7.30/2.84 YES 7.30/2.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 7.30/2.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.30/2.85 7.30/2.85 7.30/2.85 Left Termination of the query pattern 7.30/2.85 7.30/2.85 gcd(g,g,a) 7.30/2.85 7.30/2.85 w.r.t. the given Prolog program could successfully be proven: 7.30/2.85 7.30/2.85 (0) Prolog 7.30/2.85 (1) PrologToPiTRSProof [SOUND, 0 ms] 7.30/2.85 (2) PiTRS 7.30/2.85 (3) DependencyPairsProof [EQUIVALENT, 8 ms] 7.30/2.85 (4) PiDP 7.30/2.85 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 7.30/2.85 (6) AND 7.30/2.85 (7) PiDP 7.30/2.85 (8) UsableRulesProof [EQUIVALENT, 0 ms] 7.30/2.85 (9) PiDP 7.30/2.85 (10) PiDPToQDPProof [EQUIVALENT, 16 ms] 7.30/2.85 (11) QDP 7.30/2.85 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.30/2.85 (13) YES 7.30/2.85 (14) PiDP 7.30/2.85 (15) UsableRulesProof [EQUIVALENT, 0 ms] 7.30/2.85 (16) PiDP 7.30/2.85 (17) PiDPToQDPProof [SOUND, 0 ms] 7.30/2.85 (18) QDP 7.30/2.85 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.30/2.85 (20) YES 7.30/2.85 (21) PiDP 7.30/2.85 (22) UsableRulesProof [EQUIVALENT, 0 ms] 7.30/2.85 (23) PiDP 7.30/2.85 (24) PiDPToQDPProof [EQUIVALENT, 0 ms] 7.30/2.85 (25) QDP 7.30/2.85 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 7.30/2.85 (27) YES 7.30/2.85 (28) PiDP 7.30/2.85 (29) UsableRulesProof [EQUIVALENT, 0 ms] 7.30/2.85 (30) PiDP 7.30/2.85 (31) PiDPToQDPProof [SOUND, 1 ms] 7.30/2.85 (32) QDP 7.30/2.85 (33) QDPQMonotonicMRRProof [EQUIVALENT, 47 ms] 7.30/2.85 (34) QDP 7.30/2.85 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 7.30/2.85 (36) TRUE 7.30/2.85 7.30/2.85 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (0) 7.30/2.85 Obligation: 7.30/2.85 Clauses: 7.30/2.85 7.30/2.85 gcd(X, Y, D) :- ','(le(X, Y), gcd_le(X, Y, D)). 7.30/2.85 gcd(X, Y, D) :- ','(gt(X, Y), gcd_le(Y, X, D)). 7.30/2.85 gcd_le(0, Y, Y). 7.30/2.85 gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D)). 7.30/2.85 gt(s(X), s(Y)) :- gt(X, Y). 7.30/2.85 gt(s(X), 0). 7.30/2.85 le(s(X), s(Y)) :- le(X, Y). 7.30/2.85 le(0, s(Y)). 7.30/2.85 le(0, 0). 7.30/2.85 add(s(X), Y, s(Z)) :- add(X, Y, Z). 7.30/2.85 add(0, X, X). 7.30/2.85 7.30/2.85 7.30/2.85 Query: gcd(g,g,a) 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (1) PrologToPiTRSProof (SOUND) 7.30/2.85 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 7.30/2.85 7.30/2.85 gcd_in_3: (b,b,f) 7.30/2.85 7.30/2.85 le_in_2: (b,b) 7.30/2.85 7.30/2.85 gcd_le_in_3: (b,b,f) 7.30/2.85 7.30/2.85 add_in_3: (b,f,b) 7.30/2.85 7.30/2.85 gt_in_2: (b,b) 7.30/2.85 7.30/2.85 Transforming Prolog into the following Term Rewriting System: 7.30/2.85 7.30/2.85 Pi-finite rewrite system: 7.30/2.85 The TRS R consists of the following rules: 7.30/2.85 7.30/2.85 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.85 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.85 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.85 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.85 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.85 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.85 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.85 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.85 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.85 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 7.30/2.85 The argument filtering Pi contains the following mapping: 7.30/2.85 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.85 7.30/2.85 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.85 7.30/2.85 s(x1) = s(x1) 7.30/2.85 7.30/2.85 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.85 7.30/2.85 0 = 0 7.30/2.85 7.30/2.85 le_out_gg(x1, x2) = le_out_gg 7.30/2.85 7.30/2.85 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.85 7.30/2.85 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.85 7.30/2.85 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.85 7.30/2.85 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.85 7.30/2.85 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.85 7.30/2.85 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.85 7.30/2.85 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.85 7.30/2.85 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.85 7.30/2.85 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.85 7.30/2.85 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.85 7.30/2.85 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.85 7.30/2.85 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.85 7.30/2.85 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.85 7.30/2.85 7.30/2.85 7.30/2.85 7.30/2.85 7.30/2.85 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 7.30/2.85 7.30/2.85 7.30/2.85 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (2) 7.30/2.85 Obligation: 7.30/2.85 Pi-finite rewrite system: 7.30/2.85 The TRS R consists of the following rules: 7.30/2.85 7.30/2.85 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.85 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.85 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.85 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.85 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.85 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.85 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.85 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.85 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.85 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 7.30/2.85 The argument filtering Pi contains the following mapping: 7.30/2.85 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.85 7.30/2.85 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.85 7.30/2.85 s(x1) = s(x1) 7.30/2.85 7.30/2.85 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.85 7.30/2.85 0 = 0 7.30/2.85 7.30/2.85 le_out_gg(x1, x2) = le_out_gg 7.30/2.85 7.30/2.85 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.85 7.30/2.85 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.85 7.30/2.85 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.85 7.30/2.85 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.85 7.30/2.85 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.85 7.30/2.85 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.85 7.30/2.85 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.85 7.30/2.85 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.85 7.30/2.85 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.85 7.30/2.85 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.85 7.30/2.85 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.85 7.30/2.85 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.85 7.30/2.85 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.85 7.30/2.85 7.30/2.85 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (3) DependencyPairsProof (EQUIVALENT) 7.30/2.85 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 7.30/2.85 Pi DP problem: 7.30/2.85 The TRS P consists of the following rules: 7.30/2.85 7.30/2.85 GCD_IN_GGA(X, Y, D) -> U1_GGA(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> LE_IN_GG(X, Y) 7.30/2.85 LE_IN_GG(s(X), s(Y)) -> U8_GG(X, Y, le_in_gg(X, Y)) 7.30/2.85 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.85 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> GCD_LE_IN_GGA(X, Y, D) 7.30/2.85 GCD_LE_IN_GGA(s(X), Y, D) -> U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 GCD_LE_IN_GGA(s(X), Y, D) -> ADD_IN_GAG(s(X), Z, Y) 7.30/2.85 ADD_IN_GAG(s(X), Y, s(Z)) -> U9_GAG(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 ADD_IN_GAG(s(X), Y, s(Z)) -> ADD_IN_GAG(X, Y, Z) 7.30/2.85 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> GCD_IN_GGA(s(X), Z, D) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> U3_GGA(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> GT_IN_GG(X, Y) 7.30/2.85 GT_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, gt_in_gg(X, Y)) 7.30/2.85 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.85 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> GCD_LE_IN_GGA(Y, X, D) 7.30/2.85 7.30/2.85 The TRS R consists of the following rules: 7.30/2.85 7.30/2.85 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.85 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.85 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.85 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.85 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.85 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.85 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.85 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.85 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.85 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 7.30/2.85 The argument filtering Pi contains the following mapping: 7.30/2.85 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.85 7.30/2.85 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.85 7.30/2.85 s(x1) = s(x1) 7.30/2.85 7.30/2.85 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.85 7.30/2.85 0 = 0 7.30/2.85 7.30/2.85 le_out_gg(x1, x2) = le_out_gg 7.30/2.85 7.30/2.85 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.85 7.30/2.85 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.85 7.30/2.85 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.85 7.30/2.85 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.85 7.30/2.85 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.85 7.30/2.85 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.85 7.30/2.85 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.85 7.30/2.85 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.85 7.30/2.85 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.85 7.30/2.85 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.85 7.30/2.85 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.85 7.30/2.85 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.85 7.30/2.85 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.85 7.30/2.85 GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2) 7.30/2.85 7.30/2.85 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 7.30/2.85 7.30/2.85 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 7.30/2.85 7.30/2.85 U8_GG(x1, x2, x3) = U8_GG(x3) 7.30/2.85 7.30/2.85 U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) 7.30/2.85 7.30/2.85 GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2) 7.30/2.85 7.30/2.85 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) 7.30/2.85 7.30/2.85 ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3) 7.30/2.85 7.30/2.85 U9_GAG(x1, x2, x3, x4) = U9_GAG(x4) 7.30/2.85 7.30/2.85 U6_GGA(x1, x2, x3, x4) = U6_GGA(x4) 7.30/2.85 7.30/2.85 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 7.30/2.85 7.30/2.85 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 7.30/2.85 7.30/2.85 U7_GG(x1, x2, x3) = U7_GG(x3) 7.30/2.85 7.30/2.85 U4_GGA(x1, x2, x3, x4) = U4_GGA(x4) 7.30/2.85 7.30/2.85 7.30/2.85 We have to consider all (P,R,Pi)-chains 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (4) 7.30/2.85 Obligation: 7.30/2.85 Pi DP problem: 7.30/2.85 The TRS P consists of the following rules: 7.30/2.85 7.30/2.85 GCD_IN_GGA(X, Y, D) -> U1_GGA(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> LE_IN_GG(X, Y) 7.30/2.85 LE_IN_GG(s(X), s(Y)) -> U8_GG(X, Y, le_in_gg(X, Y)) 7.30/2.85 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.85 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> U2_GGA(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> GCD_LE_IN_GGA(X, Y, D) 7.30/2.85 GCD_LE_IN_GGA(s(X), Y, D) -> U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 GCD_LE_IN_GGA(s(X), Y, D) -> ADD_IN_GAG(s(X), Z, Y) 7.30/2.85 ADD_IN_GAG(s(X), Y, s(Z)) -> U9_GAG(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 ADD_IN_GAG(s(X), Y, s(Z)) -> ADD_IN_GAG(X, Y, Z) 7.30/2.85 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_GGA(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> GCD_IN_GGA(s(X), Z, D) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> U3_GGA(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 GCD_IN_GGA(X, Y, D) -> GT_IN_GG(X, Y) 7.30/2.85 GT_IN_GG(s(X), s(Y)) -> U7_GG(X, Y, gt_in_gg(X, Y)) 7.30/2.85 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.85 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> U4_GGA(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> GCD_LE_IN_GGA(Y, X, D) 7.30/2.85 7.30/2.85 The TRS R consists of the following rules: 7.30/2.85 7.30/2.85 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.85 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.85 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.85 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.85 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.85 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.85 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.85 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.85 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.85 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 7.30/2.85 The argument filtering Pi contains the following mapping: 7.30/2.85 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.85 7.30/2.85 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.85 7.30/2.85 s(x1) = s(x1) 7.30/2.85 7.30/2.85 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.85 7.30/2.85 0 = 0 7.30/2.85 7.30/2.85 le_out_gg(x1, x2) = le_out_gg 7.30/2.85 7.30/2.85 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.85 7.30/2.85 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.85 7.30/2.85 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.85 7.30/2.85 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.85 7.30/2.85 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.85 7.30/2.85 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.85 7.30/2.85 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.85 7.30/2.85 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.85 7.30/2.85 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.85 7.30/2.85 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.85 7.30/2.85 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.85 7.30/2.85 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.85 7.30/2.85 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.85 7.30/2.85 GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2) 7.30/2.85 7.30/2.85 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 7.30/2.85 7.30/2.85 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 7.30/2.85 7.30/2.85 U8_GG(x1, x2, x3) = U8_GG(x3) 7.30/2.85 7.30/2.85 U2_GGA(x1, x2, x3, x4) = U2_GGA(x4) 7.30/2.85 7.30/2.85 GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2) 7.30/2.85 7.30/2.85 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) 7.30/2.85 7.30/2.85 ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3) 7.30/2.85 7.30/2.85 U9_GAG(x1, x2, x3, x4) = U9_GAG(x4) 7.30/2.85 7.30/2.85 U6_GGA(x1, x2, x3, x4) = U6_GGA(x4) 7.30/2.85 7.30/2.85 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 7.30/2.85 7.30/2.85 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 7.30/2.85 7.30/2.85 U7_GG(x1, x2, x3) = U7_GG(x3) 7.30/2.85 7.30/2.85 U4_GGA(x1, x2, x3, x4) = U4_GGA(x4) 7.30/2.85 7.30/2.85 7.30/2.85 We have to consider all (P,R,Pi)-chains 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (5) DependencyGraphProof (EQUIVALENT) 7.30/2.85 The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 9 less nodes. 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (6) 7.30/2.85 Complex Obligation (AND) 7.30/2.85 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (7) 7.30/2.85 Obligation: 7.30/2.85 Pi DP problem: 7.30/2.85 The TRS P consists of the following rules: 7.30/2.85 7.30/2.85 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.85 7.30/2.85 The TRS R consists of the following rules: 7.30/2.85 7.30/2.85 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.85 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.85 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.85 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.85 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.85 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.85 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.85 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.85 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.85 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.85 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.85 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.85 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.85 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.85 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.85 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.85 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.85 7.30/2.85 The argument filtering Pi contains the following mapping: 7.30/2.85 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.85 7.30/2.85 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.85 7.30/2.85 s(x1) = s(x1) 7.30/2.85 7.30/2.85 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.85 7.30/2.85 0 = 0 7.30/2.85 7.30/2.85 le_out_gg(x1, x2) = le_out_gg 7.30/2.85 7.30/2.85 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.85 7.30/2.85 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.85 7.30/2.85 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.85 7.30/2.85 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.85 7.30/2.85 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.85 7.30/2.85 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.85 7.30/2.85 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.85 7.30/2.85 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.85 7.30/2.85 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.85 7.30/2.85 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.85 7.30/2.85 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.85 7.30/2.85 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.85 7.30/2.85 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.85 7.30/2.85 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.85 7.30/2.85 GT_IN_GG(x1, x2) = GT_IN_GG(x1, x2) 7.30/2.85 7.30/2.85 7.30/2.85 We have to consider all (P,R,Pi)-chains 7.30/2.85 ---------------------------------------- 7.30/2.85 7.30/2.85 (8) UsableRulesProof (EQUIVALENT) 7.30/2.85 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (9) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 Pi is empty. 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (10) PiDPToQDPProof (EQUIVALENT) 7.30/2.86 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (11) 7.30/2.86 Obligation: 7.30/2.86 Q DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 Q is empty. 7.30/2.86 We have to consider all (P,Q,R)-chains. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (12) QDPSizeChangeProof (EQUIVALENT) 7.30/2.86 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. 7.30/2.86 7.30/2.86 From the DPs we obtained the following set of size-change graphs: 7.30/2.86 *GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y) 7.30/2.86 The graph contains the following edges 1 > 1, 2 > 2 7.30/2.86 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (13) 7.30/2.86 YES 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (14) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 ADD_IN_GAG(s(X), Y, s(Z)) -> ADD_IN_GAG(X, Y, Z) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.86 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.86 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.86 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.86 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.86 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.86 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.86 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.86 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.86 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.86 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.86 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.86 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.86 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.86 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 7.30/2.86 The argument filtering Pi contains the following mapping: 7.30/2.86 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.86 7.30/2.86 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.86 7.30/2.86 s(x1) = s(x1) 7.30/2.86 7.30/2.86 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.86 7.30/2.86 0 = 0 7.30/2.86 7.30/2.86 le_out_gg(x1, x2) = le_out_gg 7.30/2.86 7.30/2.86 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.86 7.30/2.86 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.86 7.30/2.86 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.86 7.30/2.86 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.86 7.30/2.86 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.86 7.30/2.86 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.86 7.30/2.86 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.86 7.30/2.86 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.86 7.30/2.86 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.86 7.30/2.86 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.86 7.30/2.86 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.86 7.30/2.86 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.86 7.30/2.86 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.86 7.30/2.86 ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3) 7.30/2.86 7.30/2.86 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (15) UsableRulesProof (EQUIVALENT) 7.30/2.86 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (16) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 ADD_IN_GAG(s(X), Y, s(Z)) -> ADD_IN_GAG(X, Y, Z) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 The argument filtering Pi contains the following mapping: 7.30/2.86 s(x1) = s(x1) 7.30/2.86 7.30/2.86 ADD_IN_GAG(x1, x2, x3) = ADD_IN_GAG(x1, x3) 7.30/2.86 7.30/2.86 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (17) PiDPToQDPProof (SOUND) 7.30/2.86 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (18) 7.30/2.86 Obligation: 7.30/2.86 Q DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 ADD_IN_GAG(s(X), s(Z)) -> ADD_IN_GAG(X, Z) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 Q is empty. 7.30/2.86 We have to consider all (P,Q,R)-chains. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (19) QDPSizeChangeProof (EQUIVALENT) 7.30/2.86 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. 7.30/2.86 7.30/2.86 From the DPs we obtained the following set of size-change graphs: 7.30/2.86 *ADD_IN_GAG(s(X), s(Z)) -> ADD_IN_GAG(X, Z) 7.30/2.86 The graph contains the following edges 1 > 1, 2 > 2 7.30/2.86 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (20) 7.30/2.86 YES 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (21) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.86 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.86 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.86 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.86 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.86 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.86 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.86 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.86 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.86 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.86 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.86 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.86 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.86 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.86 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 7.30/2.86 The argument filtering Pi contains the following mapping: 7.30/2.86 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.86 7.30/2.86 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.86 7.30/2.86 s(x1) = s(x1) 7.30/2.86 7.30/2.86 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.86 7.30/2.86 0 = 0 7.30/2.86 7.30/2.86 le_out_gg(x1, x2) = le_out_gg 7.30/2.86 7.30/2.86 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.86 7.30/2.86 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.86 7.30/2.86 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.86 7.30/2.86 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.86 7.30/2.86 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.86 7.30/2.86 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.86 7.30/2.86 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.86 7.30/2.86 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.86 7.30/2.86 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.86 7.30/2.86 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.86 7.30/2.86 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.86 7.30/2.86 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.86 7.30/2.86 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.86 7.30/2.86 LE_IN_GG(x1, x2) = LE_IN_GG(x1, x2) 7.30/2.86 7.30/2.86 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (22) UsableRulesProof (EQUIVALENT) 7.30/2.86 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (23) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 Pi is empty. 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (24) PiDPToQDPProof (EQUIVALENT) 7.30/2.86 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (25) 7.30/2.86 Obligation: 7.30/2.86 Q DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.86 7.30/2.86 R is empty. 7.30/2.86 Q is empty. 7.30/2.86 We have to consider all (P,Q,R)-chains. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (26) QDPSizeChangeProof (EQUIVALENT) 7.30/2.86 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. 7.30/2.86 7.30/2.86 From the DPs we obtained the following set of size-change graphs: 7.30/2.86 *LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y) 7.30/2.86 The graph contains the following edges 1 > 1, 2 > 2 7.30/2.86 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (27) 7.30/2.86 YES 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (28) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> GCD_LE_IN_GGA(X, Y, D) 7.30/2.86 GCD_LE_IN_GGA(s(X), Y, D) -> U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.86 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> GCD_IN_GGA(s(X), Z, D) 7.30/2.86 GCD_IN_GGA(X, Y, D) -> U1_GGA(X, Y, D, le_in_gg(X, Y)) 7.30/2.86 GCD_IN_GGA(X, Y, D) -> U3_GGA(X, Y, D, gt_in_gg(X, Y)) 7.30/2.86 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> GCD_LE_IN_GGA(Y, X, D) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 gcd_in_gga(X, Y, D) -> U1_gga(X, Y, D, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.86 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.86 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.86 U1_gga(X, Y, D, le_out_gg(X, Y)) -> U2_gga(X, Y, D, gcd_le_in_gga(X, Y, D)) 7.30/2.86 gcd_le_in_gga(0, Y, Y) -> gcd_le_out_gga(0, Y, Y) 7.30/2.86 gcd_le_in_gga(s(X), Y, D) -> U5_gga(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.86 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.86 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.86 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.86 U5_gga(X, Y, D, add_out_gag(s(X), Z, Y)) -> U6_gga(X, Y, D, gcd_in_gga(s(X), Z, D)) 7.30/2.86 gcd_in_gga(X, Y, D) -> U3_gga(X, Y, D, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.86 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.86 U3_gga(X, Y, D, gt_out_gg(X, Y)) -> U4_gga(X, Y, D, gcd_le_in_gga(Y, X, D)) 7.30/2.86 U4_gga(X, Y, D, gcd_le_out_gga(Y, X, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 U6_gga(X, Y, D, gcd_out_gga(s(X), Z, D)) -> gcd_le_out_gga(s(X), Y, D) 7.30/2.86 U2_gga(X, Y, D, gcd_le_out_gga(X, Y, D)) -> gcd_out_gga(X, Y, D) 7.30/2.86 7.30/2.86 The argument filtering Pi contains the following mapping: 7.30/2.86 gcd_in_gga(x1, x2, x3) = gcd_in_gga(x1, x2) 7.30/2.86 7.30/2.86 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.86 7.30/2.86 s(x1) = s(x1) 7.30/2.86 7.30/2.86 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.86 7.30/2.86 0 = 0 7.30/2.86 7.30/2.86 le_out_gg(x1, x2) = le_out_gg 7.30/2.86 7.30/2.86 U2_gga(x1, x2, x3, x4) = U2_gga(x4) 7.30/2.86 7.30/2.86 gcd_le_in_gga(x1, x2, x3) = gcd_le_in_gga(x1, x2) 7.30/2.86 7.30/2.86 gcd_le_out_gga(x1, x2, x3) = gcd_le_out_gga(x3) 7.30/2.86 7.30/2.86 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x4) 7.30/2.86 7.30/2.86 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.86 7.30/2.86 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.86 7.30/2.86 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.86 7.30/2.86 U6_gga(x1, x2, x3, x4) = U6_gga(x4) 7.30/2.86 7.30/2.86 U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4) 7.30/2.86 7.30/2.86 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.86 7.30/2.86 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.86 7.30/2.86 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.86 7.30/2.86 U4_gga(x1, x2, x3, x4) = U4_gga(x4) 7.30/2.86 7.30/2.86 gcd_out_gga(x1, x2, x3) = gcd_out_gga(x3) 7.30/2.86 7.30/2.86 GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2) 7.30/2.86 7.30/2.86 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 7.30/2.86 7.30/2.86 GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2) 7.30/2.86 7.30/2.86 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) 7.30/2.86 7.30/2.86 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 7.30/2.86 7.30/2.86 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (29) UsableRulesProof (EQUIVALENT) 7.30/2.86 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (30) 7.30/2.86 Obligation: 7.30/2.86 Pi DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 U1_GGA(X, Y, D, le_out_gg(X, Y)) -> GCD_LE_IN_GGA(X, Y, D) 7.30/2.86 GCD_LE_IN_GGA(s(X), Y, D) -> U5_GGA(X, Y, D, add_in_gag(s(X), Z, Y)) 7.30/2.86 U5_GGA(X, Y, D, add_out_gag(s(X), Z, Y)) -> GCD_IN_GGA(s(X), Z, D) 7.30/2.86 GCD_IN_GGA(X, Y, D) -> U1_GGA(X, Y, D, le_in_gg(X, Y)) 7.30/2.86 GCD_IN_GGA(X, Y, D) -> U3_GGA(X, Y, D, gt_in_gg(X, Y)) 7.30/2.86 U3_GGA(X, Y, D, gt_out_gg(X, Y)) -> GCD_LE_IN_GGA(Y, X, D) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 add_in_gag(s(X), Y, s(Z)) -> U9_gag(X, Y, Z, add_in_gag(X, Y, Z)) 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(X, Y, le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg(0, s(Y)) 7.30/2.86 le_in_gg(0, 0) -> le_out_gg(0, 0) 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(X, Y, gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg(s(X), 0) 7.30/2.86 U9_gag(X, Y, Z, add_out_gag(X, Y, Z)) -> add_out_gag(s(X), Y, s(Z)) 7.30/2.86 U8_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y)) 7.30/2.86 U7_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y)) 7.30/2.86 add_in_gag(0, X, X) -> add_out_gag(0, X, X) 7.30/2.86 7.30/2.86 The argument filtering Pi contains the following mapping: 7.30/2.86 le_in_gg(x1, x2) = le_in_gg(x1, x2) 7.30/2.86 7.30/2.86 s(x1) = s(x1) 7.30/2.86 7.30/2.86 U8_gg(x1, x2, x3) = U8_gg(x3) 7.30/2.86 7.30/2.86 0 = 0 7.30/2.86 7.30/2.86 le_out_gg(x1, x2) = le_out_gg 7.30/2.86 7.30/2.86 add_in_gag(x1, x2, x3) = add_in_gag(x1, x3) 7.30/2.86 7.30/2.86 U9_gag(x1, x2, x3, x4) = U9_gag(x4) 7.30/2.86 7.30/2.86 add_out_gag(x1, x2, x3) = add_out_gag(x2) 7.30/2.86 7.30/2.86 gt_in_gg(x1, x2) = gt_in_gg(x1, x2) 7.30/2.86 7.30/2.86 U7_gg(x1, x2, x3) = U7_gg(x3) 7.30/2.86 7.30/2.86 gt_out_gg(x1, x2) = gt_out_gg 7.30/2.86 7.30/2.86 GCD_IN_GGA(x1, x2, x3) = GCD_IN_GGA(x1, x2) 7.30/2.86 7.30/2.86 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 7.30/2.86 7.30/2.86 GCD_LE_IN_GGA(x1, x2, x3) = GCD_LE_IN_GGA(x1, x2) 7.30/2.86 7.30/2.86 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x4) 7.30/2.86 7.30/2.86 U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4) 7.30/2.86 7.30/2.86 7.30/2.86 We have to consider all (P,R,Pi)-chains 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (31) PiDPToQDPProof (SOUND) 7.30/2.86 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (32) 7.30/2.86 Obligation: 7.30/2.86 Q DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 U1_GGA(X, Y, le_out_gg) -> GCD_LE_IN_GGA(X, Y) 7.30/2.86 GCD_LE_IN_GGA(s(X), Y) -> U5_GGA(X, add_in_gag(s(X), Y)) 7.30/2.86 U5_GGA(X, add_out_gag(Z)) -> GCD_IN_GGA(s(X), Z) 7.30/2.86 GCD_IN_GGA(X, Y) -> U1_GGA(X, Y, le_in_gg(X, Y)) 7.30/2.86 GCD_IN_GGA(X, Y) -> U3_GGA(X, Y, gt_in_gg(X, Y)) 7.30/2.86 U3_GGA(X, Y, gt_out_gg) -> GCD_LE_IN_GGA(Y, X) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 add_in_gag(s(X), s(Z)) -> U9_gag(add_in_gag(X, Z)) 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg 7.30/2.86 le_in_gg(0, 0) -> le_out_gg 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg 7.30/2.86 U9_gag(add_out_gag(Y)) -> add_out_gag(Y) 7.30/2.86 U8_gg(le_out_gg) -> le_out_gg 7.30/2.86 U7_gg(gt_out_gg) -> gt_out_gg 7.30/2.86 add_in_gag(0, X) -> add_out_gag(X) 7.30/2.86 7.30/2.86 The set Q consists of the following terms: 7.30/2.86 7.30/2.86 add_in_gag(x0, x1) 7.30/2.86 le_in_gg(x0, x1) 7.30/2.86 gt_in_gg(x0, x1) 7.30/2.86 U9_gag(x0) 7.30/2.86 U8_gg(x0) 7.30/2.86 U7_gg(x0) 7.30/2.86 7.30/2.86 We have to consider all (P,Q,R)-chains. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (33) QDPQMonotonicMRRProof (EQUIVALENT) 7.30/2.86 By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain. 7.30/2.86 7.30/2.86 7.30/2.86 Strictly oriented rules of the TRS R: 7.30/2.86 7.30/2.86 add_in_gag(s(X), s(Z)) -> U9_gag(add_in_gag(X, Z)) 7.30/2.86 7.30/2.86 Used ordering: Polynomial interpretation [POLO]: 7.30/2.86 7.30/2.86 POL(0) = 0 7.30/2.86 POL(GCD_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 7.30/2.86 POL(GCD_LE_IN_GGA(x_1, x_2)) = 2*x_1 + 2*x_2 7.30/2.86 POL(U1_GGA(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 7.30/2.86 POL(U3_GGA(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 7.30/2.86 POL(U5_GGA(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 7.30/2.86 POL(U7_gg(x_1)) = 2 7.30/2.86 POL(U8_gg(x_1)) = 2 7.30/2.86 POL(U9_gag(x_1)) = x_1 7.30/2.86 POL(add_in_gag(x_1, x_2)) = x_2 7.30/2.86 POL(add_out_gag(x_1)) = x_1 7.30/2.86 POL(gt_in_gg(x_1, x_2)) = 2 + 2*x_1 7.30/2.86 POL(gt_out_gg) = 0 7.30/2.86 POL(le_in_gg(x_1, x_2)) = 1 + 2*x_1 + x_2 7.30/2.86 POL(le_out_gg) = 0 7.30/2.86 POL(s(x_1)) = 1 + x_1 7.30/2.86 7.30/2.86 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (34) 7.30/2.86 Obligation: 7.30/2.86 Q DP problem: 7.30/2.86 The TRS P consists of the following rules: 7.30/2.86 7.30/2.86 U1_GGA(X, Y, le_out_gg) -> GCD_LE_IN_GGA(X, Y) 7.30/2.86 GCD_LE_IN_GGA(s(X), Y) -> U5_GGA(X, add_in_gag(s(X), Y)) 7.30/2.86 U5_GGA(X, add_out_gag(Z)) -> GCD_IN_GGA(s(X), Z) 7.30/2.86 GCD_IN_GGA(X, Y) -> U1_GGA(X, Y, le_in_gg(X, Y)) 7.30/2.86 GCD_IN_GGA(X, Y) -> U3_GGA(X, Y, gt_in_gg(X, Y)) 7.30/2.86 U3_GGA(X, Y, gt_out_gg) -> GCD_LE_IN_GGA(Y, X) 7.30/2.86 7.30/2.86 The TRS R consists of the following rules: 7.30/2.86 7.30/2.86 le_in_gg(s(X), s(Y)) -> U8_gg(le_in_gg(X, Y)) 7.30/2.86 le_in_gg(0, s(Y)) -> le_out_gg 7.30/2.86 le_in_gg(0, 0) -> le_out_gg 7.30/2.86 gt_in_gg(s(X), s(Y)) -> U7_gg(gt_in_gg(X, Y)) 7.30/2.86 gt_in_gg(s(X), 0) -> gt_out_gg 7.30/2.86 U9_gag(add_out_gag(Y)) -> add_out_gag(Y) 7.30/2.86 U8_gg(le_out_gg) -> le_out_gg 7.30/2.86 U7_gg(gt_out_gg) -> gt_out_gg 7.30/2.86 add_in_gag(0, X) -> add_out_gag(X) 7.30/2.86 7.30/2.86 The set Q consists of the following terms: 7.30/2.86 7.30/2.86 add_in_gag(x0, x1) 7.30/2.86 le_in_gg(x0, x1) 7.30/2.86 gt_in_gg(x0, x1) 7.30/2.86 U9_gag(x0) 7.30/2.86 U8_gg(x0) 7.30/2.86 U7_gg(x0) 7.30/2.86 7.30/2.86 We have to consider all (P,Q,R)-chains. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (35) DependencyGraphProof (EQUIVALENT) 7.30/2.86 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes. 7.30/2.86 ---------------------------------------- 7.30/2.86 7.30/2.86 (36) 7.30/2.86 TRUE 7.30/2.90 EOF