": 450, "to": 615, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 613, "to": 407, "label": "INSTANCE with matching:\nT47 -> T92\nT50 -> T95\nT48 -> T93\nT49 -> T94" }, { "from": 616, "to": 618, "label": "SPLIT 1" }, { "from": 616, "to": 619, "label": "SPLIT 2\nreplacements:X100 -> T103" }, { "from": 618, "to": 620, "label": "CASE" }, { "from": 619, "to": 3, "label": "INSTANCE with matching:\nT1 -> d(e(T103))\nT2 -> T101" }, { "from": 620, "to": 621, "label": "PARALLEL" }, { "from": 620, "to": 622, "label": "PARALLEL" }, { "from": 621, "to": 623, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT102 -> e(t),\nX100 -> const(1)" }, { "from": 621, "to": 624, "label": "EVAL-BACKTRACK" }, { "from": 622, "to": 626, "label": "PARALLEL" }, { "from": 622, "to": 627, "label": "PARALLEL" }, { "from": 623, "to": 625, "label": "SUCCESS" }, { "from": 626, "to": 628, "label": "EVAL with clause\np(d(e(const(X105))), const(0)).\nand substitutionX105 -> T108,\nT102 -> e(const(T108)),\nX100 -> const(0)" }, { "from": 626, "to": 629, "label": "EVAL-BACKTRACK" }, { "from": 627, "to": 631, "label": "PARALLEL" }, { "from": 627, "to": 632, "label": "PARALLEL" }, { "from": 628, "to": 630, "label": "SUCCESS" }, { "from": 631, "to": 633, "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": 631, "to": 634, "label": "EVAL-BACKTRACK" }, { "from": 632, "to": 655, "label": "PARALLEL" }, { "from": 632, "to": 656, "label": "PARALLEL" }, { "from": 633, "to": 653, "label": "SPLIT 1" }, { "from": 633, "to": 654, "label": "SPLIT 2\nreplacements:X134 -> T121,\nT120 -> T122" }, { "from": 653, "to": 618, "label": "INSTANCE with matching:\nT102 -> e(T119)\nX100 -> X134" }, { "from": 654, "to": 618, "label": "INSTANCE with matching:\nT102 -> e(T122)\nX100 -> X135" }, { "from": 655, "to": 657, "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": 655, "to": 658, "label": "EVAL-BACKTRACK" }, { "from": 656, "to": 659, "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": 656, "to": 660, "label": "EVAL-BACKTRACK" }, { "from": 657, "to": 633, "label": "INSTANCE with matching:\nT119 -> T133\nX134 -> X165\nT120 -> T134\nX135 -> X164" }, { "from": 659, "to": 661, "label": "SPLIT 1" }, { "from": 659, "to": 662, "label": "SPLIT 2\nreplacements:X175 -> T139" }, { "from": 661, "to": 618, "label": "INSTANCE with matching:\nT102 -> T138\nX100 -> X175" }, { "from": 662, "to": 618, "label": "INSTANCE with matching:\nT102 -> e(T139)\nX100 -> X176" } ], "type": "Graph" } } ---------------------------------------- (77) Complex Obligation (AND) ---------------------------------------- (78) Obligation: Rules: f661_out -> f662_in :|: TRUE f662_out -> f659_out :|: TRUE f659_in -> f661_in :|: TRUE f631_in -> f633_in :|: TRUE f633_out -> f631_out :|: TRUE f634_out -> f631_out :|: TRUE f631_in -> f634_in :|: TRUE f656_in -> f659_in :|: TRUE f659_out -> f656_out :|: TRUE f660_out -> f656_out :|: TRUE f656_in -> f660_in :|: TRUE f618_in -> f620_in :|: TRUE f620_out -> f618_out :|: TRUE f621_out -> f620_out :|: TRUE f620_in -> f621_in :|: TRUE f620_in -> f622_in :|: TRUE f622_out -> f620_out :|: TRUE f655_in -> f658_in :|: TRUE f658_out -> f655_out :|: TRUE f655_in -> f657_in :|: TRUE f657_out -> f655_out :|: TRUE f633_in -> f653_in :|: TRUE f653_out -> f654_in :|: TRUE f654_out -> f633_out :|: TRUE f654_in -> f618_in :|: TRUE f618_out -> f654_out :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f655_out -> f632_out :|: TRUE f656_out -> f632_out :|: TRUE f633_out -> f657_out :|: TRUE f657_in -> f633_in :|: TRUE f662_in -> f618_in :|: TRUE f618_out -> f662_out :|: TRUE f622_in -> f627_in :|: TRUE f622_in -> f626_in :|: TRUE f627_out -> f622_out :|: TRUE f626_out -> f622_out :|: TRUE f627_in -> f631_in :|: TRUE f632_out -> f627_out :|: TRUE f631_out -> f627_out :|: TRUE f627_in -> f632_in :|: TRUE f653_in -> f618_in :|: TRUE f618_out -> f653_out :|: TRUE f618_out -> f661_out :|: TRUE f661_in -> f618_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x) -> f4_in(x) :|: TRUE f4_in(x1) -> f6_in(x1) :|: TRUE f7_out(x2) -> f4_out(x2) :|: TRUE f6_out(x3) -> f4_out(x3) :|: TRUE f4_in(x4) -> f7_in(x4) :|: TRUE f7_in(x5) -> f14_in(x5) :|: TRUE f14_out(x6) -> f7_out(x6) :|: TRUE f7_in(x7) -> f15_in(x7) :|: TRUE f15_out(x8) -> f7_out(x8) :|: TRUE f15_in(x9) -> f22_in(x9) :|: TRUE f15_in(x10) -> f23_in(x10) :|: TRUE f22_out(x11) -> f15_out(x11) :|: TRUE f23_out(x12) -> f15_out(x12) :|: TRUE f403_out(x13) -> f23_out(x13) :|: TRUE f23_in(x14) -> f405_in(x14) :|: TRUE f405_out(x15) -> f23_out(x15) :|: TRUE f23_in(x16) -> f403_in(x16) :|: TRUE f405_in(T101) -> f616_in(T101) :|: TRUE f616_out(x17) -> f405_out(x17) :|: TRUE f405_in(x18) -> f617_in :|: TRUE f617_out -> f405_out(x19) :|: TRUE f616_in(x20) -> f618_in :|: TRUE f619_out(x21) -> f616_out(x21) :|: TRUE f618_out -> f619_in(x22) :|: TRUE Start term: f3_in(T2) ---------------------------------------- (79) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f659_in -> f661_in :|: TRUE f631_in -> f633_in :|: TRUE f656_in -> f659_in :|: TRUE f618_in -> f620_in :|: TRUE f620_in -> f622_in :|: TRUE f655_in -> f657_in :|: TRUE f633_in -> f653_in :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f657_in -> f633_in :|: TRUE f622_in -> f627_in :|: TRUE f627_in -> f631_in :|: TRUE f627_in -> f632_in :|: TRUE f653_in -> f618_in :|: TRUE f661_in -> f618_in :|: TRUE ---------------------------------------- (80) Obligation: Rules: f659_in -> f661_in :|: TRUE f631_in -> f633_in :|: TRUE f656_in -> f659_in :|: TRUE f618_in -> f620_in :|: TRUE f620_in -> f622_in :|: TRUE f655_in -> f657_in :|: TRUE f633_in -> f653_in :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f657_in -> f633_in :|: TRUE f622_in -> f627_in :|: TRUE f627_in -> f631_in :|: TRUE f627_in -> f632_in :|: TRUE f653_in -> f618_in :|: TRUE f661_in -> f618_in :|: TRUE ---------------------------------------- (81) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (82) Obligation: Rules: f618_in -> f618_in :|: TRUE ---------------------------------------- (83) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (84) Obligation: Rules: f618_in -> f618_in :|: TRUE ---------------------------------------- (85) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f618_in -> f618_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (86) Obligation: Termination digraph: Nodes: (1) f618_in -> f618_in :|: TRUE Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (87) FilterProof (EQUIVALENT) Used the following sort dictionary for filtering: f618_in() Replaced non-predefined constructor symbols by 0. ---------------------------------------- (88) Obligation: Rules: f618_in -> f618_in :|: TRUE ---------------------------------------- (89) IntTRSPeriodicNontermProof (COMPLETE) Normalized system to the following form: f(pc) -> f(1) :|: pc = 1 && TRUE Witness term starting non-terminating reduction: f(1) ---------------------------------------- (90) NO ---------------------------------------- (91) Obligation: Rules: f416_out(T48, T49) -> f413_out(T48, T49) :|: TRUE f413_in(x, x1) -> f416_in(x, x1) :|: TRUE f419_in(x2, x3) -> f427_in(x2, x3) :|: TRUE f427_out(x4, x5) -> f419_out(x4, x5) :|: TRUE f419_in(x6, x7) -> f428_in(x6, x7) :|: TRUE f428_out(x8, x9) -> f419_out(x8, x9) :|: TRUE f440_out(x10, x11) -> f428_out(x10, x11) :|: TRUE f428_in(x12, x13) -> f440_in(x12, x13) :|: TRUE f439_out(x14, x15) -> f428_out(x14, x15) :|: TRUE f428_in(x16, x17) -> f439_in(x16, x17) :|: TRUE f450_out(x18, x19) -> f440_out(x18, x19) :|: TRUE f440_in(x20, x21) -> f449_in(x20, x21) :|: TRUE f449_out(x22, x23) -> f440_out(x22, x23) :|: TRUE f440_in(x24, x25) -> f450_in(x24, x25) :|: TRUE f407_out(T72, T74, T73, T75) -> f443_out(T72, T74, T73, T75) :|: TRUE f443_in(x26, x27, x28, x29) -> f407_in(x26, x27, x28, x29) :|: TRUE f413_out(x30, x31) -> f407_out(x32, x33, x30, x31) :|: TRUE f411_out(x34, x35) -> f413_in(x36, x37) :|: TRUE f407_in(x38, x39, x40, x41) -> f411_in(x38, x39) :|: TRUE f416_in(x42, x43) -> f418_in(x42, x43) :|: TRUE f419_out(x44, x45) -> f416_out(x44, x45) :|: TRUE f418_out(x46, x47) -> f416_out(x46, x47) :|: TRUE f416_in(x48, x49) -> f419_in(x48, x49) :|: TRUE f443_out(x50, x51, x52, x53) -> f439_out(x50 + x52, x51 + x53) :|: TRUE f439_in(x54, x55) -> f444_in :|: TRUE f444_out -> f439_out(x56, x57) :|: TRUE f439_in(x58 + x59, x60 + x61) -> f443_in(x58, x60, x59, x61) :|: TRUE f613_in(T92, T95, T93, T94) -> f407_in(T92, T95, T93, T94) :|: TRUE f407_out(x62, x63, x64, x65) -> f613_out(x62, x63, x64, x65) :|: TRUE f413_out(T47, T50) -> f411_out(T47, T50) :|: TRUE f411_in(x66, x67) -> f413_in(x66, x67) :|: TRUE f449_in(x68 * x69, x68 * x70 + x69 * x71) -> f613_in(x68, x71, x69, x70) :|: TRUE f614_out -> f449_out(x72, x73) :|: TRUE f613_out(x74, x75, x76, x77) -> f449_out(x74 * x76, x74 * x77 + x76 * x75) :|: TRUE f449_in(x78, x79) -> f614_in :|: TRUE f4_out(T2) -> f3_out(T2) :|: TRUE f3_in(x80) -> f4_in(x80) :|: TRUE f4_in(x81) -> f6_in(x81) :|: TRUE f7_out(x82) -> f4_out(x82) :|: TRUE f6_out(x83) -> f4_out(x83) :|: TRUE f4_in(x84) -> f7_in(x84) :|: TRUE f7_in(x85) -> f14_in(x85) :|: TRUE f14_out(x86) -> f7_out(x86) :|: TRUE f7_in(x87) -> f15_in(x87) :|: TRUE f15_out(x88) -> f7_out(x88) :|: TRUE f15_in(x89) -> f22_in(x89) :|: TRUE f15_in(x90) -> f23_in(x90) :|: TRUE f22_out(x91) -> f15_out(x91) :|: TRUE f23_out(x92) -> f15_out(x92) :|: TRUE f403_out(x93) -> f23_out(x93) :|: TRUE f23_in(x94) -> f405_in(x94) :|: TRUE f405_out(x95) -> f23_out(x95) :|: TRUE f23_in(x96) -> f403_in(x96) :|: TRUE f409_out -> f403_out(x97) :|: TRUE f403_in(x98 * x99 + x100 * x101) -> f407_in(x98, x101, x100, x99) :|: TRUE f407_out(x102, x103, x104, x105) -> f403_out(x102 * x105 + x104 * x103) :|: TRUE f403_in(x106) -> f409_in :|: TRUE Start term: f3_in(T2) ---------------------------------------- (92) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f413_in(x, x1) -> f416_in(x, x1) :|: TRUE f419_in(x6, x7) -> f428_in(x6, x7) :|: TRUE f428_in(x12, x13) -> f440_in(x12, x13) :|: TRUE f428_in(x16, x17) -> f439_in(x16, x17) :|: TRUE f440_in(x20, x21) -> f449_in(x20, x21) :|: TRUE f443_in(x26, x27, x28, x29) -> f407_in(x26, x27, x28, x29) :|: TRUE f407_in(x38, x39, x40, x41) -> f411_in(x38, x39) :|: TRUE f416_in(x48, x49) -> f419_in(x48, x49) :|: TRUE f439_in(x58 + x59, x60 + x61) -> f443_in(x58, x60, x59, x61) :|: TRUE f613_in(T92, T95, T93, T94) -> f407_in(T92, T95, T93, T94) :|: TRUE f411_in(x66, x67) -> f413_in(x66, x67) :|: TRUE f449_in(x68 * x69, x68 * x70 + x69 * x71) -> f613_in(x68, x71, x69, x70) :|: TRUE ---------------------------------------- (93) Obligation: Rules: f413_in(x, x1) -> f416_in(x, x1) :|: TRUE f419_in(x6, x7) -> f428_in(x6, x7) :|: TRUE f428_in(x12, x13) -> f440_in(x12, x13) :|: TRUE f428_in(x16, x17) -> f439_in(x16, x17) :|: TRUE f440_in(x20, x21) -> f449_in(x20, x21) :|: TRUE f443_in(x26, x27, x28, x29) -> f407_in(x26, x27, x28, x29) :|: TRUE f407_in(x38, x39, x40, x41) -> f411_in(x38, x39) :|: TRUE f416_in(x48, x49) -> f419_in(x48, x49) :|: TRUE f439_in(x58 + x59, x60 + x61) -> f443_in(x58, x60, x59, x61) :|: TRUE f613_in(T92, T95, T93, T94) -> f407_in(T92, T95, T93, T94) :|: TRUE f411_in(x66, x67) -> f413_in(x66, x67) :|: TRUE f449_in(x68 * x69, x68 * x70 + x69 * x71) -> f613_in(x68, x71, x69, x70) :|: TRUE ---------------------------------------- (94) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (95) Obligation: Rules: f413_in(sum~x58:0~x59:0, sum~x60:0~x61:0) -> f413_in(x58:0, x60:0) :|: TRUE && sum~x58:0~x59:0 = x58:0 + x59:0 && sum~x60:0~x61:0 = x60:0 + x61:0 f413_in(times~x68:0~x69:0, sum~times~x68:0~x70:0~times~x69:0~x71:0) -> f413_in(x68:0, x71:0) :|: TRUE && times~x68:0~x69:0 = x68:0 * x69:0 && sum~times~x68:0~x70:0~times~x69:0~x71:0 = x68:0 * x70:0 + x69:0 * x71:0 ---------------------------------------- (96) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (97) Obligation: Rules: f413_in(sum~x58:0~x59:0, sum~x60:0~x61:0) -> f413_in(x58:0, x60:0) :|: TRUE && sum~x58:0~x59:0 = x58:0 + x59:0 && sum~x60:0~x61:0 = x60:0 + x61:0 f413_in(times~x68:0~x69:0, sum~times~x68:0~x70:0~times~x69:0~x71:0) -> f413_in(x68:0, x71:0) :|: TRUE && times~x68:0~x69:0 = x68:0 * x69:0 && sum~times~x68:0~x70:0~times~x69:0~x71:0 = x68:0 * x70:0 + x69:0 * x71:0 ---------------------------------------- (98) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f413_in(sum~x58:0~x59:0, sum~x60:0~x61:0) -> f413_in(x58:0, x60:0) :|: TRUE && sum~x58:0~x59:0 = x58:0 + x59:0 && sum~x60:0~x61:0 = x60:0 + x61:0 (2) f413_in(times~x68:0~x69:0, sum~times~x68:0~x70:0~times~x69:0~x71:0) -> f413_in(x68:0, x71:0) :|: TRUE && times~x68:0~x69:0 = x68:0 * x69:0 && sum~times~x68:0~x70:0~times~x69:0~x71:0 = x68:0 * x70:0 + x69:0 * x71:0 Arcs: (1) -> (1), (2) (2) -> (1), (2) This digraph is fully evaluated! ---------------------------------------- (99) Obligation: Termination digraph: Nodes: (1) f413_in(sum~x58:0~x59:0, sum~x60:0~x61:0) -> f413_in(x58:0, x60:0) :|: TRUE && sum~x58:0~x59:0 = x58:0 + x59:0 && sum~x60:0~x61:0 = x60:0 + x61:0 (2) f413_in(times~x68:0~x69:0, sum~times~x68:0~x70:0~times~x69:0~x71:0) -> f413_in(x68:0, x71:0) :|: TRUE && times~x68:0~x69:0 = x68:0 * x69:0 && sum~times~x68:0~x70:0~times~x69:0~x71:0 = x68:0 * x70:0 + x69:0 * x71:0 Arcs: (1) -> (1), (2) (2) -> (1), (2) This digraph is fully evaluated! ---------------------------------------- (100) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (101) Obligation: Rules: f413_in(times~x68:0:0~x69:0:0, sum~times~x68:0:0~x70:0:0~times~x69:0:0~x71:0:0) -> f413_in(x68:0:0, x71:0:0) :|: TRUE && times~x68:0:0~x69:0:0 = x68:0:0 * x69:0:0 && sum~times~x68:0:0~x70:0:0~times~x69:0:0~x71:0:0 = x68:0:0 * x70:0:0 + x69:0:0 * x71:0:0 f413_in(sum~x58:0:0~x59:0:0, sum~x60:0:0~x61:0:0) -> f413_in(x58:0:0, x60:0:0) :|: TRUE && sum~x58:0:0~x59:0:0 = x58:0:0 + x59:0:0 && sum~x60:0:0~x61:0:0 = x60:0:0 + x61:0:0 ---------------------------------------- (102) IRSwTToIntTRSProof (SOUND) Applied path-length measure to transform intTRS with terms to intTRS. ---------------------------------------- (103) Obligation: Rules: f413_in(times~x2~x4, sum~times~x2~x5~times~x4~x3) -> f413_in(x2, x3) :|: TRUE && times~x2~x4 = x2 * x4 && sum~times~x2~x5~times~x4~x3 = x2 * x5 + x4 * x3 f413_in(sum~x8~x10, sum~x9~x11) -> f413_in(x8, x9) :|: TRUE && sum~x8~x10 = x8 + x10 && sum~x9~x11 = x9 + x11 ---------------------------------------- (104) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (105) Obligation: Rules: f413_in(times~x2:0~x4:0, sum~times~x2:0~x5:0~times~x4:0~x3:0) -> f413_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 f413_in(sum~x8:0~x10:0, sum~x9:0~x11:0) -> f413_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: f15_in(T2) -> f22_in(T2) :|: TRUE f15_in(x) -> f23_in(x) :|: TRUE f22_out(x1) -> f15_out(x1) :|: TRUE f23_out(x2) -> f15_out(x2) :|: TRUE f339_out(T26) -> f340_in(T27) :|: TRUE f340_out(x3) -> f320_out(x4, x3) :|: TRUE f320_in(x5, x6) -> f339_in(x5) :|: TRUE f623_in -> f623_out :|: TRUE f631_in -> f633_in :|: TRUE f633_out -> f631_out :|: TRUE f634_out -> f631_out :|: TRUE f631_in -> f634_in :|: TRUE f656_in -> f659_in :|: TRUE f659_out -> f656_out :|: TRUE f660_out -> f656_out :|: TRUE f656_in -> f660_in :|: TRUE f618_in -> f620_in :|: TRUE f620_out -> f618_out :|: TRUE f4_out(x7) -> f3_out(x7) :|: TRUE f3_in(x8) -> f4_in(x8) :|: TRUE f340_in(x9) -> f3_in(x9) :|: TRUE f3_out(x10) -> f340_out(x10) :|: TRUE f655_in -> f658_in :|: TRUE f658_out -> f655_out :|: TRUE f655_in -> f657_in :|: TRUE f657_out -> f655_out :|: TRUE f633_in -> f653_in :|: TRUE f653_out -> f654_in :|: TRUE f654_out -> f633_out :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f655_out -> f632_out :|: TRUE f656_out -> f632_out :|: TRUE f627_in -> f631_in :|: TRUE f632_out -> f627_out :|: TRUE f631_out -> f627_out :|: TRUE f627_in -> f632_in :|: TRUE f22_in(x11 + x12) -> f320_in(x11, x12) :|: TRUE f320_out(x13, x14) -> f22_out(x13 + x14) :|: TRUE f22_in(x15) -> f322_in :|: TRUE f322_out -> f22_out(x16) :|: TRUE f661_out -> f662_in :|: TRUE f662_out -> f659_out :|: TRUE f659_in -> f661_in :|: TRUE f405_in(T101) -> f616_in(T101) :|: TRUE f616_out(x17) -> f405_out(x17) :|: TRUE f405_in(x18) -> f617_in :|: TRUE f617_out -> f405_out(x19) :|: TRUE f616_in(x20) -> f618_in :|: TRUE f619_out(x21) -> f616_out(x21) :|: TRUE f618_out -> f619_in(x22) :|: TRUE f621_in -> f623_in :|: TRUE f624_out -> f621_out :|: TRUE f621_in -> f624_in :|: TRUE f623_out -> f621_out :|: TRUE f628_in -> f628_out :|: TRUE f4_in(x23) -> f6_in(x23) :|: TRUE f7_out(x24) -> f4_out(x24) :|: TRUE f6_out(x25) -> f4_out(x25) :|: TRUE f4_in(x26) -> f7_in(x26) :|: TRUE f7_in(x27) -> f14_in(x27) :|: TRUE f14_out(x28) -> f7_out(x28) :|: TRUE f7_in(x29) -> f15_in(x29) :|: TRUE f15_out(x30) -> f7_out(x30) :|: TRUE f621_out -> f620_out :|: TRUE f620_in -> f621_in :|: TRUE f620_in -> f622_in :|: TRUE f622_out -> f620_out :|: TRUE f626_in -> f628_in :|: TRUE f628_out -> f626_out :|: TRUE f626_in -> f629_in :|: TRUE f629_out -> f626_out :|: TRUE f403_out(x31) -> f23_out(x31) :|: TRUE f23_in(x32) -> f405_in(x32) :|: TRUE f405_out(x33) -> f23_out(x33) :|: TRUE f23_in(x34) -> f403_in(x34) :|: TRUE f633_out -> f657_out :|: TRUE f657_in -> f633_in :|: TRUE f654_in -> f618_in :|: TRUE f618_out -> f654_out :|: TRUE f662_in -> f618_in :|: TRUE f618_out -> f662_out :|: TRUE f622_in -> f627_in :|: TRUE f622_in -> f626_in :|: TRUE f627_out -> f622_out :|: TRUE f626_out -> f622_out :|: TRUE f619_in(x35) -> f3_in(x35) :|: TRUE f3_out(x36) -> f619_out(x36) :|: TRUE f653_in -> f618_in :|: TRUE f618_out -> f653_out :|: TRUE f339_in(x37) -> f3_in(x37) :|: TRUE f3_out(x38) -> f339_out(x38) :|: TRUE f618_out -> f661_out :|: TRUE f661_in -> f618_in :|: TRUE Start term: f3_in(T2) ---------------------------------------- (109) IRSwTSimpleDependencyGraphProof (EQUIVALENT) Constructed simple dependency graph. Simplified to the following IRSwTs: intTRSProblem: f15_in(T2) -> f22_in(T2) :|: TRUE f15_in(x) -> f23_in(x) :|: TRUE f320_in(x5, x6) -> f339_in(x5) :|: TRUE f623_in -> f623_out :|: TRUE f631_in -> f633_in :|: TRUE f633_out -> f631_out :|: TRUE f656_in -> f659_in :|: TRUE f659_out -> f656_out :|: TRUE f618_in -> f620_in :|: TRUE f620_out -> f618_out :|: TRUE f3_in(x8) -> f4_in(x8) :|: TRUE f655_in -> f657_in :|: TRUE f657_out -> f655_out :|: TRUE f633_in -> f653_in :|: TRUE f653_out -> f654_in :|: TRUE f654_out -> f633_out :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f655_out -> f632_out :|: TRUE f656_out -> f632_out :|: TRUE f627_in -> f631_in :|: TRUE f632_out -> f627_out :|: TRUE f631_out -> f627_out :|: TRUE f627_in -> f632_in :|: TRUE f22_in(x11 + x12) -> f320_in(x11, x12) :|: TRUE f661_out -> f662_in :|: TRUE f662_out -> f659_out :|: TRUE f659_in -> f661_in :|: TRUE f405_in(T101) -> f616_in(T101) :|: TRUE f616_in(x20) -> f618_in :|: TRUE f618_out -> f619_in(x22) :|: TRUE f621_in -> f623_in :|: TRUE f623_out -> f621_out :|: TRUE f628_in -> f628_out :|: TRUE f4_in(x26) -> f7_in(x26) :|: TRUE f7_in(x29) -> f15_in(x29) :|: TRUE f621_out -> f620_out :|: TRUE f620_in -> f621_in :|: TRUE f620_in -> f622_in :|: TRUE f622_out -> f620_out :|: TRUE f626_in -> f628_in :|: TRUE f628_out -> f626_out :|: TRUE f23_in(x32) -> f405_in(x32) :|: TRUE f633_out -> f657_out :|: TRUE f657_in -> f633_in :|: TRUE f654_in -> f618_in :|: TRUE f618_out -> f654_out :|: TRUE f662_in -> f618_in :|: TRUE f618_out -> f662_out :|: TRUE f622_in -> f627_in :|: TRUE f622_in -> f626_in :|: TRUE f627_out -> f622_out :|: TRUE f626_out -> f622_out :|: TRUE f619_in(x35) -> f3_in(x35) :|: TRUE f653_in -> f618_in :|: TRUE f618_out -> f653_out :|: TRUE f339_in(x37) -> f3_in(x37) :|: TRUE f618_out -> f661_out :|: TRUE f661_in -> f618_in :|: TRUE ---------------------------------------- (110) Obligation: Rules: f15_in(T2) -> f22_in(T2) :|: TRUE f15_in(x) -> f23_in(x) :|: TRUE f320_in(x5, x6) -> f339_in(x5) :|: TRUE f623_in -> f623_out :|: TRUE f631_in -> f633_in :|: TRUE f633_out -> f631_out :|: TRUE f656_in -> f659_in :|: TRUE f659_out -> f656_out :|: TRUE f618_in -> f620_in :|: TRUE f620_out -> f618_out :|: TRUE f3_in(x8) -> f4_in(x8) :|: TRUE f655_in -> f657_in :|: TRUE f657_out -> f655_out :|: TRUE f633_in -> f653_in :|: TRUE f653_out -> f654_in :|: TRUE f654_out -> f633_out :|: TRUE f632_in -> f656_in :|: TRUE f632_in -> f655_in :|: TRUE f655_out -> f632_out :|: TRUE f656_out -> f632_out :|: TRUE f627_in -> f631_in :|: TRUE f632_out -> f627_out :|: TRUE f631_out -> f627_out :|: TRUE f627_in -> f632_in :|: TRUE f22_in(x11 + x12) -> f320_in(x11, x12) :|: TRUE f661_out -> f662_in :|: TRUE f662_out -> f659_out :|: TRUE f659_in -> f661_in :|: TRUE f405_in(T101) -> f616_in(T101) :|: TRUE f616_in(x20) -> f618_in :|: TRUE f618_out -> f619_in(x22) :|: TRUE f621_in -> f623_in :|: TRUE f623_out -> f621_out :|: TRUE f628_in -> f628_out :|: TRUE f4_in(x26) -> f7_in(x26) :|: TRUE f7_in(x29) -> f15_in(x29) :|: TRUE f621_out -> f620_out :|: TRUE f620_in -> f621_in :|: TRUE f620_in -> f622_in :|: TRUE f622_out -> f620_out :|: TRUE f626_in -> f628_in :|: TRUE f628_out -> f626_out :|: TRUE f23_in(x32) -> f405_in(x32) :|: TRUE f633_out -> f657_out :|: TRUE f657_in -> f633_in :|: TRUE f654_in -> f618_in :|: TRUE f618_out -> f654_out :|: TRUE f662_in -> f618_in :|: TRUE f618_out -> f662_out :|: TRUE f622_in -> f627_in :|: TRUE f622_in -> f626_in :|: TRUE f627_out -> f622_out :|: TRUE f626_out -> f622_out :|: TRUE f619_in(x35) -> f3_in(x35) :|: TRUE f653_in -> f618_in :|: TRUE f618_out -> f653_out :|: TRUE f339_in(x37) -> f3_in(x37) :|: TRUE f618_out -> f661_out :|: TRUE f661_in -> f618_in :|: TRUE ---------------------------------------- (111) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (112) Obligation: Rules: f620_out -> f3_in(x22:0) :|: TRUE f3_in(sum~x11:0~x12:0) -> f3_in(x11:0) :|: TRUE && sum~x11:0~x12:0 = x11:0 + x12:0 f618_in -> f618_in :|: TRUE f620_out -> f618_in :|: TRUE f618_in -> f620_out :|: TRUE f620_out -> f620_out :|: TRUE f3_in(x8:0) -> f618_in :|: TRUE ---------------------------------------- (113) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (114) Obligation: Rules: f620_out -> f3_in(x22:0) :|: TRUE f3_in(sum~x11:0~x12:0) -> f3_in(x11:0) :|: TRUE && sum~x11:0~x12:0 = x11:0 + x12:0 f618_in -> f618_in :|: TRUE f620_out -> f618_in :|: TRUE f618_in -> f620_out :|: TRUE f620_out -> f620_out :|: TRUE f3_in(x8:0) -> f618_in :|: TRUE ---------------------------------------- (115) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) f620_out -> f3_in(x22:0) :|: TRUE (2) f3_in(sum~x11:0~x12:0) -> f3_in(x11:0) :|: TRUE && sum~x11:0~x12:0 = x11:0 + x12:0 (3) f618_in -> f618_in :|: TRUE (4) f620_out -> f618_in :|: TRUE (5) f618_in -> f620_out :|: TRUE (6) f620_out -> f620_out :|: TRUE (7) f3_in(x8:0) -> f618_in :|: TRUE Arcs: (1) -> (2), (7) (2) -> (2), (7) (3) -> (3), (5) (4) -> (3), (5) (5) -> (1), (4), (6) (6) -> (1), (4), (6) (7) -> (3), (5) This digraph is fully evaluated! ---------------------------------------- (116) Obligation: Termination digraph: Nodes: (1) f620_out -> f3_in(x22:0) :|: TRUE (2) f618_in -> f620_out :|: TRUE (3) f618_in -> f618_in :|: TRUE (4) f3_in(x8:0) -> f618_in :|: TRUE (5) f3_in(sum~x11:0~x12:0) -> f3_in(x11:0) :|: TRUE && sum~x11:0~x12:0 = x11:0 + x12:0 (6) f620_out -> f618_in :|: TRUE (7) f620_out -> f620_out :|: TRUE Arcs: (1) -> (4), (5) (2) -> (1), (6), (7) (3) -> (2), (3) (4) -> (2), (3) (5) -> (4), (5) (6) -> (2), (3) (7) -> (1), (6), (7) This digraph is fully evaluated! ---------------------------------------- (117) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (118) Obligation: Rules: f618_in -> f620_out :|: TRUE f620_out -> f620_out :|: TRUE f618_in -> f618_in :|: TRUE f620_out -> f618_in :|: TRUE f3_in(sum~x11:0:0~x12:0:0) -> f3_in(x11:0:0) :|: TRUE && sum~x11:0:0~x12:0:0 = x11:0:0 + x12:0:0 f3_in(x8:0:0) -> f618_in :|: TRUE f620_out -> f3_in(x22:0:0) :|: TRUE ---------------------------------------- (119) PrologToDTProblemTransformerProof (SOUND) Built DT problem from termination graph DT10. { "root": 50, "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": { "120": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (const (1)))) (const (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "126": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "800": { "goal": [{ "clause": 0, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" }], "kb": { "nonunifying": [ [ "(p T1 T342)", "(p (d (e (t))) (const (1)))" ], [ "(p T1 T342)", "(p (d (e (const X235))) (const (0)))" ], [ "(p T1 T342)", "(p (d (e (+ X329 X330))) (+ X331 X332))" ], [ "(p T1 T342)", "(p (d (e (* X412 X413))) (+ (* X412 X414) (* X413 X415)))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T342"], "free": [ "X235", "X329", "X330", "X331", "X332", "X412", "X413", "X414", "X415", "X475" ], "exprvars": [] } }, "801": { "goal": [ { "clause": 1, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" }, { "clause": 2, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" }, { "clause": 3, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" }, { "clause": 4, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" } ], "kb": { "nonunifying": [ [ "(p T1 T342)", "(p (d (e (t))) (const (1)))" ], [ "(p T1 T342)", "(p (d (e (const X235))) (const (0)))" ], [ "(p T1 T342)", "(p (d (e (+ X329 X330))) (+ X331 X332))" ], [ "(p T1 T342)", "(p (d (e (* X412 X413))) (+ (* X412 X414) (* X413 X415)))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T342"], "free": [ "X235", "X329", "X330", "X331", "X332", "X412", "X413", "X414", "X415", "X475" ], "exprvars": [] } }, "802": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (const (1)))) T342)" }], "kb": { "nonunifying": [ [ "(p T1 T342)", "(p (d (e (t))) (const (1)))" ], [ "(p T1 T342)", "(p (d (e (const X235))) (const (0)))" ], [ "(p T1 T342)", "(p (d (e (+ X329 X330))) (+ X331 X332))" ], [ "(p T1 T342)", "(p (d (e (* X412 X413))) (+ (* X412 X414) (* X413 X415)))" ] ], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T342"], "free": [ "X235", "X329", "X330", "X331", "X332", "X412", "X413", "X414", "X415" ], "exprvars": [] } }, "803": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "804": { "goal": [{ "clause": 1, "scope": 8, "term": "(',' (p (d T343) X475) (p (d (e X475)) T342))" }], "kb": { "nonunifying": [ [ "(p T1 T342)", "(p (d (e (t))) (const (1)))" ], [ "(p T1 T342)", "(p (d (e (const X235))) (const (0)))" ], [ "(p T1 T342)", "(p (d (e (+ X329 X330))) (+ X331 X332))" ], [ "(p T1 T342)", 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"PlainIntegerRelationState", "relations": [] }, "ground": ["T182"], "free": [], "exprvars": [] } }, "451": { "goal": [{ "clause": -1, "scope": -1, "term": "(',' (p (d (e T59)) X114) (p (d (e T60)) X113))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [ "X113", "X114" ], "exprvars": [] } }, "693": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "452": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "694": { "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": [] } }, "453": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "695": { "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": [] } }, "454": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T29)) X49)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X49"], "exprvars": [] } }, "696": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e T191)) T182)" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": ["T182"], "free": [], "exprvars": [] } }, "213": { "goal": [{ "clause": 1, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "455": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (+ T62 T61))) (const (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "697": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "214": { "goal": [ { "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": [] } }, "456": { "goal": [{ "clause": 3, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "698": { "goal": [{ "clause": 2, "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": [] } }, "215": { "goal": [{ "clause": -1, "scope": -1, "term": "(p (d (e (const (0)))) (const (1)))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "457": { "goal": [{ "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": [] } }, "699": { "goal": [ { "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": [] } }, "216": { "goal": [], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": [], "exprvars": [] } }, "219": { "goal": [{ "clause": 2, "scope": 2, "term": "(',' (p (d T10) X14) (p (d (e X14)) (const (1))))" }], "kb": { "nonunifying": [], "intvars": {}, "arithmetic": { "type": "PlainIntegerRelationState", "relations": [] }, "ground": [], "free": ["X14"], "exprvars": [] } }, "220": { "goal": [ { "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": [] } }, "100": { "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": [] } }, "103": { "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": [] } } }, "edges": [ { "from": 50, "to": 51, "label": "CASE" }, { "from": 51, "to": 52, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT1 -> d(e(t)),\nT2 -> const(1)" }, { "from": 51, "to": 53, "label": "EVAL-BACKTRACK" }, { "from": 52, "to": 54, "label": "SUCCESS" }, { "from": 53, "to": 641, "label": "EVAL with clause\np(d(e(const(X235))), const(0)).\nand substitutionX235 -> T122,\nT1 -> d(e(const(T122))),\nT2 -> const(0)" }, { "from": 53, "to": 642, "label": "EVAL-BACKTRACK" }, { "from": 54, "to": 55, "label": "BACKTRACK\nfor clause: p(d(e(const(A))), const(0))because of non-unification" }, { "from": 55, "to": 56, "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": 56, "to": 57, "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": 57, "to": 70, "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": 57, "to": 71, "label": "EVAL-BACKTRACK" }, { "from": 70, "to": 80, "label": "CASE" }, { "from": 80, "to": 100, "label": "PARALLEL" }, { "from": 80, "to": 103, "label": "PARALLEL" }, { "from": 100, "to": 120, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT10 -> e(t),\nX14 -> const(1)" }, { "from": 100, "to": 126, "label": "EVAL-BACKTRACK" }, { "from": 103, "to": 213, "label": "PARALLEL" }, { "from": 103, "to": 214, "label": "PARALLEL" }, { "from": 120, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> const(1)" }, { "from": 213, "to": 215, "label": "EVAL with clause\np(d(e(const(X19))), const(0)).\nand substitutionX19 -> T15,\nT10 -> e(const(T15)),\nX14 -> const(0)" }, { "from": 213, "to": 216, "label": "EVAL-BACKTRACK" }, { "from": 214, "to": 219, "label": "PARALLEL" }, { "from": 214, "to": 220, "label": "PARALLEL" }, { "from": 215, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> const(1)" }, { "from": 219, "to": 382, "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": 219, "to": 383, "label": "EVAL-BACKTRACK" }, { "from": 220, "to": 456, "label": "PARALLEL" }, { "from": 220, "to": 457, "label": "PARALLEL" }, { "from": 382, "to": 384, "label": "SPLIT 1" }, { "from": 382, "to": 385, "label": "SPLIT 2\nreplacements:X48 -> T28,\nT27 -> T29" }, { "from": 384, "to": 386, "label": "CASE" }, { "from": 385, "to": 454, "label": "SPLIT 1" }, { "from": 385, "to": 455, "label": "SPLIT 2\nreplacements:X49 -> T61,\nT28 -> T62" }, { "from": 386, "to": 387, "label": "PARALLEL" }, { "from": 386, "to": 388, "label": "PARALLEL" }, { "from": 387, "to": 389, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT26 -> t,\nX48 -> const(1)" }, { "from": 387, "to": 390, "label": "EVAL-BACKTRACK" }, { "from": 388, "to": 392, "label": "PARALLEL" }, { "from": 388, "to": 393, "label": "PARALLEL" }, { "from": 389, "to": 391, "label": "SUCCESS" }, { "from": 392, "to": 394, "label": "EVAL with clause\np(d(e(const(X54))), const(0)).\nand substitutionX54 -> T34,\nT26 -> const(T34),\nX48 -> const(0)" }, { "from": 392, "to": 395, "label": "EVAL-BACKTRACK" }, { "from": 393, "to": 397, "label": "PARALLEL" }, { "from": 393, "to": 398, "label": "PARALLEL" }, { "from": 394, "to": 396, "label": "SUCCESS" }, { "from": 397, "to": 399, "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": 397, "to": 400, "label": "EVAL-BACKTRACK" }, { "from": 398, "to": 447, "label": "PARALLEL" }, { "from": 398, "to": 448, "label": "PARALLEL" }, { "from": 399, "to": 401, "label": "SPLIT 1" }, { "from": 399, "to": 402, "label": "SPLIT 2\nreplacements:X83 -> T47,\nT46 -> T48" }, { "from": 401, "to": 384, "label": "INSTANCE with matching:\nT26 -> T45\nX48 -> X83" }, { "from": 402, "to": 384, "label": "INSTANCE with matching:\nT26 -> T48\nX48 -> X84" }, { "from": 447, "to": 451, "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": 447, "to": 452, "label": "EVAL-BACKTRACK" }, { "from": 448, "to": 453, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 451, "to": 399, "label": "INSTANCE with matching:\nT45 -> T59\nX83 -> X114\nT46 -> T60\nX84 -> X113" }, { "from": 454, "to": 384, "label": "INSTANCE with matching:\nT26 -> T29\nX48 -> X49" }, { "from": 455, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(T62, T61)))\nT2 -> const(1)" }, { "from": 456, "to": 506, "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": 456, "to": 507, "label": "EVAL-BACKTRACK" }, { "from": 457, "to": 512, "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": 457, "to": 513, "label": "EVAL-BACKTRACK" }, { "from": 506, "to": 508, "label": "SPLIT 1" }, { "from": 506, "to": 509, "label": "SPLIT 2\nreplacements:X146 -> T75,\nT74 -> T76,\nT73 -> T77" }, { "from": 508, "to": 384, "label": "INSTANCE with matching:\nT26 -> T73\nX48 -> X146" }, { "from": 509, "to": 510, "label": "SPLIT 1" }, { "from": 509, "to": 511, "label": "SPLIT 2\nreplacements:X145 -> T78,\nT77 -> T79,\nT76 -> T80,\nT75 -> T81" }, { "from": 510, "to": 384, "label": "INSTANCE with matching:\nT26 -> T76\nX48 -> X145" }, { "from": 511, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T79, T78), *(T80, T81))))\nT2 -> const(1)" }, { "from": 512, "to": 546, "label": "SPLIT 1" }, { "from": 512, "to": 547, "label": "SPLIT 2\nreplacements:X156 -> T86" }, { "from": 546, "to": 548, "label": "CASE" }, { "from": 547, "to": 70, "label": "INSTANCE with matching:\nT10 -> e(T86)\nX14 -> X157" }, { "from": 548, "to": 549, "label": "PARALLEL" }, { "from": 548, "to": 550, "label": "PARALLEL" }, { "from": 549, "to": 551, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT85 -> e(t),\nX156 -> const(1)" }, { "from": 549, "to": 552, "label": "EVAL-BACKTRACK" }, { "from": 550, "to": 554, "label": "PARALLEL" }, { "from": 550, "to": 555, "label": "PARALLEL" }, { "from": 551, "to": 553, "label": "SUCCESS" }, { "from": 554, "to": 556, "label": "EVAL with clause\np(d(e(const(X162))), const(0)).\nand substitutionX162 -> T91,\nT85 -> e(const(T91)),\nX156 -> const(0)" }, { "from": 554, "to": 557, "label": "EVAL-BACKTRACK" }, { "from": 555, "to": 607, "label": "PARALLEL" }, { "from": 555, "to": 608, "label": "PARALLEL" }, { "from": 556, "to": 558, "label": "SUCCESS" }, { "from": 607, "to": 609, "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": 607, "to": 610, "label": "EVAL-BACKTRACK" }, { "from": 608, "to": 611, "label": "PARALLEL" }, { "from": 608, "to": 612, "label": "PARALLEL" }, { "from": 609, "to": 399, "label": "INSTANCE with matching:\nT45 -> T102\nX83 -> X191\nT46 -> T103\nX84 -> X192" }, { "from": 611, "to": 635, "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": 611, "to": 636, "label": "EVAL-BACKTRACK" }, { "from": 612, "to": 637, "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": 612, "to": 638, "label": "EVAL-BACKTRACK" }, { "from": 635, "to": 399, "label": "INSTANCE with matching:\nT45 -> T114\nX83 -> X222\nT46 -> T115\nX84 -> X221" }, { "from": 637, "to": 639, "label": "SPLIT 1" }, { "from": 637, "to": 640, "label": "SPLIT 2\nreplacements:X232 -> T120" }, { "from": 639, "to": 546, "label": "INSTANCE with matching:\nT85 -> T119\nX156 -> X232" }, { "from": 640, "to": 384, "label": "INSTANCE with matching:\nT26 -> T120\nX48 -> X233" }, { "from": 641, "to": 643, "label": "SUCCESS" }, { "from": 642, "to": 687, "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": 642, "to": 688, "label": "EVAL-BACKTRACK" }, { "from": 643, "to": 644, "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": 644, "to": 645, "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": 645, "to": 646, "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": 645, "to": 647, "label": "EVAL-BACKTRACK" }, { "from": 646, "to": 648, "label": "CASE" }, { "from": 648, "to": 649, "label": "PARALLEL" }, { "from": 648, "to": 650, "label": "PARALLEL" }, { "from": 649, "to": 651, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT129 -> e(t),\nX248 -> const(1)" }, { "from": 649, "to": 652, "label": "EVAL-BACKTRACK" }, { "from": 650, "to": 663, "label": "PARALLEL" }, { "from": 650, "to": 664, "label": "PARALLEL" }, { "from": 651, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> const(0)" }, { "from": 663, "to": 665, "label": "EVAL with clause\np(d(e(const(X253))), const(0)).\nand substitutionX253 -> T134,\nT129 -> e(const(T134)),\nX248 -> const(0)" }, { "from": 663, "to": 666, "label": "EVAL-BACKTRACK" }, { "from": 664, "to": 667, "label": "PARALLEL" }, { "from": 664, "to": 668, "label": "PARALLEL" }, { "from": 665, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> const(0)" }, { "from": 667, "to": 669, "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": 667, "to": 670, "label": "EVAL-BACKTRACK" }, { "from": 668, "to": 675, "label": "PARALLEL" }, { "from": 668, "to": 676, "label": "PARALLEL" }, { "from": 669, "to": 671, "label": "SPLIT 1" }, { "from": 669, "to": 672, "label": "SPLIT 2\nreplacements:X282 -> T147,\nT146 -> T148" }, { "from": 671, "to": 384, "label": "INSTANCE with matching:\nT26 -> T145\nX48 -> X282" }, { "from": 672, "to": 673, "label": "SPLIT 1" }, { "from": 672, "to": 674, "label": "SPLIT 2\nreplacements:X283 -> T149,\nT147 -> T150" }, { "from": 673, "to": 384, "label": "INSTANCE with matching:\nT26 -> T148\nX48 -> X283" }, { "from": 674, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(T150, T149)))\nT2 -> const(0)" }, { "from": 675, "to": 677, "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": 675, "to": 678, "label": "EVAL-BACKTRACK" }, { "from": 676, "to": 683, "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": 676, "to": 684, "label": "EVAL-BACKTRACK" }, { "from": 677, "to": 679, "label": "SPLIT 1" }, { "from": 677, "to": 680, "label": "SPLIT 2\nreplacements:X313 -> T163,\nT162 -> T164,\nT161 -> T165" }, { "from": 679, "to": 384, "label": "INSTANCE with matching:\nT26 -> T161\nX48 -> X313" }, { "from": 680, "to": 681, "label": "SPLIT 1" }, { "from": 680, "to": 682, "label": "SPLIT 2\nreplacements:X312 -> T166,\nT165 -> T167,\nT164 -> T168,\nT163 -> T169" }, { "from": 681, "to": 384, "label": "INSTANCE with matching:\nT26 -> T164\nX48 -> X312" }, { "from": 682, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T167, T166), *(T168, T169))))\nT2 -> const(0)" }, { "from": 683, "to": 685, "label": "SPLIT 1" }, { "from": 683, "to": 686, "label": "SPLIT 2\nreplacements:X323 -> T174" }, { "from": 685, "to": 546, "label": "INSTANCE with matching:\nT85 -> T173\nX156 -> X323" }, { "from": 686, "to": 646, "label": "INSTANCE with matching:\nT129 -> e(T174)\nX248 -> X324" }, { "from": 687, "to": 689, "label": "CASE" }, { "from": 688, "to": 774, "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": 688, "to": 775, "label": "EVAL-BACKTRACK" }, { "from": 689, "to": 690, "label": "PARALLEL" }, { "from": 689, "to": 691, "label": "PARALLEL" }, { "from": 690, "to": 692, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT183 -> t,\nT181 -> const(1),\nT184 -> T185" }, { "from": 690, "to": 693, "label": "EVAL-BACKTRACK" }, { "from": 691, "to": 694, "label": "PARALLEL" }, { "from": 691, "to": 695, "label": "PARALLEL" }, { "from": 692, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T185))\nT2 -> T182" }, { "from": 694, "to": 696, "label": "EVAL with clause\np(d(e(const(X337))), const(0)).\nand substitutionX337 -> T190,\nT183 -> const(T190),\nT181 -> const(0),\nT184 -> T191" }, { "from": 694, "to": 697, "label": "EVAL-BACKTRACK" }, { "from": 695, "to": 698, "label": "PARALLEL" }, { "from": 695, "to": 699, "label": "PARALLEL" }, { "from": 696, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T191))\nT2 -> T182" }, { "from": 698, "to": 700, "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": 698, "to": 701, "label": "EVAL-BACKTRACK" }, { "from": 699, "to": 706, "label": "PARALLEL" }, { "from": 699, "to": 707, "label": "PARALLEL" }, { "from": 700, "to": 702, "label": "SPLIT 1" }, { "from": 700, "to": 703, "label": "SPLIT 2\nnew knowledge:\nT210 is ground\nreplacements:T213 -> T215,\nT214 -> T216" }, { "from": 702, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T212))\nT2 -> T210" }, { "from": 703, "to": 704, "label": "SPLIT 1" }, { "from": 703, "to": 705, "label": "SPLIT 2\nnew knowledge:\nT211 is ground\nreplacements:T216 -> T217" }, { "from": 704, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T215))\nT2 -> T211" }, { "from": 705, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T217))\nT2 -> T182" }, { "from": 706, "to": 708, "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": 706, "to": 709, "label": "EVAL-BACKTRACK" }, { "from": 707, "to": 710, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 708, "to": 700, "label": "INSTANCE with matching:\nT212 -> T234\nT210 -> T237\nT213 -> T235\nT211 -> T236\nT214 -> T238" }, { "from": 710, "to": 711, "label": "FAILURE" }, { "from": 711, "to": 766, "label": "PARALLEL" }, { "from": 711, "to": 767, "label": "PARALLEL" }, { "from": 766, "to": 768, "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": 766, "to": 769, "label": "EVAL-BACKTRACK" }, { "from": 767, "to": 770, "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": 767, "to": 771, "label": "EVAL-BACKTRACK" }, { "from": 768, "to": 703, "label": "INSTANCE with matching:\nT215 -> T255\nT211 -> T258\nT216 -> T256\nT182 -> T257" }, { "from": 770, "to": 772, "label": "SPLIT 1" }, { "from": 770, "to": 773, "label": "SPLIT 2\nreplacements:X407 -> T269" }, { "from": 772, "to": 546, "label": "INSTANCE with matching:\nT85 -> T268\nX156 -> X407" }, { "from": 773, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T269))\nT2 -> +(T266, T267)" }, { "from": 774, "to": 776, "label": "CASE" }, { "from": 775, "to": 797, "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": 775, "to": 798, "label": "EVAL-BACKTRACK" }, { "from": 776, "to": 777, "label": "PARALLEL" }, { "from": 776, "to": 778, "label": "PARALLEL" }, { "from": 777, "to": 779, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT274 -> t,\nT277 -> const(1)" }, { "from": 777, "to": 780, "label": "EVAL-BACKTRACK" }, { "from": 778, "to": 781, "label": "PARALLEL" }, { "from": 778, "to": 782, "label": "PARALLEL" }, { "from": 779, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T275))\nT2 -> T276" }, { "from": 781, "to": 783, "label": "EVAL with clause\np(d(e(const(X420))), const(0)).\nand substitutionX420 -> T282,\nT274 -> const(T282),\nT277 -> const(0)" }, { "from": 781, "to": 784, "label": "EVAL-BACKTRACK" }, { "from": 782, "to": 785, "label": "PARALLEL" }, { "from": 782, "to": 786, "label": "PARALLEL" }, { "from": 783, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(T275))\nT2 -> T276" }, { "from": 785, "to": 787, "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": 785, "to": 788, "label": "EVAL-BACKTRACK" }, { "from": 786, "to": 789, "label": "PARALLEL" }, { "from": 786, "to": 790, "label": "PARALLEL" }, { "from": 787, "to": 700, "label": "INSTANCE with matching:\nT212 -> T299\nT210 -> T301\nT213 -> T300\nT211 -> T302\nT214 -> T275\nT182 -> T276" }, { "from": 789, "to": 791, "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": 789, "to": 792, "label": "EVAL-BACKTRACK" }, { "from": 790, "to": 793, "label": "BACKTRACK\nfor clause: p(d(d(X)), DDX) :- ','(p(d(X), DX), p(d(e(DX)), DDX))because of non-unification" }, { "from": 791, "to": 700, "label": "INSTANCE with matching:\nT212 -> T319\nT210 -> T322\nT213 -> T320\nT211 -> T321\nT214 -> T275\nT182 -> T276" }, { "from": 793, "to": 794, "label": "FAILURE" }, { "from": 794, "to": 795, "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": 794, "to": 796, "label": "EVAL-BACKTRACK" }, { "from": 795, "to": 770, "label": "INSTANCE with matching:\nT268 -> T338\nX407 -> X470\nT266 -> *(T334, T335)\nT267 -> *(T336, T337)" }, { "from": 797, "to": 799, "label": "CASE" }, { "from": 799, "to": 800, "label": "PARALLEL" }, { "from": 799, "to": 801, "label": "PARALLEL" }, { "from": 800, "to": 802, "label": "EVAL with clause\np(d(e(t)), const(1)).\nand substitutionT343 -> e(t),\nX475 -> const(1)" }, { "from": 800, "to": 803, "label": "EVAL-BACKTRACK" }, { "from": 801, "to": 804, "label": "PARALLEL" }, { "from": 801, "to": 805, "label": "PARALLEL" }, { "from": 802, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(1)))\nT2 -> T342" }, { "from": 804, "to": 806, "label": "EVAL with clause\np(d(e(const(X480))), const(0)).\nand substitutionX480 -> T348,\nT343 -> e(const(T348)),\nX475 -> const(0)" }, { "from": 804, "to": 807, "label": "EVAL-BACKTRACK" }, { "from": 805, "to": 808, "label": "PARALLEL" }, { "from": 805, "to": 809, "label": "PARALLEL" }, { "from": 806, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(const(0)))\nT2 -> T342" }, { "from": 808, "to": 810, "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": 808, "to": 814, "label": "EVAL-BACKTRACK" }, { "from": 809, "to": 819, "label": "PARALLEL" }, { "from": 809, "to": 820, "label": "PARALLEL" }, { "from": 810, "to": 815, "label": "SPLIT 1" }, { "from": 810, "to": 816, "label": "SPLIT 2\nreplacements:X509 -> T361,\nT360 -> T362,\nT1 -> T363" }, { "from": 815, "to": 384, "label": "INSTANCE with matching:\nT26 -> T359\nX48 -> X509" }, { "from": 816, "to": 817, "label": "SPLIT 1" }, { "from": 816, "to": 818, "label": "SPLIT 2\nreplacements:X510 -> T364,\nT361 -> T365,\nT363 -> T366" }, { "from": 817, "to": 384, "label": "INSTANCE with matching:\nT26 -> T362\nX48 -> X510" }, { "from": 818, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(T365, T364)))\nT2 -> T342" }, { "from": 819, "to": 821, "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": 819, "to": 822, "label": "EVAL-BACKTRACK" }, { "from": 820, "to": 827, "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": 820, "to": 828, "label": "EVAL-BACKTRACK" }, { "from": 821, "to": 823, "label": "SPLIT 1" }, { "from": 821, "to": 824, "label": "SPLIT 2\nreplacements:X540 -> T379,\nT378 -> T380,\nT377 -> T381,\nT1 -> T382" }, { "from": 823, "to": 384, "label": "INSTANCE with matching:\nT26 -> T377\nX48 -> X540" }, { "from": 824, "to": 825, "label": "SPLIT 1" }, { "from": 824, "to": 826, "label": "SPLIT 2\nreplacements:X539 -> T383,\nT381 -> T384,\nT380 -> T385,\nT379 -> T386,\nT382 -> T387" }, { "from": 825, "to": 384, "label": "INSTANCE with matching:\nT26 -> T380\nX48 -> X539" }, { "from": 826, "to": 50, "label": "INSTANCE with matching:\nT1 -> d(e(+(*(T384, T383), *(T385, T386))))\nT2 -> T342" }, { "from": 827, "to": 829, "label": "SPLIT 1" }, { "from": 827, "to": 830, "label": "SPLIT 2\nreplacements:X550 -> T392,\nT1 -> T393" }, { "from": 829, "to": 546, "label": "INSTANCE with matching:\nT85 -> T391\nX156 -> X550" }, { "from": 830, "to": 797, "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) 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 ---------------------------------------- (131) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (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: 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 ---------------------------------------- (133) 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 ---------------------------------------- (134) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (135) 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 ---------------------------------------- (136) 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 ---------------------------------------- (137) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (138) 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 ---------------------------------------- (139) 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 ---------------------------------------- (140) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (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(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 ---------------------------------------- (142) 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 ---------------------------------------- (143) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (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(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