4.52/1.94 YES 4.90/2.04 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.90/2.04 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.90/2.04 4.90/2.04 4.90/2.04 Left Termination of the query pattern 4.90/2.04 4.90/2.04 insert(g,g,a) 4.90/2.04 4.90/2.04 w.r.t. the given Prolog program could successfully be proven: 4.90/2.04 4.90/2.04 (0) Prolog 4.90/2.04 (1) PrologToPiTRSProof [SOUND, 0 ms] 4.90/2.04 (2) PiTRS 4.90/2.04 (3) DependencyPairsProof [EQUIVALENT, 17 ms] 4.90/2.04 (4) PiDP 4.90/2.04 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.90/2.04 (6) AND 4.90/2.04 (7) PiDP 4.90/2.04 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.90/2.04 (9) PiDP 4.90/2.04 (10) PiDPToQDPProof [EQUIVALENT, 12 ms] 4.90/2.04 (11) QDP 4.90/2.04 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.90/2.04 (13) YES 4.90/2.04 (14) PiDP 4.90/2.04 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.90/2.04 (16) PiDP 4.90/2.04 (17) PiDPToQDPProof [SOUND, 0 ms] 4.90/2.04 (18) QDP 4.90/2.04 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.90/2.04 (20) YES 4.90/2.04 4.90/2.04 4.90/2.04 ---------------------------------------- 4.90/2.04 4.90/2.04 (0) 4.90/2.04 Obligation: 4.90/2.04 Clauses: 4.90/2.04 4.90/2.04 insert(X, void, tree(X, void, void)). 4.90/2.04 insert(X, tree(X, Left, Right), tree(X, Left, Right)). 4.90/2.04 insert(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), insert(X, Left, Left1)). 4.90/2.05 insert(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), insert(X, Right, Right1)). 4.90/2.05 less(0, s(X1)). 4.90/2.05 less(s(X), s(Y)) :- less(X, Y). 4.90/2.05 4.90/2.05 4.90/2.05 Query: insert(g,g,a) 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (1) PrologToPiTRSProof (SOUND) 4.90/2.05 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.90/2.05 4.90/2.05 insert_in_3: (b,b,f) 4.90/2.05 4.90/2.05 less_in_2: (b,b) 4.90/2.05 4.90/2.05 Transforming Prolog into the following Term Rewriting System: 4.90/2.05 4.90/2.05 Pi-finite rewrite system: 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 4.90/2.05 4.90/2.05 4.90/2.05 4.90/2.05 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.90/2.05 4.90/2.05 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (2) 4.90/2.05 Obligation: 4.90/2.05 Pi-finite rewrite system: 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (3) DependencyPairsProof (EQUIVALENT) 4.90/2.05 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 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)) 4.90/2.05 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) 4.90/2.05 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)) 4.90/2.05 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 4.90/2.05 4.90/2.05 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 4.90/2.05 4.90/2.05 U5_GG(x1, x2, x3) = U5_GG(x3) 4.90/2.05 4.90/2.05 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x3, x6) 4.90/2.05 4.90/2.05 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (4) 4.90/2.05 Obligation: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> LESS_IN_GG(X, Y) 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 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)) 4.90/2.05 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> LESS_IN_GG(Y, X) 4.90/2.05 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)) 4.90/2.05 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 4.90/2.05 4.90/2.05 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 4.90/2.05 4.90/2.05 U5_GG(x1, x2, x3) = U5_GG(x3) 4.90/2.05 4.90/2.05 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x2, x3, x6) 4.90/2.05 4.90/2.05 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (5) DependencyGraphProof (EQUIVALENT) 4.90/2.05 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (6) 4.90/2.05 Complex Obligation (AND) 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (7) 4.90/2.05 Obligation: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 4.90/2.05 4.90/2.05 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (8) UsableRulesProof (EQUIVALENT) 4.90/2.05 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (9) 4.90/2.05 Obligation: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 4.90/2.05 R is empty. 4.90/2.05 Pi is empty. 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (10) PiDPToQDPProof (EQUIVALENT) 4.90/2.05 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (11) 4.90/2.05 Obligation: 4.90/2.05 Q DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 4.90/2.05 R is empty. 4.90/2.05 Q is empty. 4.90/2.05 We have to consider all (P,Q,R)-chains. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (12) QDPSizeChangeProof (EQUIVALENT) 4.90/2.05 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. 4.90/2.05 4.90/2.05 From the DPs we obtained the following set of size-change graphs: 4.90/2.05 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 4.90/2.05 The graph contains the following edges 1 > 1, 2 > 2 4.90/2.05 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (13) 4.90/2.05 YES 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (14) 4.90/2.05 Obligation: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 insert_in_gga(X, void, tree(X, void, void)) -> insert_out_gga(X, void, tree(X, void, void)) 4.90/2.05 insert_in_gga(X, tree(X, Left, Right), tree(X, Left, Right)) -> insert_out_gga(X, tree(X, Left, Right), tree(X, Left, Right)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_gga(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 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)) 4.90/2.05 insert_in_gga(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_gga(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 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)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 insert_in_gga(x1, x2, x3) = insert_in_gga(x1, x2) 4.90/2.05 4.90/2.05 void = void 4.90/2.05 4.90/2.05 insert_out_gga(x1, x2, x3) = insert_out_gga(x3) 4.90/2.05 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x4, x6) 4.90/2.05 4.90/2.05 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x2, x3, x6) 4.90/2.05 4.90/2.05 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 4.90/2.05 4.90/2.05 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (15) UsableRulesProof (EQUIVALENT) 4.90/2.05 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (16) 4.90/2.05 Obligation: 4.90/2.05 Pi DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 U1_GGA(X, Y, Left, Right, Left1, less_out_gg(X, Y)) -> INSERT_IN_GGA(X, Left, Left1) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left1, Right)) -> U1_GGA(X, Y, Left, Right, Left1, less_in_gg(X, Y)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right), tree(Y, Left, Right1)) -> U3_GGA(X, Y, Left, Right, Right1, less_in_gg(Y, X)) 4.90/2.05 U3_GGA(X, Y, Left, Right, Right1, less_out_gg(Y, X)) -> INSERT_IN_GGA(X, Right, Right1) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg(0, s(X1)) 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 4.90/2.05 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 4.90/2.05 4.90/2.05 The argument filtering Pi contains the following mapping: 4.90/2.05 tree(x1, x2, x3) = tree(x1, x2, x3) 4.90/2.05 4.90/2.05 less_in_gg(x1, x2) = less_in_gg(x1, x2) 4.90/2.05 4.90/2.05 0 = 0 4.90/2.05 4.90/2.05 s(x1) = s(x1) 4.90/2.05 4.90/2.05 less_out_gg(x1, x2) = less_out_gg 4.90/2.05 4.90/2.05 U5_gg(x1, x2, x3) = U5_gg(x3) 4.90/2.05 4.90/2.05 INSERT_IN_GGA(x1, x2, x3) = INSERT_IN_GGA(x1, x2) 4.90/2.05 4.90/2.05 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 4.90/2.05 4.90/2.05 4.90/2.05 We have to consider all (P,R,Pi)-chains 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (17) PiDPToQDPProof (SOUND) 4.90/2.05 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (18) 4.90/2.05 Obligation: 4.90/2.05 Q DP problem: 4.90/2.05 The TRS P consists of the following rules: 4.90/2.05 4.90/2.05 U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) 4.90/2.05 INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) 4.90/2.05 U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) 4.90/2.05 4.90/2.05 The TRS R consists of the following rules: 4.90/2.05 4.90/2.05 less_in_gg(0, s(X1)) -> less_out_gg 4.90/2.05 less_in_gg(s(X), s(Y)) -> U5_gg(less_in_gg(X, Y)) 4.90/2.05 U5_gg(less_out_gg) -> less_out_gg 4.90/2.05 4.90/2.05 The set Q consists of the following terms: 4.90/2.05 4.90/2.05 less_in_gg(x0, x1) 4.90/2.05 U5_gg(x0) 4.90/2.05 4.90/2.05 We have to consider all (P,Q,R)-chains. 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (19) QDPSizeChangeProof (EQUIVALENT) 4.90/2.05 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. 4.90/2.05 4.90/2.05 From the DPs we obtained the following set of size-change graphs: 4.90/2.05 *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U1_GGA(X, Y, Left, Right, less_in_gg(X, Y)) 4.90/2.05 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 4.90/2.05 4.90/2.05 4.90/2.05 *INSERT_IN_GGA(X, tree(Y, Left, Right)) -> U3_GGA(X, Y, Left, Right, less_in_gg(Y, X)) 4.90/2.05 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4 4.90/2.05 4.90/2.05 4.90/2.05 *U1_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Left) 4.90/2.05 The graph contains the following edges 1 >= 1, 3 >= 2 4.90/2.05 4.90/2.05 4.90/2.05 *U3_GGA(X, Y, Left, Right, less_out_gg) -> INSERT_IN_GGA(X, Right) 4.90/2.05 The graph contains the following edges 1 >= 1, 4 >= 2 4.90/2.05 4.90/2.05 4.90/2.05 ---------------------------------------- 4.90/2.05 4.90/2.05 (20) 4.90/2.05 YES 4.98/2.08 EOF