5.99/2.24 YES 5.99/2.28 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.99/2.28 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.99/2.28 5.99/2.28 5.99/2.28 Left Termination of the query pattern 5.99/2.28 5.99/2.28 merge(g,g,a) 5.99/2.28 5.99/2.28 w.r.t. the given Prolog program could successfully be proven: 5.99/2.28 5.99/2.28 (0) Prolog 5.99/2.28 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.99/2.28 (2) PiTRS 5.99/2.28 (3) DependencyPairsProof [EQUIVALENT, 11 ms] 5.99/2.28 (4) PiDP 5.99/2.28 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.99/2.28 (6) AND 5.99/2.28 (7) PiDP 5.99/2.28 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.99/2.28 (9) PiDP 5.99/2.28 (10) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.99/2.28 (11) QDP 5.99/2.28 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.99/2.28 (13) YES 5.99/2.28 (14) PiDP 5.99/2.28 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.99/2.28 (16) PiDP 5.99/2.28 (17) PiDPToQDPProof [EQUIVALENT, 0 ms] 5.99/2.28 (18) QDP 5.99/2.28 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.99/2.28 (20) YES 5.99/2.28 (21) PiDP 5.99/2.28 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.99/2.28 (23) PiDP 5.99/2.28 (24) PiDPToQDPProof [SOUND, 0 ms] 5.99/2.28 (25) QDP 5.99/2.28 (26) MRRProof [EQUIVALENT, 40 ms] 5.99/2.28 (27) QDP 5.99/2.28 (28) DependencyGraphProof [EQUIVALENT, 0 ms] 5.99/2.28 (29) QDP 5.99/2.28 (30) UsableRulesProof [EQUIVALENT, 0 ms] 5.99/2.28 (31) QDP 5.99/2.28 (32) QReductionProof [EQUIVALENT, 0 ms] 5.99/2.28 (33) QDP 5.99/2.28 (34) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.99/2.28 (35) YES 5.99/2.28 5.99/2.28 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (0) 5.99/2.28 Obligation: 5.99/2.28 Clauses: 5.99/2.28 5.99/2.28 merge(X, [], X). 5.99/2.28 merge([], X, X). 5.99/2.28 merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)). 5.99/2.28 merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)). 5.99/2.28 gt(s(X), s(Y)) :- gt(X, Y). 5.99/2.28 gt(s(X), zero). 5.99/2.28 le(s(X), s(Y)) :- le(X, Y). 5.99/2.28 le(zero, s(Y)). 5.99/2.28 le(zero, zero). 5.99/2.28 5.99/2.28 5.99/2.28 Query: merge(g,g,a) 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (1) PrologToPiTRSProof (SOUND) 5.99/2.28 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.99/2.28 5.99/2.28 merge_in_3: (b,b,f) 5.99/2.28 5.99/2.28 le_in_2: (b,b) 5.99/2.28 5.99/2.28 gt_in_2: (b,b) 5.99/2.28 5.99/2.28 Transforming Prolog into the following Term Rewriting System: 5.99/2.28 5.99/2.28 Pi-finite rewrite system: 5.99/2.28 The TRS R consists of the following rules: 5.99/2.28 5.99/2.28 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.28 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.28 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.28 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.28 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.28 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.28 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.28 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.28 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.28 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.28 5.99/2.28 The argument filtering Pi contains the following mapping: 5.99/2.28 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.28 5.99/2.28 [] = [] 5.99/2.28 5.99/2.28 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.28 5.99/2.28 .(x1, x2) = .(x1, x2) 5.99/2.28 5.99/2.28 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.28 5.99/2.28 s(x1) = s(x1) 5.99/2.28 5.99/2.28 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 zero = zero 5.99/2.28 5.99/2.28 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.28 5.99/2.28 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 5.99/2.28 5.99/2.28 5.99/2.28 5.99/2.28 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.99/2.28 5.99/2.28 5.99/2.28 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (2) 5.99/2.28 Obligation: 5.99/2.28 Pi-finite rewrite system: 5.99/2.28 The TRS R consists of the following rules: 5.99/2.28 5.99/2.28 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.28 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.28 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.28 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.28 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.28 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.28 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.28 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.28 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.28 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.28 5.99/2.28 The argument filtering Pi contains the following mapping: 5.99/2.28 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.28 5.99/2.28 [] = [] 5.99/2.28 5.99/2.28 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.28 5.99/2.28 .(x1, x2) = .(x1, x2) 5.99/2.28 5.99/2.28 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.28 5.99/2.28 s(x1) = s(x1) 5.99/2.28 5.99/2.28 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 zero = zero 5.99/2.28 5.99/2.28 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.28 5.99/2.28 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 5.99/2.28 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (3) DependencyPairsProof (EQUIVALENT) 5.99/2.28 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.99/2.28 Pi DP problem: 5.99/2.28 The TRS P consists of the following rules: 5.99/2.28 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> U1_GGA(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> LE_IN_GG(A, B) 5.99/2.28 LE_IN_GG(s(X), s(Y)) -> U6_GG(X, Y, le_in_gg(X, Y)) 5.99/2.28 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.28 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_GGA(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y), Z) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> U3_GGA(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> GT_IN_GG(A, B) 5.99/2.28 GT_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, gt_in_gg(X, Y)) 5.99/2.28 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.28 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_GGA(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y, Z) 5.99/2.28 5.99/2.28 The TRS R consists of the following rules: 5.99/2.28 5.99/2.28 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.28 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.28 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.28 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.28 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.28 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.28 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.28 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.28 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.28 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.28 5.99/2.28 The argument filtering Pi contains the following mapping: 5.99/2.28 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.28 5.99/2.28 [] = [] 5.99/2.28 5.99/2.28 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.28 5.99/2.28 .(x1, x2) = .(x1, x2) 5.99/2.28 5.99/2.28 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.28 5.99/2.28 s(x1) = s(x1) 5.99/2.28 5.99/2.28 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 zero = zero 5.99/2.28 5.99/2.28 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.28 5.99/2.28 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 5.99/2.28 5.99/2.28 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 5.99/2.28 5.99/2.28 U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) 5.99/2.28 5.99/2.28 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 5.99/2.28 5.99/2.28 U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) 5.99/2.28 5.99/2.28 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 5.99/2.28 We have to consider all (P,R,Pi)-chains 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (4) 5.99/2.28 Obligation: 5.99/2.28 Pi DP problem: 5.99/2.28 The TRS P consists of the following rules: 5.99/2.28 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> U1_GGA(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> LE_IN_GG(A, B) 5.99/2.28 LE_IN_GG(s(X), s(Y)) -> U6_GG(X, Y, le_in_gg(X, Y)) 5.99/2.28 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.28 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_GGA(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y), Z) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> U3_GGA(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> GT_IN_GG(A, B) 5.99/2.28 GT_IN_GG(s(X), s(Y)) -> U5_GG(X, Y, gt_in_gg(X, Y)) 5.99/2.28 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.28 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_GGA(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y, Z) 5.99/2.28 5.99/2.28 The TRS R consists of the following rules: 5.99/2.28 5.99/2.28 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.28 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.28 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.28 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.28 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.28 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.28 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.28 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.28 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.28 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.28 5.99/2.28 The argument filtering Pi contains the following mapping: 5.99/2.28 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.28 5.99/2.28 [] = [] 5.99/2.28 5.99/2.28 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.28 5.99/2.28 .(x1, x2) = .(x1, x2) 5.99/2.28 5.99/2.28 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.28 5.99/2.28 s(x1) = s(x1) 5.99/2.28 5.99/2.28 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 zero = zero 5.99/2.28 5.99/2.28 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.28 5.99/2.28 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.28 5.99/2.28 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 5.99/2.28 5.99/2.28 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 5.99/2.28 5.99/2.28 U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3) 5.99/2.28 5.99/2.28 U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 5.99/2.28 5.99/2.28 U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3) 5.99/2.28 5.99/2.28 U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 5.99/2.28 We have to consider all (P,R,Pi)-chains 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (5) DependencyGraphProof (EQUIVALENT) 5.99/2.28 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 6 less nodes. 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (6) 5.99/2.28 Complex Obligation (AND) 5.99/2.28 5.99/2.28 ---------------------------------------- 5.99/2.28 5.99/2.28 (7) 5.99/2.28 Obligation: 5.99/2.28 Pi DP problem: 5.99/2.28 The TRS P consists of the following rules: 5.99/2.28 5.99/2.28 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.28 5.99/2.28 The TRS R consists of the following rules: 5.99/2.28 5.99/2.28 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.28 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.28 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.28 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.28 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.28 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.28 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.28 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.28 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.28 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.28 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.28 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.28 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.28 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.28 5.99/2.28 The argument filtering Pi contains the following mapping: 5.99/2.28 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.28 5.99/2.28 [] = [] 5.99/2.28 5.99/2.28 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.28 5.99/2.28 .(x1, x2) = .(x1, x2) 5.99/2.28 5.99/2.28 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.28 5.99/2.28 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.28 5.99/2.28 s(x1) = s(x1) 5.99/2.28 5.99/2.28 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.28 5.99/2.28 zero = zero 5.99/2.28 5.99/2.28 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.29 5.99/2.29 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 5.99/2.29 5.99/2.29 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (8) UsableRulesProof (EQUIVALENT) 5.99/2.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (9) 5.99/2.29 Obligation: 5.99/2.29 Pi DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.29 5.99/2.29 R is empty. 5.99/2.29 Pi is empty. 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (10) PiDPToQDPProof (EQUIVALENT) 5.99/2.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (11) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.29 5.99/2.29 R is empty. 5.99/2.29 Q is empty. 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (12) QDPSizeChangeProof (EQUIVALENT) 5.99/2.29 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. 5.99/2.29 5.99/2.29 From the DPs we obtained the following set of size-change graphs: 5.99/2.29 *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 5.99/2.29 The graph contains the following edges 1 > 1, 2 > 2 5.99/2.29 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (13) 5.99/2.29 YES 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (14) 5.99/2.29 Obligation: 5.99/2.29 Pi DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.29 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.29 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.29 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.29 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.29 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.29 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.29 5.99/2.29 The argument filtering Pi contains the following mapping: 5.99/2.29 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.29 5.99/2.29 [] = [] 5.99/2.29 5.99/2.29 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.29 5.99/2.29 .(x1, x2) = .(x1, x2) 5.99/2.29 5.99/2.29 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.29 5.99/2.29 s(x1) = s(x1) 5.99/2.29 5.99/2.29 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 zero = zero 5.99/2.29 5.99/2.29 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.29 5.99/2.29 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 5.99/2.29 5.99/2.29 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (15) UsableRulesProof (EQUIVALENT) 5.99/2.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (16) 5.99/2.29 Obligation: 5.99/2.29 Pi DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.29 5.99/2.29 R is empty. 5.99/2.29 Pi is empty. 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (17) PiDPToQDPProof (EQUIVALENT) 5.99/2.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (18) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.29 5.99/2.29 R is empty. 5.99/2.29 Q is empty. 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (19) QDPSizeChangeProof (EQUIVALENT) 5.99/2.29 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. 5.99/2.29 5.99/2.29 From the DPs we obtained the following set of size-change graphs: 5.99/2.29 *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 5.99/2.29 The graph contains the following edges 1 > 1, 2 > 2 5.99/2.29 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (20) 5.99/2.29 YES 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (21) 5.99/2.29 Obligation: 5.99/2.29 Pi DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y), Z) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> U1_GGA(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> U3_GGA(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.29 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y, Z) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 merge_in_gga(X, [], X) -> merge_out_gga(X, [], X) 5.99/2.29 merge_in_gga([], X, X) -> merge_out_gga([], X, X) 5.99/2.29 merge_in_gga(.(A, X), .(B, Y), .(A, Z)) -> U1_gga(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.29 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U1_gga(A, X, B, Y, Z, le_out_gg(A, B)) -> U2_gga(A, X, B, Y, Z, merge_in_gga(X, .(B, Y), Z)) 5.99/2.29 merge_in_gga(.(A, X), .(B, Y), .(B, Z)) -> U3_gga(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 U3_gga(A, X, B, Y, Z, gt_out_gg(A, B)) -> U4_gga(A, X, B, Y, Z, merge_in_gga(.(A, X), Y, Z)) 5.99/2.29 U4_gga(A, X, B, Y, Z, merge_out_gga(.(A, X), Y, Z)) -> merge_out_gga(.(A, X), .(B, Y), .(B, Z)) 5.99/2.29 U2_gga(A, X, B, Y, Z, merge_out_gga(X, .(B, Y), Z)) -> merge_out_gga(.(A, X), .(B, Y), .(A, Z)) 5.99/2.29 5.99/2.29 The argument filtering Pi contains the following mapping: 5.99/2.29 merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2) 5.99/2.29 5.99/2.29 [] = [] 5.99/2.29 5.99/2.29 merge_out_gga(x1, x2, x3) = merge_out_gga(x1, x2, x3) 5.99/2.29 5.99/2.29 .(x1, x2) = .(x1, x2) 5.99/2.29 5.99/2.29 U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.29 5.99/2.29 s(x1) = s(x1) 5.99/2.29 5.99/2.29 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 zero = zero 5.99/2.29 5.99/2.29 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.29 5.99/2.29 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.29 5.99/2.29 U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 5.99/2.29 5.99/2.29 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (22) UsableRulesProof (EQUIVALENT) 5.99/2.29 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (23) 5.99/2.29 Obligation: 5.99/2.29 Pi DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 U1_GGA(A, X, B, Y, Z, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y), Z) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y), .(A, Z)) -> U1_GGA(A, X, B, Y, Z, le_in_gg(A, B)) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y), .(B, Z)) -> U3_GGA(A, X, B, Y, Z, gt_in_gg(A, B)) 5.99/2.29 U3_GGA(A, X, B, Y, Z, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y, Z) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.29 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The argument filtering Pi contains the following mapping: 5.99/2.29 .(x1, x2) = .(x1, x2) 5.99/2.29 5.99/2.29 le_in_gg(x1, x2) = le_in_gg(x1, x2) 5.99/2.29 5.99/2.29 s(x1) = s(x1) 5.99/2.29 5.99/2.29 U6_gg(x1, x2, x3) = U6_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 zero = zero 5.99/2.29 5.99/2.29 le_out_gg(x1, x2) = le_out_gg(x1, x2) 5.99/2.29 5.99/2.29 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 5.99/2.29 5.99/2.29 U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3) 5.99/2.29 5.99/2.29 gt_out_gg(x1, x2) = gt_out_gg(x1, x2) 5.99/2.29 5.99/2.29 MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2) 5.99/2.29 5.99/2.29 U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6) 5.99/2.29 5.99/2.29 5.99/2.29 We have to consider all (P,R,Pi)-chains 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (24) PiDPToQDPProof (SOUND) 5.99/2.29 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (25) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U3_GGA(A, X, B, Y, gt_in_gg(A, B)) 5.99/2.29 U3_GGA(A, X, B, Y, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.29 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The set Q consists of the following terms: 5.99/2.29 5.99/2.29 le_in_gg(x0, x1) 5.99/2.29 gt_in_gg(x0, x1) 5.99/2.29 U6_gg(x0, x1, x2) 5.99/2.29 U5_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (26) MRRProof (EQUIVALENT) 5.99/2.29 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 5.99/2.29 5.99/2.29 Strictly oriented dependency pairs: 5.99/2.29 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U3_GGA(A, X, B, Y, gt_in_gg(A, B)) 5.99/2.29 5.99/2.29 Strictly oriented rules of the TRS R: 5.99/2.29 5.99/2.29 le_in_gg(zero, s(Y)) -> le_out_gg(zero, s(Y)) 5.99/2.29 le_in_gg(zero, zero) -> le_out_gg(zero, zero) 5.99/2.29 5.99/2.29 Used ordering: Polynomial interpretation [POLO]: 5.99/2.29 5.99/2.29 POL(.(x_1, x_2)) = 1 + 2*x_1 + x_2 5.99/2.29 POL(MERGE_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 5.99/2.29 POL(U1_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + x_1 + 2*x_2 + 2*x_3 + 2*x_4 + x_5 5.99/2.29 POL(U3_GGA(x_1, x_2, x_3, x_4, x_5)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 + 2*x_4 + 2*x_5 5.99/2.29 POL(U5_gg(x_1, x_2, x_3)) = x_1 + x_2 + x_3 5.99/2.29 POL(U6_gg(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 5.99/2.29 POL(gt_in_gg(x_1, x_2)) = x_1 + x_2 5.99/2.29 POL(gt_out_gg(x_1, x_2)) = x_1 + x_2 5.99/2.29 POL(le_in_gg(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 5.99/2.29 POL(le_out_gg(x_1, x_2)) = 2*x_1 + 2*x_2 5.99/2.29 POL(s(x_1)) = 2*x_1 5.99/2.29 POL(zero) = 1 5.99/2.29 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (27) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 U3_GGA(A, X, B, Y, gt_out_gg(A, B)) -> MERGE_IN_GGA(.(A, X), Y) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The set Q consists of the following terms: 5.99/2.29 5.99/2.29 le_in_gg(x0, x1) 5.99/2.29 gt_in_gg(x0, x1) 5.99/2.29 U6_gg(x0, x1, x2) 5.99/2.29 U5_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (28) DependencyGraphProof (EQUIVALENT) 5.99/2.29 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (29) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), s(Y)) -> U5_gg(X, Y, gt_in_gg(X, Y)) 5.99/2.29 gt_in_gg(s(X), zero) -> gt_out_gg(s(X), zero) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 U5_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The set Q consists of the following terms: 5.99/2.29 5.99/2.29 le_in_gg(x0, x1) 5.99/2.29 gt_in_gg(x0, x1) 5.99/2.29 U6_gg(x0, x1, x2) 5.99/2.29 U5_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (30) UsableRulesProof (EQUIVALENT) 5.99/2.29 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (31) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The set Q consists of the following terms: 5.99/2.29 5.99/2.29 le_in_gg(x0, x1) 5.99/2.29 gt_in_gg(x0, x1) 5.99/2.29 U6_gg(x0, x1, x2) 5.99/2.29 U5_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (32) QReductionProof (EQUIVALENT) 5.99/2.29 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 5.99/2.29 5.99/2.29 gt_in_gg(x0, x1) 5.99/2.29 U5_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (33) 5.99/2.29 Obligation: 5.99/2.29 Q DP problem: 5.99/2.29 The TRS P consists of the following rules: 5.99/2.29 5.99/2.29 MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 5.99/2.29 The TRS R consists of the following rules: 5.99/2.29 5.99/2.29 le_in_gg(s(X), s(Y)) -> U6_gg(X, Y, le_in_gg(X, Y)) 5.99/2.29 U6_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 5.99/2.29 5.99/2.29 The set Q consists of the following terms: 5.99/2.29 5.99/2.29 le_in_gg(x0, x1) 5.99/2.29 U6_gg(x0, x1, x2) 5.99/2.29 5.99/2.29 We have to consider all (P,Q,R)-chains. 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (34) QDPSizeChangeProof (EQUIVALENT) 5.99/2.29 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. 5.99/2.29 5.99/2.29 From the DPs we obtained the following set of size-change graphs: 5.99/2.29 *U1_GGA(A, X, B, Y, le_out_gg(A, B)) -> MERGE_IN_GGA(X, .(B, Y)) 5.99/2.29 The graph contains the following edges 2 >= 1 5.99/2.29 5.99/2.29 5.99/2.29 *MERGE_IN_GGA(.(A, X), .(B, Y)) -> U1_GGA(A, X, B, Y, le_in_gg(A, B)) 5.99/2.29 The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4 5.99/2.29 5.99/2.29 5.99/2.29 ---------------------------------------- 5.99/2.29 5.99/2.29 (35) 5.99/2.29 YES 6.19/2.32 EOF