/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) Overlay + Local Confluence [EQUIVALENT, 3 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) QDP (10) QReductionProof [EQUIVALENT, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) QDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) QDP (17) QReductionProof [EQUIVALENT, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) QDP (24) QReductionProof [EQUIVALENT, 0 ms] (25) QDP (26) QDPQMonotonicMRRProof [EQUIVALENT, 66 ms] (27) QDP (28) TransformationProof [EQUIVALENT, 0 ms] (29) QDP (30) DependencyGraphProof [EQUIVALENT, 0 ms] (31) QDP (32) TransformationProof [EQUIVALENT, 0 ms] (33) QDP (34) TransformationProof [EQUIVALENT, 0 ms] (35) QDP (36) TransformationProof [EQUIVALENT, 0 ms] (37) QDP (38) TransformationProof [EQUIVALENT, 0 ms] (39) QDP (40) TransformationProof [EQUIVALENT, 0 ms] (41) QDP (42) TransformationProof [EQUIVALENT, 0 ms] (43) QDP (44) TransformationProof [EQUIVALENT, 0 ms] (45) QDP (46) TransformationProof [EQUIVALENT, 0 ms] (47) QDP (48) DependencyGraphProof [EQUIVALENT, 0 ms] (49) AND (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) UsableRulesProof [EQUIVALENT, 0 ms] (56) QDP (57) QReductionProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 0 ms] (64) QDP (65) DependencyGraphProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) UsableRulesProof [EQUIVALENT, 0 ms] (70) QDP (71) QReductionProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) UsableRulesProof [EQUIVALENT, 0 ms] (76) QDP (77) QReductionProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 0 ms] (80) QDP (81) DependencyGraphProof [EQUIVALENT, 0 ms] (82) QDP (83) UsableRulesProof [EQUIVALENT, 0 ms] (84) QDP (85) QReductionProof [EQUIVALENT, 0 ms] (86) QDP (87) TransformationProof [EQUIVALENT, 0 ms] (88) QDP (89) TransformationProof [EQUIVALENT, 0 ms] (90) QDP (91) TransformationProof [EQUIVALENT, 0 ms] (92) QDP (93) TransformationProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 0 ms] (96) QDP (97) TransformationProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 0 ms] (100) QDP (101) TransformationProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) QDPQMonotonicMRRProof [EQUIVALENT, 93 ms] (106) QDP (107) DependencyGraphProof [EQUIVALENT, 0 ms] (108) QDP (109) QDPQMonotonicMRRProof [EQUIVALENT, 36 ms] (110) QDP (111) UsableRulesProof [EQUIVALENT, 0 ms] (112) QDP (113) QReductionProof [EQUIVALENT, 0 ms] (114) QDP (115) QDPSizeChangeProof [EQUIVALENT, 0 ms] (116) YES (117) QDP (118) TransformationProof [EQUIVALENT, 0 ms] (119) QDP (120) TransformationProof [EQUIVALENT, 0 ms] (121) QDP (122) TransformationProof [EQUIVALENT, 0 ms] (123) QDP (124) TransformationProof [EQUIVALENT, 0 ms] (125) QDP (126) TransformationProof [EQUIVALENT, 0 ms] (127) QDP (128) DependencyGraphProof [EQUIVALENT, 0 ms] (129) QDP (130) TransformationProof [EQUIVALENT, 0 ms] (131) QDP (132) TransformationProof [EQUIVALENT, 0 ms] (133) QDP (134) TransformationProof [EQUIVALENT, 0 ms] (135) QDP (136) DependencyGraphProof [EQUIVALENT, 0 ms] (137) QDP (138) TransformationProof [EQUIVALENT, 0 ms] (139) QDP (140) TransformationProof [EQUIVALENT, 0 ms] (141) QDP (142) TransformationProof [EQUIVALENT, 0 ms] (143) QDP (144) TransformationProof [EQUIVALENT, 0 ms] (145) QDP (146) TransformationProof [EQUIVALENT, 0 ms] (147) QDP (148) UsableRulesProof [EQUIVALENT, 0 ms] (149) QDP (150) QReductionProof [EQUIVALENT, 0 ms] (151) QDP (152) TransformationProof [EQUIVALENT, 0 ms] (153) QDP (154) TransformationProof [EQUIVALENT, 0 ms] (155) QDP (156) TransformationProof [EQUIVALENT, 0 ms] (157) QDP (158) TransformationProof [EQUIVALENT, 0 ms] (159) QDP (160) DependencyGraphProof [EQUIVALENT, 0 ms] (161) QDP (162) TransformationProof [EQUIVALENT, 0 ms] (163) QDP (164) UsableRulesProof [EQUIVALENT, 0 ms] (165) QDP (166) QReductionProof [EQUIVALENT, 0 ms] (167) QDP (168) TransformationProof [EQUIVALENT, 0 ms] (169) QDP (170) UsableRulesProof [EQUIVALENT, 0 ms] (171) QDP (172) QReductionProof [EQUIVALENT, 0 ms] (173) QDP (174) TransformationProof [EQUIVALENT, 0 ms] (175) QDP (176) DependencyGraphProof [EQUIVALENT, 0 ms] (177) QDP (178) UsableRulesProof [EQUIVALENT, 0 ms] (179) QDP (180) QReductionProof [EQUIVALENT, 0 ms] (181) QDP (182) TransformationProof [EQUIVALENT, 0 ms] (183) QDP (184) TransformationProof [EQUIVALENT, 0 ms] (185) QDP (186) TransformationProof [EQUIVALENT, 0 ms] (187) QDP (188) TransformationProof [EQUIVALENT, 0 ms] (189) QDP (190) TransformationProof [EQUIVALENT, 0 ms] (191) QDP (192) TransformationProof [EQUIVALENT, 0 ms] (193) QDP (194) TransformationProof [EQUIVALENT, 0 ms] (195) QDP (196) TransformationProof [EQUIVALENT, 0 ms] (197) QDP (198) TransformationProof [EQUIVALENT, 0 ms] (199) QDP (200) TransformationProof [EQUIVALENT, 0 ms] (201) QDP (202) TransformationProof [EQUIVALENT, 0 ms] (203) QDP (204) TransformationProof [EQUIVALENT, 0 ms] (205) QDP (206) TransformationProof [EQUIVALENT, 0 ms] (207) QDP (208) TransformationProof [EQUIVALENT, 0 ms] (209) QDP (210) TransformationProof [EQUIVALENT, 0 ms] (211) QDP (212) TransformationProof [EQUIVALENT, 0 ms] (213) QDP (214) TransformationProof [EQUIVALENT, 0 ms] (215) QDP (216) TransformationProof [EQUIVALENT, 0 ms] (217) QDP (218) TransformationProof [EQUIVALENT, 0 ms] (219) QDP (220) TransformationProof [EQUIVALENT, 4 ms] (221) QDP (222) TransformationProof [EQUIVALENT, 0 ms] (223) QDP (224) TransformationProof [EQUIVALENT, 0 ms] (225) QDP (226) QDPQMonotonicMRRProof [EQUIVALENT, 142 ms] (227) QDP (228) DependencyGraphProof [EQUIVALENT, 0 ms] (229) AND (230) QDP (231) UsableRulesProof [EQUIVALENT, 0 ms] (232) QDP (233) QReductionProof [EQUIVALENT, 0 ms] (234) QDP (235) QDPSizeChangeProof [EQUIVALENT, 0 ms] (236) YES (237) QDP (238) UsableRulesProof [EQUIVALENT, 0 ms] (239) QDP (240) QReductionProof [EQUIVALENT, 0 ms] (241) QDP (242) QDPSizeChangeProof [EQUIVALENT, 0 ms] (243) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) Q is empty. ---------------------------------------- (1) Overlay + Local Confluence (EQUIVALENT) The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: EQ(s(x), s(y)) -> EQ(x, y) UNION(edge(x, y, i), h) -> UNION(i, h) REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) REACH(x, y, i, h) -> EQ(x, y) REACH(x, y, i, h) -> ISEMPTY(i) REACH(x, y, i, h) -> EQ(x, from(i)) REACH(x, y, i, h) -> FROM(i) REACH(x, y, i, h) -> EQ(y, to(i)) REACH(x, y, i, h) -> TO(i) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(false, b3, x, y, i, h) -> REST(i) IF3(false, b3, x, y, i, h) -> FROM(i) IF3(false, b3, x, y, i, h) -> TO(i) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> OR(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REST(i) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) IF4(false, x, y, i, h) -> TO(i) IF4(false, x, y, i, h) -> UNION(rest(i), h) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 13 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: UNION(edge(x, y, i), h) -> UNION(i, h) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: UNION(edge(x, y, i), h) -> UNION(i, h) R is empty. The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: UNION(edge(x, y, i), h) -> UNION(i, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *UNION(edge(x, y, i), h) -> UNION(i, h) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: EQ(s(x), s(y)) -> EQ(x, y) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: EQ(s(x), s(y)) -> EQ(x, y) R is empty. The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: EQ(s(x), s(y)) -> EQ(x, y) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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: *EQ(s(x), s(y)) -> EQ(x, y) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) or(true, y) -> true or(false, y) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty reach(x, y, i, h) -> if1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) if1(true, b1, b2, b3, x, y, i, h) -> true if1(false, b1, b2, b3, x, y, i, h) -> if2(b1, b2, b3, x, y, i, h) if2(true, b2, b3, x, y, i, h) -> false if2(false, b2, b3, x, y, i, h) -> if3(b2, b3, x, y, i, h) if3(false, b3, x, y, i, h) -> reach(x, y, rest(i), edge(from(i), to(i), h)) if3(true, b3, x, y, i, h) -> if4(b3, x, y, i, h) if4(true, x, y, i, h) -> true if4(false, x, y, i, h) -> or(reach(x, y, rest(i), h), reach(to(i), y, union(rest(i), h), empty)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) or(true, x0) or(false, x0) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. or(true, x0) or(false, x0) reach(x0, x1, x2, x3) if1(true, x0, x1, x2, x3, x4, x5, x6) if1(false, x0, x1, x2, x3, x4, x5, x6) if2(true, x0, x1, x2, x3, x4, x5) if2(false, x0, x1, x2, x3, x4, x5) if3(false, x0, x1, x2, x3, x4) if3(true, x0, x1, x2, x3, x4) if4(true, x0, x1, x2, x3) if4(false, x0, x1, x2, x3) ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(empty) -> true isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented rules of the TRS R: isEmpty(empty) -> true Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)) = 2*x_2 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 2*x_1 POL(IF3(x_1, x_2, x_3, x_4, x_5, x_6)) = 2 POL(IF4(x_1, x_2, x_3, x_4, x_5)) = 2 POL(REACH(x_1, x_2, x_3, x_4)) = 2 POL(edge(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 POL(empty) = 0 POL(eq(x_1, x_2)) = 2 + 2*x_1 POL(false) = 1 POL(from(x_1)) = 2 + 2*x_1 POL(isEmpty(x_1)) = 1 POL(rest(x_1)) = 2 + 2*x_1 POL(s(x_1)) = x_1 POL(to(x_1)) = x_1 POL(true) = 0 POL(union(x_1, x_2)) = 2*x_1 + 2*x_2 ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule REACH(x, y, i, h) -> IF1(eq(x, y), isEmpty(i), eq(x, from(i)), eq(y, to(i)), x, y, i, h) at position [0] we obtained the following new rules [LPAR04]: (REACH(0, 0, y2, y3) -> IF1(true, isEmpty(y2), eq(0, from(y2)), eq(0, to(y2)), 0, 0, y2, y3),REACH(0, 0, y2, y3) -> IF1(true, isEmpty(y2), eq(0, from(y2)), eq(0, to(y2)), 0, 0, y2, y3)) (REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3),REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3)) (REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3),REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3)) (REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3),REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3)) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) REACH(0, 0, y2, y3) -> IF1(true, isEmpty(y2), eq(0, from(y2)), eq(0, to(y2)), 0, 0, y2, y3) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, x, y, i, h) -> REACH(to(i), y, union(rest(i), h), empty) at position [0] we obtained the following new rules [LPAR04]: (IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(rest(edge(x0, x1, x2)), y3), empty),IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(rest(edge(x0, x1, x2)), y3), empty)) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(rest(edge(x0, x1, x2)), y3), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(rest(edge(x0, x1, x2)), y3), empty) at position [2,0] we obtained the following new rules [LPAR04]: (IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty),IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty)) ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF1(false, b1, b2, b3, x, y, i, h) -> IF2(b1, b2, b3, x, y, i, h) we obtained the following new rules [LPAR04]: (IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2),IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2)) (IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2),IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2)) (IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3),IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3)) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF2(false, b2, b3, x, y, i, h) -> IF3(b2, b3, x, y, i, h) we obtained the following new rules [LPAR04]: (IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5),IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5)) (IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5),IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5)) (IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6),IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6)) ---------------------------------------- (39) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(false, b3, x, y, i, h) -> REACH(x, y, rest(i), edge(from(i), to(i), h)) we obtained the following new rules [LPAR04]: (IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)),IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4))) (IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)),IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4))) (IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)),IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5))) ---------------------------------------- (41) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (42) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(true, b3, x, y, i, h) -> IF4(b3, x, y, i, h) we obtained the following new rules [LPAR04]: (IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4),IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4)) (IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4),IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4)) (IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5),IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5)) ---------------------------------------- (43) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (44) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF4(false, x, y, i, h) -> REACH(x, y, rest(i), h) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3),IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3)) (IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3),IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3)) (IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4),IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4)) ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF4(false, y0, y1, edge(x0, x1, x2), y3) -> REACH(x1, y1, union(x2, y3), empty) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty),IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty)) (IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty),IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty)) (IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty),IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty)) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (49) Complex Obligation (AND) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF1(false, y_0, y_2, y_4, s(z0), 0, z1, z2) -> IF2(y_0, y_2, y_4, s(z0), 0, z1, z2) we obtained the following new rules [LPAR04]: (IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5),IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5)) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule REACH(s(x0), 0, y2, y3) -> IF1(false, isEmpty(y2), eq(s(x0), from(y2)), eq(0, to(y2)), s(x0), 0, y2, y3) at position [1] we obtained the following new rules [LPAR04]: (REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2),REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2)) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. isEmpty(empty) isEmpty(edge(x0, x1, x2)) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), from(edge(x0, x1, x2))), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) at position [2,1] we obtained the following new rules [LPAR04]: (REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2),REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2)) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, to(edge(x0, x1, x2))), s(y0), 0, edge(x0, x1, x2), y2) at position [3,1] we obtained the following new rules [LPAR04]: (REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2),REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2)) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF3(false, z1, s(z2), 0, z3, z4) -> REACH(s(z2), 0, rest(z3), edge(from(z3), to(z3), z4)) at position [2] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)),IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3))) (IF3(false, y0, s(y1), 0, empty, y3) -> REACH(s(y1), 0, empty, edge(from(empty), to(empty), y3)),IF3(false, y0, s(y1), 0, empty, y3) -> REACH(s(y1), 0, empty, edge(from(empty), to(empty), y3))) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) IF3(false, y0, s(y1), 0, empty, y3) -> REACH(s(y1), 0, empty, edge(from(empty), to(empty), y3)) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) at position [3,0] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3)),IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3))) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3)) The TRS R consists of the following rules: from(edge(x, y, i)) -> x eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) to(edge(x, y, i)) -> y eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. from(edge(x0, x1, x2)) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, to(edge(x0, x1, x2)), y3)) at position [3,1] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)),IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3))) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. to(edge(x0, x1, x2)) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, s(z1), 0, z2, z3) -> REACH(s(z1), 0, rest(z2), z3) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2),IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2)) (IF4(false, s(y0), 0, empty, y2) -> REACH(s(y0), 0, empty, y2),IF4(false, s(y0), 0, empty, y2) -> REACH(s(y0), 0, empty, y2)) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF4(false, s(y0), 0, empty, y2) -> REACH(s(y0), 0, empty, y2) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. rest(edge(x0, x1, x2)) rest(empty) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, s(z1), 0, edge(x2, x3, x4), z3) -> REACH(x3, 0, union(x4, z3), empty) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty),IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty)) (IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty),IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty)) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF1(false, false, x1, x2, s(x3), 0, x4, x5) -> IF2(false, x1, x2, s(x3), 0, x4, x5) we obtained the following new rules [LPAR04]: (IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4),IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4)) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF2(false, z1, z2, s(z3), 0, z4, z5) -> IF3(z1, z2, s(z3), 0, z4, z5) we obtained the following new rules [LPAR04]: (IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6),IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6)) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(true, z1, s(z2), 0, z3, z4) -> IF4(z1, s(z2), 0, z3, z4) we obtained the following new rules [LPAR04]: (IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6),IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6)) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, s(y0), 0, edge(x0, x1, x2), y2) -> REACH(s(y0), 0, x2, y2) we obtained the following new rules [LPAR04]: (IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4),IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4)) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(false, y0, s(y1), 0, edge(x0, x1, x2), y3) -> REACH(s(y1), 0, x2, edge(x0, x1, y3)) we obtained the following new rules [LPAR04]: (IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)),IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5))) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, s(y0), 0, edge(y1, y2, empty), x0) -> REACH(y2, 0, x0, empty) we obtained the following new rules [LPAR04]: (IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty),IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty)) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF2(false, z0, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF3(z0, z1, s(z2), 0, edge(z3, z4, z5), z6) we obtained the following new rules [LPAR04]: (IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6),IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6)) (IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6),IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6)) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(true, z1, s(z2), 0, edge(z3, z4, z5), z6) -> IF4(z1, s(z2), 0, edge(z3, z4, z5), z6) we obtained the following new rules [LPAR04]: (IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5),IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5)) (IF3(true, false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) -> IF4(false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)),IF3(true, false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) -> IF4(false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) IF3(true, false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) -> IF4(false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: IF4(false, s(x0), 0, edge(x1, s(y_0), empty), edge(y_1, y_2, y_3)) -> REACH(s(y_0), 0, edge(y_1, y_2, y_3), empty) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)) = 2*x_1 + 2*x_2 + 2*x_6 + x_7 + x_8 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 2*x_1 + 2*x_5 + x_6 + x_7 POL(IF3(x_1, x_2, x_3, x_4, x_5, x_6)) = 2*x_4 + x_5 + x_6 POL(IF4(x_1, x_2, x_3, x_4, x_5)) = 2*x_1 + 2*x_3 + x_4 + x_5 POL(REACH(x_1, x_2, x_3, x_4)) = 2*x_2 + x_3 + x_4 POL(edge(x_1, x_2, x_3)) = 2*x_2 + x_3 POL(empty) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(s(x_1)) = 2 POL(true) = 0 POL(union(x_1, x_2)) = x_1 + x_2 ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) IF3(true, false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) -> IF4(false, s(x1), 0, edge(x2, s(y_2), empty), edge(y_3, y_4, y_5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: IF4(false, s(y0), 0, edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, 0, edge(x0, x1, union(x2, x3)), empty) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)) = 1 + 2*x_1 + 2*x_2 + 2*x_6 + x_7 + x_8 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 1 + 2*x_1 + 2*x_5 + x_6 + x_7 POL(IF3(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + 2*x_4 + x_5 + x_6 POL(IF4(x_1, x_2, x_3, x_4, x_5)) = 1 + 2*x_1 + 2*x_3 + x_4 + x_5 POL(REACH(x_1, x_2, x_3, x_4)) = x_1 + 2*x_2 + x_3 + x_4 POL(edge(x_1, x_2, x_3)) = 2*x_2 + x_3 POL(empty) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(s(x_1)) = 1 POL(true) = 0 POL(union(x_1, x_2)) = x_1 + x_2 ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. union(empty, x0) union(edge(x0, x1, x2), x3) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) 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: *REACH(s(y0), 0, edge(x0, x1, x2), y2) -> IF1(false, false, eq(s(y0), x0), eq(0, x1), s(y0), 0, edge(x0, x1, x2), y2) The graph contains the following edges 1 >= 5, 2 >= 6, 3 >= 7, 4 >= 8 *IF3(true, false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) -> IF4(false, s(x1), 0, edge(x2, x3, edge(y_3, y_4, y_5)), x5) The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5 *IF1(false, false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, s(z0), 0, edge(z1, z2, z3), z4) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6, 8 >= 7 *IF2(false, true, x1, s(x2), 0, edge(x3, x4, x5), x6) -> IF3(true, x1, s(x2), 0, edge(x3, x4, x5), x6) The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6 *IF2(false, false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF3(false, x1, s(x2), 0, edge(x3, x4, edge(y_4, y_5, y_6)), x6) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6 *IF4(false, s(x0), 0, edge(x1, x2, edge(y_1, y_2, y_3)), x4) -> REACH(s(x0), 0, edge(y_1, y_2, y_3), x4) The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3, 5 >= 4 *IF3(false, x0, s(x1), 0, edge(x2, x3, edge(y_1, y_2, y_3)), x5) -> REACH(s(x1), 0, edge(y_1, y_2, y_3), edge(x2, x3, x5)) The graph contains the following edges 3 >= 1, 4 >= 2, 5 > 3 ---------------------------------------- (116) YES ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF1(false, y_0, y_2, y_4, 0, s(z0), z1, z2) -> IF2(y_0, y_2, y_4, 0, s(z0), z1, z2) we obtained the following new rules [LPAR04]: (IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5),IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5)) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule REACH(0, s(x0), y2, y3) -> IF1(false, isEmpty(y2), eq(0, from(y2)), eq(s(x0), to(y2)), 0, s(x0), y2, y3) at position [1] we obtained the following new rules [LPAR04]: (REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, from(edge(x0, x1, x2))), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2),REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, from(edge(x0, x1, x2))), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2)) ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, from(edge(x0, x1, x2))), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, from(edge(x0, x1, x2))), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2) at position [2,1] we obtained the following new rules [LPAR04]: (REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2),REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2)) ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), to(edge(x0, x1, x2))), 0, s(y0), edge(x0, x1, x2), y2) at position [3,1] we obtained the following new rules [LPAR04]: (REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2),REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2)) ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF3(false, z1, 0, s(z2), z3, z4) -> REACH(0, s(z2), rest(z3), edge(from(z3), to(z3), z4)) at position [2] we obtained the following new rules [LPAR04]: (IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)),IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3))) (IF3(false, y0, 0, s(y1), empty, y3) -> REACH(0, s(y1), empty, edge(from(empty), to(empty), y3)),IF3(false, y0, 0, s(y1), empty, y3) -> REACH(0, s(y1), empty, edge(from(empty), to(empty), y3))) ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) IF3(false, y0, 0, s(y1), empty, y3) -> REACH(0, s(y1), empty, edge(from(empty), to(empty), y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y3)) at position [3,0] we obtained the following new rules [LPAR04]: (IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, to(edge(x0, x1, x2)), y3)),IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, to(edge(x0, x1, x2)), y3))) ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, to(edge(x0, x1, x2)), y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, to(edge(x0, x1, x2)), y3)) at position [3,1] we obtained the following new rules [LPAR04]: (IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)),IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3))) ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, 0, s(z1), z2, z3) -> REACH(0, s(z1), rest(z2), z3) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2),IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2)) (IF4(false, 0, s(y0), empty, y2) -> REACH(0, s(y0), empty, y2),IF4(false, 0, s(y0), empty, y2) -> REACH(0, s(y0), empty, y2)) ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), empty, y2) -> REACH(0, s(y0), empty, y2) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF1(false, false, x1, x2, 0, s(x3), x4, x5) -> IF2(false, x1, x2, 0, s(x3), x4, x5) we obtained the following new rules [LPAR04]: (IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4),IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4)) ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (140) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF2(false, z1, z2, 0, s(z3), z4, z5) -> IF3(z1, z2, 0, s(z3), z4, z5) we obtained the following new rules [LPAR04]: (IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6),IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6)) ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(true, z1, 0, s(z2), z3, z4) -> IF4(z1, 0, s(z2), z3, z4) we obtained the following new rules [LPAR04]: (IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6),IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6)) ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (144) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF1(false, y_1, y_3, y_5, s(z0), s(z1), z2, z3) -> IF2(y_1, y_3, y_5, s(z0), s(z1), z2, z3) we obtained the following new rules [LPAR04]: (IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6),IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6)) ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule REACH(s(x0), s(x1), y2, y3) -> IF1(eq(x0, x1), isEmpty(y2), eq(s(x0), from(y2)), eq(s(x1), to(y2)), s(x0), s(x1), y2, y3) at position [1] we obtained the following new rules [LPAR04]: (REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3),REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3)) ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: to(edge(x, y, i)) -> y rest(edge(x, y, i)) -> i rest(empty) -> empty union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) from(edge(x, y, i)) -> x eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) isEmpty(edge(x, y, i)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (148) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) isEmpty(empty) isEmpty(edge(x0, x1, x2)) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (150) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. isEmpty(empty) isEmpty(edge(x0, x1, x2)) ---------------------------------------- (151) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (152) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), from(edge(x0, x1, x2))), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) at position [2,1] we obtained the following new rules [LPAR04]: (REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3),REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3)) ---------------------------------------- (153) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (154) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), to(edge(x0, x1, x2))), s(y0), s(y1), edge(x0, x1, x2), y3) at position [3,1] we obtained the following new rules [LPAR04]: (REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3),REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3)) ---------------------------------------- (155) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (156) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, 0, s(z1), edge(x2, x3, x4), z3) -> REACH(x3, s(z1), union(x4, z3), empty) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty),IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty)) (IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty),IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty)) ---------------------------------------- (157) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (158) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF3(false, z1, s(z2), s(z3), z4, z5) -> REACH(s(z2), s(z3), rest(z4), edge(from(z4), to(z4), z5)) at position [2] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y4)),IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y4))) (IF3(false, y0, s(y1), s(y2), empty, y4) -> REACH(s(y1), s(y2), empty, edge(from(empty), to(empty), y4)),IF3(false, y0, s(y1), s(y2), empty, y4) -> REACH(s(y1), s(y2), empty, edge(from(empty), to(empty), y4))) ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y4)) IF3(false, y0, s(y1), s(y2), empty, y4) -> REACH(s(y1), s(y2), empty, edge(from(empty), to(empty), y4)) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (160) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y4)) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(from(edge(x0, x1, x2)), to(edge(x0, x1, x2)), y4)) at position [3,0] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4)),IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4))) ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4)) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) from(edge(x, y, i)) -> x to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) from(edge(x0, x1, x2)) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. from(edge(x0, x1, x2)) ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (168) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, to(edge(x0, x1, x2)), y4)) at position [3,1] we obtained the following new rules [LPAR04]: (IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)),IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4))) ---------------------------------------- (169) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) The TRS R consists of the following rules: to(edge(x, y, i)) -> y union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (170) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) to(edge(x0, x1, x2)) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (172) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. to(edge(x0, x1, x2)) ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (174) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, s(z1), s(z2), z3, z4) -> REACH(s(z1), s(z2), rest(z3), z4) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3),IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3)) (IF4(false, s(y0), s(y1), empty, y3) -> REACH(s(y0), s(y1), empty, y3),IF4(false, s(y0), s(y1), empty, y3) -> REACH(s(y0), s(y1), empty, y3)) ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF4(false, s(y0), s(y1), empty, y3) -> REACH(s(y0), s(y1), empty, y3) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (176) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) rest(edge(x, y, i)) -> i rest(empty) -> empty The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (178) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) rest(edge(x0, x1, x2)) rest(empty) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (180) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. rest(edge(x0, x1, x2)) rest(empty) ---------------------------------------- (181) Obligation: Q DP problem: The TRS P consists of the following rules: IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (182) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule IF4(false, s(z1), s(z2), edge(x2, x3, x4), z4) -> REACH(x3, s(z2), union(x4, z4), empty) at position [2] we obtained the following new rules [LPAR04]: (IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty),IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty)) (IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty),IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty)) ---------------------------------------- (183) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (184) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF1(false, false, x1, x2, s(x3), s(x4), x5, x6) -> IF2(false, x1, x2, s(x3), s(x4), x5, x6) we obtained the following new rules [LPAR04]: (IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5),IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5)) ---------------------------------------- (185) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (186) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF2(false, z1, z2, s(z3), s(z4), z5, z6) -> IF3(z1, z2, s(z3), s(z4), z5, z6) we obtained the following new rules [LPAR04]: (IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7),IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7)) ---------------------------------------- (187) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (188) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(true, z1, s(z2), s(z3), z4, z5) -> IF4(z1, s(z2), s(z3), z4, z5) we obtained the following new rules [LPAR04]: (IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7),IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7)) ---------------------------------------- (189) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (190) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF2(false, z0, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF3(z0, z1, 0, s(z2), edge(z3, z4, z5), z6) we obtained the following new rules [LPAR04]: (IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6),IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6)) (IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6),IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6)) ---------------------------------------- (191) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (192) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF1(false, false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) -> IF2(false, y_0, y_1, 0, s(z0), edge(z1, z2, z3), z4) we obtained the following new rules [LPAR04]: (IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6),IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6)) (IF1(false, false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6),IF1(false, false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6)) ---------------------------------------- (193) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (194) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule REACH(0, s(y0), edge(x0, x1, x2), y2) -> IF1(false, false, eq(0, x0), eq(s(y0), x1), 0, s(y0), edge(x0, x1, x2), y2) at position [2] we obtained the following new rules [LPAR04]: (REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4),REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4)) (REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4),REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4)) ---------------------------------------- (195) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (196) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF1(false, false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) we obtained the following new rules [LPAR04]: (IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3),IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3)) ---------------------------------------- (197) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (198) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF2(false, true, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(true, x1, 0, s(x2), edge(x3, x4, x5), x6) we obtained the following new rules [LPAR04]: (IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4),IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4)) ---------------------------------------- (199) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (200) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF3(true, z1, 0, s(z2), edge(z3, z4, z5), z6) -> IF4(z1, 0, s(z2), edge(z3, z4, z5), z6) we obtained the following new rules [LPAR04]: (IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4),IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4)) ---------------------------------------- (201) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (202) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF4(false, 0, s(y0), edge(x0, x1, x2), y2) -> REACH(0, s(y0), x2, y2) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4),IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4)) ---------------------------------------- (203) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (204) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF4(false, 0, s(y0), edge(y1, y2, empty), x0) -> REACH(y2, s(y0), x0, empty) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty),IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty)) ---------------------------------------- (205) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (206) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule IF4(false, 0, s(y0), edge(y1, y2, edge(x0, x1, x2)), x3) -> REACH(y2, s(y0), edge(x0, x1, union(x2, x3)), empty) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty),IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty)) ---------------------------------------- (207) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (208) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(false, y0, 0, s(y1), edge(x0, x1, x2), y3) -> REACH(0, s(y1), x2, edge(x0, x1, y3)) we obtained the following new rules [LPAR04]: (IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)),IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5))) (IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)),IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5))) ---------------------------------------- (209) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (210) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, s(y0), s(y1), edge(x0, x1, x2), y3) -> REACH(s(y0), s(y1), x2, y3) we obtained the following new rules [LPAR04]: (IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5),IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5)) ---------------------------------------- (211) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (212) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(false, y0, s(y1), s(y2), edge(x0, x1, x2), y4) -> REACH(s(y1), s(y2), x2, edge(x0, x1, y4)) we obtained the following new rules [LPAR04]: (IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)),IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6))) ---------------------------------------- (213) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (214) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, s(y0), s(y1), edge(y2, y3, empty), x0) -> REACH(y3, s(y1), x0, empty) we obtained the following new rules [LPAR04]: (IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty),IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty)) (IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty),IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty)) (IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty),IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty)) ---------------------------------------- (215) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (216) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF2(false, z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF3(z0, z1, s(z2), s(z3), edge(z4, z5, z6), z7) we obtained the following new rules [LPAR04]: (IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7),IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7)) (IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7),IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7)) ---------------------------------------- (217) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (218) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(true, z1, s(z2), s(z3), edge(z4, z5, z6), z7) -> IF4(z1, s(z2), s(z3), edge(z4, z5, z6), z7) we obtained the following new rules [LPAR04]: (IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6),IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6)) (IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)),IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6))) (IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)),IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4))) (IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)),IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5))) ---------------------------------------- (219) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (220) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF4(z0, 0, s(z1), edge(0, z2, z3), z4) we obtained the following new rules [LPAR04]: (IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4),IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4)) (IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4),IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4)) (IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4),IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4)) ---------------------------------------- (221) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4) IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4) IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (222) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, 0, s(z1), edge(0, z2, z3), z4) -> REACH(0, s(z1), z3, z4) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(x0), edge(0, x1, edge(0, y_1, y_2)), x3) -> REACH(0, s(x0), edge(0, y_1, y_2), x3),IF4(false, 0, s(x0), edge(0, x1, edge(0, y_1, y_2)), x3) -> REACH(0, s(x0), edge(0, y_1, y_2), x3)) (IF4(false, 0, s(x0), edge(0, x1, edge(s(y_1), y_2, y_3)), x3) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), x3),IF4(false, 0, s(x0), edge(0, x1, edge(s(y_1), y_2, y_3)), x3) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), x3)) ---------------------------------------- (223) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4) IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4) IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) IF4(false, 0, s(x0), edge(0, x1, edge(0, y_1, y_2)), x3) -> REACH(0, s(x0), edge(0, y_1, y_2), x3) IF4(false, 0, s(x0), edge(0, x1, edge(s(y_1), y_2, y_3)), x3) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), x3) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (224) TransformationProof (EQUIVALENT) By forward instantiating [JAR06] the rule IF4(false, 0, s(z1), edge(0, z2, empty), z4) -> REACH(z2, s(z1), z4, empty) we obtained the following new rules [LPAR04]: (IF4(false, 0, s(x0), edge(0, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x0), edge(y_2, y_3, y_4), empty),IF4(false, 0, s(x0), edge(0, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x0), edge(y_2, y_3, y_4), empty)) (IF4(false, 0, s(x0), edge(0, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x0), edge(0, y_1, y_2), empty),IF4(false, 0, s(x0), edge(0, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x0), edge(0, y_1, y_2), empty)) (IF4(false, 0, s(x0), edge(0, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), empty),IF4(false, 0, s(x0), edge(0, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), empty)) ---------------------------------------- (225) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4) IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4) IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) IF4(false, 0, s(x0), edge(0, x1, edge(0, y_1, y_2)), x3) -> REACH(0, s(x0), edge(0, y_1, y_2), x3) IF4(false, 0, s(x0), edge(0, x1, edge(s(y_1), y_2, y_3)), x3) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), x3) IF4(false, 0, s(x0), edge(0, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x0), edge(y_2, y_3, y_4), empty) IF4(false, 0, s(x0), edge(0, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x0), edge(0, y_1, y_2), empty) IF4(false, 0, s(x0), edge(0, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (226) QDPQMonotonicMRRProof (EQUIVALENT) By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. Strictly oriented dependency pairs: IF4(false, s(y0), s(y1), edge(y2, y3, edge(x0, x1, x2)), x3) -> REACH(y3, s(y1), edge(x0, x1, union(x2, x3)), empty) IF4(false, s(x0), s(x1), edge(x2, x3, edge(y_2, y_3, y_4)), x5) -> REACH(s(x0), s(x1), edge(y_2, y_3, y_4), x5) IF4(false, s(x0), s(x1), edge(x2, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x1), edge(y_2, y_3, y_4), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x1), edge(0, y_1, y_2), empty) IF4(false, s(x0), s(x1), edge(x2, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), empty) IF3(true, false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) -> IF4(false, s(x1), s(x2), edge(x3, x4, edge(y_4, y_5, y_6)), x6) IF3(true, false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) -> IF4(false, s(x1), s(x2), edge(x3, s(y_3), empty), edge(y_4, y_5, y_6)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(0, y_3, y_4)) IF3(true, false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) -> IF4(false, s(x1), s(x2), edge(x3, 0, empty), edge(s(y_3), y_4, y_5)) IF3(true, false, 0, s(x1), edge(0, x2, x3), x4) -> IF4(false, 0, s(x1), edge(0, x2, x3), x4) IF3(true, false, 0, s(x1), edge(0, x2, empty), x4) -> IF4(false, 0, s(x1), edge(0, x2, empty), x4) IF3(true, false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) -> IF4(false, 0, s(x1), edge(0, x2, edge(y_2, y_3, y_4)), x4) Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(IF1(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)) = 2 + 2*x_2 + x_7 + x_8 POL(IF2(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 2 + 2*x_1 + x_6 + x_7 POL(IF3(x_1, x_2, x_3, x_4, x_5, x_6)) = 2 + x_5 + x_6 POL(IF4(x_1, x_2, x_3, x_4, x_5)) = 2*x_1 + x_2 + x_4 + x_5 POL(REACH(x_1, x_2, x_3, x_4)) = 1 + x_2 + x_3 + x_4 POL(edge(x_1, x_2, x_3)) = 2 + x_3 POL(empty) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(s(x_1)) = 1 POL(true) = 0 POL(union(x_1, x_2)) = x_1 + x_2 ---------------------------------------- (227) Obligation: Q DP problem: The TRS P consists of the following rules: REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) REACH(0, s(y0), edge(0, y2, y3), y4) -> IF1(false, false, true, eq(s(y0), y2), 0, s(y0), edge(0, y2, y3), y4) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) -> IF2(false, true, y_0, 0, s(z0), edge(0, z1, z2), z3) IF2(false, true, z0, 0, s(z1), edge(0, z2, z3), z4) -> IF3(true, z0, 0, s(z1), edge(0, z2, z3), z4) IF4(false, 0, s(z1), edge(0, z2, edge(x3, x4, x5)), z4) -> REACH(z2, s(z1), edge(x3, x4, union(x5, z4)), empty) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(0, y_1, y_2)), x5) -> REACH(0, s(x1), edge(0, y_1, y_2), edge(x2, x3, x5)) IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) IF2(false, true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) -> IF3(true, x1, s(x2), s(x3), edge(x4, x5, x6), x7) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF4(false, 0, s(x0), edge(0, x1, edge(0, y_1, y_2)), x3) -> REACH(0, s(x0), edge(0, y_1, y_2), x3) IF4(false, 0, s(x0), edge(0, x1, edge(s(y_1), y_2, y_3)), x3) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), x3) IF4(false, 0, s(x0), edge(0, s(y_0), empty), edge(y_2, y_3, y_4)) -> REACH(s(y_0), s(x0), edge(y_2, y_3, y_4), empty) IF4(false, 0, s(x0), edge(0, 0, empty), edge(0, y_1, y_2)) -> REACH(0, s(x0), edge(0, y_1, y_2), empty) IF4(false, 0, s(x0), edge(0, 0, empty), edge(s(y_1), y_2, y_3)) -> REACH(0, s(x0), edge(s(y_1), y_2, y_3), empty) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (228) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 11 less nodes. ---------------------------------------- (229) Complex Obligation (AND) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. union(empty, x0) union(edge(x0, x1, x2), x3) ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) The TRS R consists of the following rules: eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) eq(0, 0) -> true eq(0, s(x)) -> false The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) 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: *REACH(0, s(y0), edge(s(x0), y2, y3), y4) -> IF1(false, false, false, eq(s(y0), y2), 0, s(y0), edge(s(x0), y2, y3), y4) The graph contains the following edges 1 >= 5, 2 >= 6, 3 >= 7, 4 >= 8 *IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF3(false, x1, 0, s(x2), edge(x3, x4, x5), x6) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6 *IF1(false, false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) -> IF2(false, false, x1, 0, s(x2), edge(x3, x4, x5), x6) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 1, 1 >= 2, 2 >= 2, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6, 8 >= 7 *IF3(false, x0, 0, s(x1), edge(x2, x3, edge(s(y_1), y_2, y_3)), x5) -> REACH(0, s(x1), edge(s(y_1), y_2, y_3), edge(x2, x3, x5)) The graph contains the following edges 3 >= 1, 4 >= 2, 5 > 3 ---------------------------------------- (236) YES ---------------------------------------- (237) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: union(empty, h) -> h union(edge(x, y, i), h) -> edge(x, y, union(i, h)) eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (238) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (239) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) union(empty, x0) union(edge(x0, x1, x2), x3) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (240) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. union(empty, x0) union(edge(x0, x1, x2), x3) ---------------------------------------- (241) Obligation: Q DP problem: The TRS P consists of the following rules: IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) The TRS R consists of the following rules: eq(0, 0) -> true eq(0, s(x)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) The set Q consists of the following terms: eq(0, 0) eq(0, s(x0)) eq(s(x0), 0) eq(s(x0), s(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (242) 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: *IF2(false, false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) -> IF3(false, x1, s(x2), s(x3), edge(x4, x5, edge(y_5, y_6, y_7)), x7) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6 *REACH(s(y0), s(y1), edge(x0, x1, x2), y3) -> IF1(eq(y0, y1), false, eq(s(y0), x0), eq(s(y1), x1), s(y0), s(y1), edge(x0, x1, x2), y3) The graph contains the following edges 1 >= 5, 2 >= 6, 3 >= 7, 4 >= 8 *IF3(false, x0, s(x1), s(x2), edge(x3, x4, edge(y_2, y_3, y_4)), x6) -> REACH(s(x1), s(x2), edge(y_2, y_3, y_4), edge(x3, x4, x6)) The graph contains the following edges 3 >= 1, 4 >= 2, 5 > 3 *IF1(false, false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) -> IF2(false, y_1, y_2, s(z0), s(z1), edge(z2, z3, z4), z5) The graph contains the following edges 1 >= 1, 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6, 8 >= 7 ---------------------------------------- (243) YES