5.47/2.23 YES 5.47/2.26 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.47/2.26 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.47/2.26 5.47/2.26 5.47/2.26 Left Termination of the query pattern 5.47/2.26 5.47/2.26 insert(a,g,a) 5.47/2.26 5.47/2.26 w.r.t. the given Prolog program could successfully be proven: 5.47/2.26 5.47/2.26 (0) Prolog 5.47/2.26 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.47/2.26 (2) PiTRS 5.47/2.26 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.47/2.26 (4) PiDP 5.47/2.26 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.47/2.26 (6) AND 5.47/2.26 (7) PiDP 5.47/2.26 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.47/2.26 (9) PiDP 5.47/2.26 (10) PiDPToQDPProof [SOUND, 3 ms] 5.47/2.26 (11) QDP 5.47/2.26 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.47/2.26 (13) YES 5.47/2.26 (14) PiDP 5.47/2.26 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.47/2.26 (16) PiDP 5.47/2.26 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.47/2.26 (18) QDP 5.47/2.26 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.47/2.26 (20) YES 5.47/2.26 (21) PiDP 5.47/2.26 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.47/2.26 (23) PiDP 5.47/2.26 (24) PiDPToQDPProof [SOUND, 0 ms] 5.47/2.26 (25) QDP 5.47/2.26 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.47/2.26 (27) YES 5.47/2.26 (28) PiDP 5.47/2.26 (29) UsableRulesProof [EQUIVALENT, 0 ms] 5.47/2.26 (30) PiDP 5.47/2.26 (31) PiDPToQDPProof [SOUND, 0 ms] 5.47/2.26 (32) QDP 5.47/2.26 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.47/2.26 (34) YES 5.47/2.26 (35) PiDP 5.47/2.26 (36) UsableRulesProof [EQUIVALENT, 0 ms] 5.47/2.26 (37) PiDP 5.47/2.26 (38) PiDPToQDPProof [SOUND, 0 ms] 5.47/2.26 (39) QDP 5.47/2.26 (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.47/2.26 (41) YES 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (0) 5.47/2.26 Obligation: 5.47/2.26 Clauses: 5.47/2.26 5.47/2.26 insert(X, void, tree(X, void, void)). 5.47/2.26 insert(X, tree(X, Left, Right), tree(X, Left, Right)). 5.47/2.26 insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)). 5.47/2.26 insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)). 5.47/2.26 less(0, s(X1)). 5.47/2.26 less(s(X), s(Y)) :- less(X, Y). 5.47/2.26 5.47/2.26 5.47/2.26 Query: insert(a,g,a) 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (1) PrologToPiTRSProof (SOUND) 5.47/2.26 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.47/2.26 5.47/2.26 insert_in_3: (f,b,f) (b,b,f) 5.47/2.26 5.47/2.26 less_in_2: (f,b) (b,b) (b,f) 5.47/2.26 5.47/2.26 Transforming Prolog into the following Term Rewriting System: 5.47/2.26 5.47/2.26 Pi-finite rewrite system: 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 5.47/2.26 5.47/2.26 5.47/2.26 5.47/2.26 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.47/2.26 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (2) 5.47/2.26 Obligation: 5.47/2.26 Pi-finite rewrite system: 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (3) DependencyPairsProof (EQUIVALENT) 5.47/2.26 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y) 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.47/2.26 U1_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_AGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 U1_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_GGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_GGA(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X) 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_AGA(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> INSERT_IN_AGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 INSERT_IN_AGA(x1, x2, x3) = INSERT_IN_AGA(x2) 5.47/2.26 5.47/2.26 U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x3, x6) 5.47/2.26 5.47/2.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.47/2.26 5.47/2.26 U5_AG(x1, x2, x3) = U5_AG(x3) 5.47/2.26 5.47/2.26 U2_AGA(x1, x2, x3, x4, x5, x6) = U2_AGA(x6) 5.47/2.26 5.47/2.26 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 5.47/2.26 5.47/2.26 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.47/2.26 5.47/2.26 U5_GG(x1, x2, x3) = U5_GG(x3) 5.47/2.26 5.47/2.26 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_AGA(x1, x2, x3, x4, x5, x6) = U3_AGA(x4, x6) 5.47/2.26 5.47/2.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.47/2.26 5.47/2.26 U5_GA(x1, x2, x3) = U5_GA(x3) 5.47/2.26 5.47/2.26 U4_AGA(x1, x2, x3, x4, x5, x6) = U4_AGA(x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (4) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_AGA(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_AG(X, Y) 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.47/2.26 U1_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_AGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 U1_AGA(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_GGA(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_GGA(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GA(Y, X) 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_AGA(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> INSERT_IN_AGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 INSERT_IN_AGA(x1, x2, x3) = INSERT_IN_AGA(x2) 5.47/2.26 5.47/2.26 U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x3, x6) 5.47/2.26 5.47/2.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.47/2.26 5.47/2.26 U5_AG(x1, x2, x3) = U5_AG(x3) 5.47/2.26 5.47/2.26 U2_AGA(x1, x2, x3, x4, x5, x6) = U2_AGA(x6) 5.47/2.26 5.47/2.26 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 5.47/2.26 5.47/2.26 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.47/2.26 5.47/2.26 U5_GG(x1, x2, x3) = U5_GG(x3) 5.47/2.26 5.47/2.26 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_AGA(x1, x2, x3, x4, x5, x6) = U3_AGA(x4, x6) 5.47/2.26 5.47/2.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.47/2.26 5.47/2.26 U5_GA(x1, x2, x3) = U5_GA(x3) 5.47/2.26 5.47/2.26 U4_AGA(x1, x2, x3, x4, x5, x6) = U4_AGA(x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (5) DependencyGraphProof (EQUIVALENT) 5.47/2.26 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (6) 5.47/2.26 Complex Obligation (AND) 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (7) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (8) UsableRulesProof (EQUIVALENT) 5.47/2.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (9) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (10) PiDPToQDPProof (SOUND) 5.47/2.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (11) 5.47/2.26 Obligation: 5.47/2.26 Q DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 Q is empty. 5.47/2.26 We have to consider all (P,Q,R)-chains. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (12) QDPSizeChangeProof (EQUIVALENT) 5.47/2.26 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. 5.47/2.26 5.47/2.26 From the DPs we obtained the following set of size-change graphs: 5.47/2.26 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 5.47/2.26 The graph contains the following edges 1 > 1 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (13) 5.47/2.26 YES 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (14) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (15) UsableRulesProof (EQUIVALENT) 5.47/2.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (16) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 Pi is empty. 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (17) PiDPToQDPProof (EQUIVALENT) 5.47/2.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (18) 5.47/2.26 Obligation: 5.47/2.26 Q DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 Q is empty. 5.47/2.26 We have to consider all (P,Q,R)-chains. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (19) QDPSizeChangeProof (EQUIVALENT) 5.47/2.26 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. 5.47/2.26 5.47/2.26 From the DPs we obtained the following set of size-change graphs: 5.47/2.26 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.47/2.26 The graph contains the following edges 1 > 1, 2 > 2 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (20) 5.47/2.26 YES 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (21) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 5.47/2.26 5.47/2.26 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (22) UsableRulesProof (EQUIVALENT) 5.47/2.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (23) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 5.47/2.26 5.47/2.26 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (24) PiDPToQDPProof (SOUND) 5.47/2.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (25) 5.47/2.26 Obligation: 5.47/2.26 Q DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) 5.47/2.26 INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) 5.47/2.26 U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(less_in_gg(X, Y)) 5.47/2.26 U5_gg(less_out_gg) -> less_out_gg 5.47/2.26 5.47/2.26 The set Q consists of the following terms: 5.47/2.26 5.47/2.26 less_in_gg(x0, x1) 5.47/2.26 U5_gg(x0) 5.47/2.26 5.47/2.26 We have to consider all (P,Q,R)-chains. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (26) QDPSizeChangeProof (EQUIVALENT) 5.47/2.26 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. 5.47/2.26 5.47/2.26 From the DPs we obtained the following set of size-change graphs: 5.47/2.26 *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) 5.47/2.26 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 5.47/2.26 5.47/2.26 5.47/2.26 *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) 5.47/2.26 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 5.47/2.26 5.47/2.26 5.47/2.26 *U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) 5.47/2.26 The graph contains the following edges 1 >= 1, 3 >= 2 5.47/2.26 5.47/2.26 5.47/2.26 *U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) 5.47/2.26 The graph contains the following edges 1 >= 1, 4 >= 2 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (27) 5.47/2.26 YES 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (28) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (29) UsableRulesProof (EQUIVALENT) 5.47/2.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (30) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (31) PiDPToQDPProof (SOUND) 5.47/2.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (32) 5.47/2.26 Obligation: 5.47/2.26 Q DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 5.47/2.26 5.47/2.26 R is empty. 5.47/2.26 Q is empty. 5.47/2.26 We have to consider all (P,Q,R)-chains. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (33) QDPSizeChangeProof (EQUIVALENT) 5.47/2.26 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. 5.47/2.26 5.47/2.26 From the DPs we obtained the following set of size-change graphs: 5.47/2.26 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 5.47/2.26 The graph contains the following edges 1 > 1 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (34) 5.47/2.26 YES 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (35) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> INSERT_IN_AGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 insert_in_aga(X, void, tree(X, void, void)) -> insert_out_aga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_aga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_aga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_aga(X, Y, Left, Right, Left1, less_in_ag(X, Y)) 5.47/2.26 less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1)) 5.47/2.26 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.47/2.26 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.47/2.26 U1_aga(X, Y, Left, Right, Left1, less_out_ag(X, Y)) -> U2_aga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 5.47/2.26 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 5.47/2.26 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 5.47/2.26 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.47/2.26 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.47/2.26 U1_gga(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> U2_gga(X, Y, Left, Right, Left1, insert_in_gga(X, Left, Left1)) 5.47/2.26 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 5.47/2.26 U3_gga(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> U4_gga(X, Y, Left, Right, Right1, insert_in_gga(X, Right, Right1)) 5.47/2.26 U4_gga(X, Y, Left, Right, Right1, insert_out_gga(X, Right, Right1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 U2_gga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 U2_aga(X, Y, Left, Right, Left1, insert_out_gga(X, Left, Left1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) 5.47/2.26 insert_in_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_aga(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 U3_aga(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> U4_aga(X, Y, Left, Right, Right1, insert_in_aga(X, Right, Right1)) 5.47/2.26 U4_aga(X, Y, Left, Right, Right1, insert_out_aga(X, Right, Right1)) -> insert_out_aga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 insert_in_aga(x1, x2, x3) = insert_in_aga(x2) 5.47/2.26 5.47/2.26 void = void 5.47/2.26 5.47/2.26 insert_out_aga(x1, x2, x3) = insert_out_aga 5.47/2.26 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6) 5.47/2.26 5.47/2.26 less_in_ag(x1, x2) = less_in_ag(x2) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 less_out_ag(x1, x2) = less_out_ag(x1) 5.47/2.26 5.47/2.26 U5_ag(x1, x2, x3) = U5_ag(x3) 5.47/2.26 5.47/2.26 U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x6) 5.47/2.26 5.47/2.26 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 5.47/2.26 5.47/2.26 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 5.47/2.26 5.47/2.26 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_out_gg(x1, x2) = less_out_gg 5.47/2.26 5.47/2.26 U5_gg(x1, x2, x3) = U5_gg(x3) 5.47/2.26 5.47/2.26 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 5.47/2.26 5.47/2.26 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.47/2.26 5.47/2.26 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 5.47/2.26 5.47/2.26 U3_aga(x1, x2, x3, x4, x5, x6) = U3_aga(x4, x6) 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 U4_aga(x1, x2, x3, x4, x5, x6) = U4_aga(x6) 5.47/2.26 5.47/2.26 INSERT_IN_AGA(x1, x2, x3) = INSERT_IN_AGA(x2) 5.47/2.26 5.47/2.26 U3_AGA(x1, x2, x3, x4, x5, x6) = U3_AGA(x4, x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (36) UsableRulesProof (EQUIVALENT) 5.47/2.26 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (37) 5.47/2.26 Obligation: 5.47/2.26 Pi DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 INSERT_IN_AGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_AGA(X, Y, Left, Right, Right1, less_in_ga(Y, X)) 5.47/2.26 U3_AGA(X, Y, Left, Right, Right1, less_out_ga(Y, X)) -> INSERT_IN_AGA(X, Right, Right1) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 less_in_ga(0, s(X1)) -> less_out_ga(0, s(X1)) 5.47/2.26 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.47/2.26 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.47/2.26 5.47/2.26 The argument filtering Pi contains the following mapping: 5.47/2.26 tree(x1, x2, x3) = tree(x1, x2, x3) 5.47/2.26 5.47/2.26 s(x1) = s(x1) 5.47/2.26 5.47/2.26 0 = 0 5.47/2.26 5.47/2.26 less_in_ga(x1, x2) = less_in_ga(x1) 5.47/2.26 5.47/2.26 less_out_ga(x1, x2) = less_out_ga 5.47/2.26 5.47/2.26 U5_ga(x1, x2, x3) = U5_ga(x3) 5.47/2.26 5.47/2.26 INSERT_IN_AGA(x1, x2, x3) = INSERT_IN_AGA(x2) 5.47/2.26 5.47/2.26 U3_AGA(x1, x2, x3, x4, x5, x6) = U3_AGA(x4, x6) 5.47/2.26 5.47/2.26 5.47/2.26 We have to consider all (P,R,Pi)-chains 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (38) PiDPToQDPProof (SOUND) 5.47/2.26 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (39) 5.47/2.26 Obligation: 5.47/2.26 Q DP problem: 5.47/2.26 The TRS P consists of the following rules: 5.47/2.26 5.47/2.26 INSERT_IN_AGA(tree(Y, Left, Right)) -> U3_AGA(Right, less_in_ga(Y)) 5.47/2.26 U3_AGA(Right, less_out_ga) -> INSERT_IN_AGA(Right) 5.47/2.26 5.47/2.26 The TRS R consists of the following rules: 5.47/2.26 5.47/2.26 less_in_ga(0) -> less_out_ga 5.47/2.26 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 5.47/2.26 U5_ga(less_out_ga) -> less_out_ga 5.47/2.26 5.47/2.26 The set Q consists of the following terms: 5.47/2.26 5.47/2.26 less_in_ga(x0) 5.47/2.26 U5_ga(x0) 5.47/2.26 5.47/2.26 We have to consider all (P,Q,R)-chains. 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (40) QDPSizeChangeProof (EQUIVALENT) 5.47/2.26 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. 5.47/2.26 5.47/2.26 From the DPs we obtained the following set of size-change graphs: 5.47/2.26 *U3_AGA(Right, less_out_ga) -> INSERT_IN_AGA(Right) 5.47/2.26 The graph contains the following edges 1 >= 1 5.47/2.26 5.47/2.26 5.47/2.26 *INSERT_IN_AGA(tree(Y, Left, Right)) -> U3_AGA(Right, less_in_ga(Y)) 5.47/2.26 The graph contains the following edges 1 > 1 5.47/2.26 5.47/2.26 5.47/2.26 ---------------------------------------- 5.47/2.26 5.47/2.26 (41) 5.47/2.26 YES 5.66/2.29 EOF