5.32/2.18 YES 5.32/2.19 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 5.32/2.19 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.32/2.19 5.32/2.19 5.32/2.19 Left Termination of the query pattern 5.32/2.19 5.32/2.19 in(a,g) 5.32/2.19 5.32/2.19 w.r.t. the given Prolog program could successfully be proven: 5.32/2.19 5.32/2.19 (0) Prolog 5.32/2.19 (1) PrologToPiTRSProof [SOUND, 30 ms] 5.32/2.19 (2) PiTRS 5.32/2.19 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 5.32/2.19 (4) PiDP 5.32/2.19 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.32/2.19 (6) AND 5.32/2.19 (7) PiDP 5.32/2.19 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.19 (9) PiDP 5.32/2.19 (10) PiDPToQDPProof [SOUND, 0 ms] 5.32/2.19 (11) QDP 5.32/2.19 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.32/2.19 (13) YES 5.32/2.19 (14) PiDP 5.32/2.19 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.19 (16) PiDP 5.32/2.19 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.32/2.19 (18) QDP 5.32/2.19 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.32/2.19 (20) YES 5.32/2.19 (21) PiDP 5.32/2.19 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.19 (23) PiDP 5.32/2.19 (24) PiDPToQDPProof [SOUND, 0 ms] 5.32/2.19 (25) QDP 5.32/2.19 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.32/2.19 (27) YES 5.32/2.19 (28) PiDP 5.32/2.19 (29) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.19 (30) PiDP 5.32/2.19 (31) PiDPToQDPProof [SOUND, 0 ms] 5.32/2.19 (32) QDP 5.32/2.19 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.32/2.19 (34) YES 5.32/2.19 (35) PiDP 5.32/2.19 (36) UsableRulesProof [EQUIVALENT, 0 ms] 5.32/2.19 (37) PiDP 5.32/2.19 (38) PiDPToQDPProof [SOUND, 0 ms] 5.32/2.19 (39) QDP 5.32/2.19 (40) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.32/2.19 (41) YES 5.32/2.19 5.32/2.19 5.32/2.19 ---------------------------------------- 5.32/2.19 5.32/2.19 (0) 5.32/2.19 Obligation: 5.32/2.19 Clauses: 5.32/2.19 5.32/2.19 in(X, tree(X, X1, X2)). 5.32/2.19 in(X, tree(Y, Left, X3)) :- ','(less(X, Y), in(X, Left)). 5.32/2.19 in(X, tree(Y, X4, Right)) :- ','(less(Y, X), in(X, Right)). 5.32/2.19 less(0, s(X5)). 5.32/2.19 less(s(X), s(Y)) :- less(X, Y). 5.32/2.19 5.32/2.19 5.32/2.19 Query: in(a,g) 5.32/2.19 ---------------------------------------- 5.32/2.19 5.32/2.19 (1) PrologToPiTRSProof (SOUND) 5.32/2.19 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.32/2.19 5.32/2.19 in_in_2: (f,b) (b,b) 5.32/2.19 5.32/2.19 less_in_2: (f,b) (b,b) (b,f) 5.32/2.19 5.32/2.19 Transforming Prolog into the following Term Rewriting System: 5.32/2.19 5.32/2.19 Pi-finite rewrite system: 5.32/2.19 The TRS R consists of the following rules: 5.32/2.19 5.32/2.19 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.19 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.19 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.19 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.19 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.19 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.19 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.19 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.19 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.19 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.19 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.19 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.19 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.19 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.19 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.19 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.19 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.19 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.19 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.19 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.19 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.19 5.32/2.19 The argument filtering Pi contains the following mapping: 5.32/2.19 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.19 5.32/2.19 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.19 5.32/2.19 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.19 5.32/2.19 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.19 5.32/2.19 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.19 5.32/2.19 s(x1) = s(x1) 5.32/2.19 5.32/2.19 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.19 5.32/2.19 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.19 5.32/2.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.19 5.32/2.19 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.19 5.32/2.19 in_out_gg(x1, x2) = in_out_gg 5.32/2.19 5.32/2.19 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.19 5.32/2.19 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.19 5.32/2.19 0 = 0 5.32/2.19 5.32/2.19 less_out_gg(x1, x2) = less_out_gg 5.32/2.19 5.32/2.19 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.19 5.32/2.19 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.19 5.32/2.19 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.19 5.32/2.19 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.19 5.32/2.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.19 5.32/2.19 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.19 5.32/2.19 less_out_ga(x1, x2) = less_out_ga 5.32/2.19 5.32/2.19 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.19 5.32/2.19 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.19 5.32/2.19 5.32/2.19 5.32/2.19 5.32/2.19 5.32/2.19 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.32/2.19 5.32/2.19 5.32/2.19 5.32/2.19 ---------------------------------------- 5.32/2.19 5.32/2.19 (2) 5.32/2.19 Obligation: 5.32/2.19 Pi-finite rewrite system: 5.32/2.19 The TRS R consists of the following rules: 5.32/2.19 5.32/2.19 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.19 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.19 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.19 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.19 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.19 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.19 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.19 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.19 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.19 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.19 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.19 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.19 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.19 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.19 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.19 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.19 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.19 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.19 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.19 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.19 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.19 5.32/2.19 The argument filtering Pi contains the following mapping: 5.32/2.19 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.19 5.32/2.19 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.19 5.32/2.19 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.19 5.32/2.19 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.19 5.32/2.19 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.19 5.32/2.19 s(x1) = s(x1) 5.32/2.19 5.32/2.19 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.19 5.32/2.19 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.19 5.32/2.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.19 5.32/2.19 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.19 5.32/2.19 in_out_gg(x1, x2) = in_out_gg 5.32/2.19 5.32/2.19 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.19 5.32/2.19 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.19 5.32/2.19 0 = 0 5.32/2.19 5.32/2.19 less_out_gg(x1, x2) = less_out_gg 5.32/2.19 5.32/2.19 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.19 5.32/2.19 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.19 5.32/2.19 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.19 5.32/2.19 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.19 5.32/2.19 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.19 5.32/2.19 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.19 5.32/2.19 less_out_ga(x1, x2) = less_out_ga 5.32/2.19 5.32/2.19 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.19 5.32/2.19 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.19 5.32/2.19 5.32/2.19 5.32/2.19 ---------------------------------------- 5.32/2.19 5.32/2.19 (3) DependencyPairsProof (EQUIVALENT) 5.32/2.19 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.32/2.19 Pi DP problem: 5.32/2.19 The TRS P consists of the following rules: 5.32/2.19 5.32/2.19 IN_IN_AG(X, tree(Y, Left, X3)) -> U1_AG(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.19 IN_IN_AG(X, tree(Y, Left, X3)) -> LESS_IN_AG(X, Y) 5.32/2.19 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 5.32/2.19 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.32/2.19 U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_AG(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.19 IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.19 IN_IN_GG(X, tree(Y, Left, X3)) -> LESS_IN_GG(X, Y) 5.32/2.19 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 5.32/2.19 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.19 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_GG(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.19 IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.19 IN_IN_GG(X, tree(Y, X4, Right)) -> LESS_IN_GG(Y, X) 5.32/2.19 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_GG(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.19 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) 5.32/2.19 IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.19 IN_IN_AG(X, tree(Y, X4, Right)) -> LESS_IN_GA(Y, X) 5.32/2.19 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 5.32/2.19 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.32/2.19 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_AG(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.19 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) 5.32/2.19 5.32/2.19 The TRS R consists of the following rules: 5.32/2.19 5.32/2.19 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.19 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.19 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.19 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.19 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.19 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.19 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.19 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.19 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.19 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.19 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.19 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.19 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.19 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.19 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.19 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.19 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.19 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.19 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.19 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.19 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.19 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.19 5.32/2.19 The argument filtering Pi contains the following mapping: 5.32/2.19 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.19 5.32/2.19 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.19 5.32/2.19 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.19 5.32/2.19 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.19 5.32/2.19 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.19 5.32/2.19 s(x1) = s(x1) 5.32/2.19 5.32/2.19 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.19 5.32/2.19 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.19 5.32/2.19 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 IN_IN_AG(x1, x2) = IN_IN_AG(x2) 5.32/2.20 5.32/2.20 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x5) 5.32/2.20 5.32/2.20 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.32/2.20 5.32/2.20 U5_AG(x1, x2, x3) = U5_AG(x3) 5.32/2.20 5.32/2.20 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) 5.32/2.20 5.32/2.20 IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) 5.32/2.20 5.32/2.20 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U5_GG(x1, x2, x3) = U5_GG(x3) 5.32/2.20 5.32/2.20 U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) 5.32/2.20 5.32/2.20 U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) 5.32/2.20 5.32/2.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) 5.32/2.20 5.32/2.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.32/2.20 5.32/2.20 U5_GA(x1, x2, x3) = U5_GA(x3) 5.32/2.20 5.32/2.20 U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (4) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 IN_IN_AG(X, tree(Y, Left, X3)) -> U1_AG(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 IN_IN_AG(X, tree(Y, Left, X3)) -> LESS_IN_AG(X, Y) 5.32/2.20 LESS_IN_AG(s(X), s(Y)) -> U5_AG(X, Y, less_in_ag(X, Y)) 5.32/2.20 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.32/2.20 U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_AG(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 U1_AG(X, Y, Left, X3, less_out_ag(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.20 IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 IN_IN_GG(X, tree(Y, Left, X3)) -> LESS_IN_GG(X, Y) 5.32/2.20 LESS_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, less_in_gg(X, Y)) 5.32/2.20 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.20 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_GG(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.20 IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 IN_IN_GG(X, tree(Y, X4, Right)) -> LESS_IN_GG(Y, X) 5.32/2.20 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_GG(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) 5.32/2.20 IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 IN_IN_AG(X, tree(Y, X4, Right)) -> LESS_IN_GA(Y, X) 5.32/2.20 LESS_IN_GA(s(X), s(Y)) -> U5_GA(X, Y, less_in_ga(X, Y)) 5.32/2.20 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.32/2.20 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_AG(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 IN_IN_AG(x1, x2) = IN_IN_AG(x2) 5.32/2.20 5.32/2.20 U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x5) 5.32/2.20 5.32/2.20 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.32/2.20 5.32/2.20 U5_AG(x1, x2, x3) = U5_AG(x3) 5.32/2.20 5.32/2.20 U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5) 5.32/2.20 5.32/2.20 IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) 5.32/2.20 5.32/2.20 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U5_GG(x1, x2, x3) = U5_GG(x3) 5.32/2.20 5.32/2.20 U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5) 5.32/2.20 5.32/2.20 U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5) 5.32/2.20 5.32/2.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) 5.32/2.20 5.32/2.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.32/2.20 5.32/2.20 U5_GA(x1, x2, x3) = U5_GA(x3) 5.32/2.20 5.32/2.20 U4_AG(x1, x2, x3, x4, x5) = U4_AG(x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (5) DependencyGraphProof (EQUIVALENT) 5.32/2.20 The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 13 less nodes. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (6) 5.32/2.20 Complex Obligation (AND) 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (7) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (8) UsableRulesProof (EQUIVALENT) 5.32/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (9) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GA(s(X), s(Y)) -> LESS_IN_GA(X, Y) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 LESS_IN_GA(x1, x2) = LESS_IN_GA(x1) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (10) PiDPToQDPProof (SOUND) 5.32/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (11) 5.32/2.20 Obligation: 5.32/2.20 Q DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 Q is empty. 5.32/2.20 We have to consider all (P,Q,R)-chains. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (12) QDPSizeChangeProof (EQUIVALENT) 5.32/2.20 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. 5.32/2.20 5.32/2.20 From the DPs we obtained the following set of size-change graphs: 5.32/2.20 *LESS_IN_GA(s(X)) -> LESS_IN_GA(X) 5.32/2.20 The graph contains the following edges 1 > 1 5.32/2.20 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (13) 5.32/2.20 YES 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (14) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (15) UsableRulesProof (EQUIVALENT) 5.32/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (16) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 Pi is empty. 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (17) PiDPToQDPProof (EQUIVALENT) 5.32/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (18) 5.32/2.20 Obligation: 5.32/2.20 Q DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 Q is empty. 5.32/2.20 We have to consider all (P,Q,R)-chains. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (19) QDPSizeChangeProof (EQUIVALENT) 5.32/2.20 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. 5.32/2.20 5.32/2.20 From the DPs we obtained the following set of size-change graphs: 5.32/2.20 *LESS_IN_GG(s(X), s(Y)) -> LESS_IN_GG(X, Y) 5.32/2.20 The graph contains the following edges 1 > 1, 2 > 2 5.32/2.20 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (20) 5.32/2.20 YES 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (21) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.20 IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) 5.32/2.20 5.32/2.20 U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (22) UsableRulesProof (EQUIVALENT) 5.32/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (23) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 U1_GG(X, Y, Left, X3, less_out_gg(X, Y)) -> IN_IN_GG(X, Left) 5.32/2.20 IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_GG(X, Y, X4, Right, less_out_gg(Y, X)) -> IN_IN_GG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2) 5.32/2.20 5.32/2.20 U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5) 5.32/2.20 5.32/2.20 U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (24) PiDPToQDPProof (SOUND) 5.32/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (25) 5.32/2.20 Obligation: 5.32/2.20 Q DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 U1_GG(X, Left, less_out_gg) -> IN_IN_GG(X, Left) 5.32/2.20 IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Left, less_in_gg(X, Y)) 5.32/2.20 IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Right, less_in_gg(Y, X)) 5.32/2.20 U3_GG(X, Right, less_out_gg) -> IN_IN_GG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(less_in_gg(X, Y)) 5.32/2.20 U5_gg(less_out_gg) -> less_out_gg 5.32/2.20 5.32/2.20 The set Q consists of the following terms: 5.32/2.20 5.32/2.20 less_in_gg(x0, x1) 5.32/2.20 U5_gg(x0) 5.32/2.20 5.32/2.20 We have to consider all (P,Q,R)-chains. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (26) QDPSizeChangeProof (EQUIVALENT) 5.32/2.20 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. 5.32/2.20 5.32/2.20 From the DPs we obtained the following set of size-change graphs: 5.32/2.20 *IN_IN_GG(X, tree(Y, Left, X3)) -> U1_GG(X, Left, less_in_gg(X, Y)) 5.32/2.20 The graph contains the following edges 1 >= 1, 2 > 2 5.32/2.20 5.32/2.20 5.32/2.20 *IN_IN_GG(X, tree(Y, X4, Right)) -> U3_GG(X, Right, less_in_gg(Y, X)) 5.32/2.20 The graph contains the following edges 1 >= 1, 2 > 2 5.32/2.20 5.32/2.20 5.32/2.20 *U1_GG(X, Left, less_out_gg) -> IN_IN_GG(X, Left) 5.32/2.20 The graph contains the following edges 1 >= 1, 2 >= 2 5.32/2.20 5.32/2.20 5.32/2.20 *U3_GG(X, Right, less_out_gg) -> IN_IN_GG(X, Right) 5.32/2.20 The graph contains the following edges 1 >= 1, 2 >= 2 5.32/2.20 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (27) 5.32/2.20 YES 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (28) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (29) UsableRulesProof (EQUIVALENT) 5.32/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (30) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 LESS_IN_AG(x1, x2) = LESS_IN_AG(x2) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (31) PiDPToQDPProof (SOUND) 5.32/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (32) 5.32/2.20 Obligation: 5.32/2.20 Q DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 5.32/2.20 5.32/2.20 R is empty. 5.32/2.20 Q is empty. 5.32/2.20 We have to consider all (P,Q,R)-chains. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (33) QDPSizeChangeProof (EQUIVALENT) 5.32/2.20 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. 5.32/2.20 5.32/2.20 From the DPs we obtained the following set of size-change graphs: 5.32/2.20 *LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y) 5.32/2.20 The graph contains the following edges 1 > 1 5.32/2.20 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (34) 5.32/2.20 YES 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (35) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 in_in_ag(X, tree(X, X1, X2)) -> in_out_ag(X, tree(X, X1, X2)) 5.32/2.20 in_in_ag(X, tree(Y, Left, X3)) -> U1_ag(X, Y, Left, X3, less_in_ag(X, Y)) 5.32/2.20 less_in_ag(0, s(X5)) -> less_out_ag(0, s(X5)) 5.32/2.20 less_in_ag(s(X), s(Y)) -> U5_ag(X, Y, less_in_ag(X, Y)) 5.32/2.20 U5_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y)) 5.32/2.20 U1_ag(X, Y, Left, X3, less_out_ag(X, Y)) -> U2_ag(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(X, X1, X2)) -> in_out_gg(X, tree(X, X1, X2)) 5.32/2.20 in_in_gg(X, tree(Y, Left, X3)) -> U1_gg(X, Y, Left, X3, less_in_gg(X, Y)) 5.32/2.20 less_in_gg(0, s(X5)) -> less_out_gg(0, s(X5)) 5.32/2.20 less_in_gg(s(X), s(Y)) -> U5_gg(X, Y, less_in_gg(X, Y)) 5.32/2.20 U5_gg(X, Y, less_out_gg(X, Y)) -> less_out_gg(s(X), s(Y)) 5.32/2.20 U1_gg(X, Y, Left, X3, less_out_gg(X, Y)) -> U2_gg(X, Y, Left, X3, in_in_gg(X, Left)) 5.32/2.20 in_in_gg(X, tree(Y, X4, Right)) -> U3_gg(X, Y, X4, Right, less_in_gg(Y, X)) 5.32/2.20 U3_gg(X, Y, X4, Right, less_out_gg(Y, X)) -> U4_gg(X, Y, X4, Right, in_in_gg(X, Right)) 5.32/2.20 U4_gg(X, Y, X4, Right, in_out_gg(X, Right)) -> in_out_gg(X, tree(Y, X4, Right)) 5.32/2.20 U2_gg(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_gg(X, tree(Y, Left, X3)) 5.32/2.20 U2_ag(X, Y, Left, X3, in_out_gg(X, Left)) -> in_out_ag(X, tree(Y, Left, X3)) 5.32/2.20 in_in_ag(X, tree(Y, X4, Right)) -> U3_ag(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 U3_ag(X, Y, X4, Right, less_out_ga(Y, X)) -> U4_ag(X, Y, X4, Right, in_in_ag(X, Right)) 5.32/2.20 U4_ag(X, Y, X4, Right, in_out_ag(X, Right)) -> in_out_ag(X, tree(Y, X4, Right)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 in_in_ag(x1, x2) = in_in_ag(x2) 5.32/2.20 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 in_out_ag(x1, x2) = in_out_ag(x1) 5.32/2.20 5.32/2.20 U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5) 5.32/2.20 5.32/2.20 less_in_ag(x1, x2) = less_in_ag(x2) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 less_out_ag(x1, x2) = less_out_ag(x1) 5.32/2.20 5.32/2.20 U5_ag(x1, x2, x3) = U5_ag(x3) 5.32/2.20 5.32/2.20 U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5) 5.32/2.20 5.32/2.20 in_in_gg(x1, x2) = in_in_gg(x1, x2) 5.32/2.20 5.32/2.20 in_out_gg(x1, x2) = in_out_gg 5.32/2.20 5.32/2.20 U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5) 5.32/2.20 5.32/2.20 less_in_gg(x1, x2) = less_in_gg(x1, x2) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_out_gg(x1, x2) = less_out_gg 5.32/2.20 5.32/2.20 U5_gg(x1, x2, x3) = U5_gg(x3) 5.32/2.20 5.32/2.20 U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5) 5.32/2.20 5.32/2.20 U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5) 5.32/2.20 5.32/2.20 U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5) 5.32/2.20 5.32/2.20 U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5) 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 U4_ag(x1, x2, x3, x4, x5) = U4_ag(x5) 5.32/2.20 5.32/2.20 IN_IN_AG(x1, x2) = IN_IN_AG(x2) 5.32/2.20 5.32/2.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (36) UsableRulesProof (EQUIVALENT) 5.32/2.20 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (37) 5.32/2.20 Obligation: 5.32/2.20 Pi DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 IN_IN_AG(X, tree(Y, X4, Right)) -> U3_AG(X, Y, X4, Right, less_in_ga(Y, X)) 5.32/2.20 U3_AG(X, Y, X4, Right, less_out_ga(Y, X)) -> IN_IN_AG(X, Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 less_in_ga(0, s(X5)) -> less_out_ga(0, s(X5)) 5.32/2.20 less_in_ga(s(X), s(Y)) -> U5_ga(X, Y, less_in_ga(X, Y)) 5.32/2.20 U5_ga(X, Y, less_out_ga(X, Y)) -> less_out_ga(s(X), s(Y)) 5.32/2.20 5.32/2.20 The argument filtering Pi contains the following mapping: 5.32/2.20 tree(x1, x2, x3) = tree(x1, x2, x3) 5.32/2.20 5.32/2.20 s(x1) = s(x1) 5.32/2.20 5.32/2.20 0 = 0 5.32/2.20 5.32/2.20 less_in_ga(x1, x2) = less_in_ga(x1) 5.32/2.20 5.32/2.20 less_out_ga(x1, x2) = less_out_ga 5.32/2.20 5.32/2.20 U5_ga(x1, x2, x3) = U5_ga(x3) 5.32/2.20 5.32/2.20 IN_IN_AG(x1, x2) = IN_IN_AG(x2) 5.32/2.20 5.32/2.20 U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5) 5.32/2.20 5.32/2.20 5.32/2.20 We have to consider all (P,R,Pi)-chains 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (38) PiDPToQDPProof (SOUND) 5.32/2.20 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (39) 5.32/2.20 Obligation: 5.32/2.20 Q DP problem: 5.32/2.20 The TRS P consists of the following rules: 5.32/2.20 5.32/2.20 IN_IN_AG(tree(Y, X4, Right)) -> U3_AG(Right, less_in_ga(Y)) 5.32/2.20 U3_AG(Right, less_out_ga) -> IN_IN_AG(Right) 5.32/2.20 5.32/2.20 The TRS R consists of the following rules: 5.32/2.20 5.32/2.20 less_in_ga(0) -> less_out_ga 5.32/2.20 less_in_ga(s(X)) -> U5_ga(less_in_ga(X)) 5.32/2.20 U5_ga(less_out_ga) -> less_out_ga 5.32/2.20 5.32/2.20 The set Q consists of the following terms: 5.32/2.20 5.32/2.20 less_in_ga(x0) 5.32/2.20 U5_ga(x0) 5.32/2.20 5.32/2.20 We have to consider all (P,Q,R)-chains. 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (40) QDPSizeChangeProof (EQUIVALENT) 5.32/2.20 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. 5.32/2.20 5.32/2.20 From the DPs we obtained the following set of size-change graphs: 5.32/2.20 *U3_AG(Right, less_out_ga) -> IN_IN_AG(Right) 5.32/2.20 The graph contains the following edges 1 >= 1 5.32/2.20 5.32/2.20 5.32/2.20 *IN_IN_AG(tree(Y, X4, Right)) -> U3_AG(Right, less_in_ga(Y)) 5.32/2.20 The graph contains the following edges 1 > 1 5.32/2.20 5.32/2.20 5.32/2.20 ---------------------------------------- 5.32/2.20 5.32/2.20 (41) 5.32/2.20 YES 5.32/2.23 EOF