5.23/2.14 YES 5.23/2.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.23/2.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.23/2.16 5.23/2.16 5.23/2.16 Left Termination of the query pattern 5.23/2.16 5.23/2.16 convert(g,g,a) 5.23/2.16 5.23/2.16 w.r.t. the given Prolog program could successfully be proven: 5.23/2.16 5.23/2.16 (0) Prolog 5.23/2.16 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.23/2.16 (2) PiTRS 5.23/2.16 (3) DependencyPairsProof [EQUIVALENT, 15 ms] 5.23/2.16 (4) PiDP 5.23/2.16 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.23/2.16 (6) AND 5.23/2.16 (7) PiDP 5.23/2.16 (8) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.16 (9) PiDP 5.23/2.16 (10) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.16 (11) QDP 5.23/2.16 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.23/2.16 (13) YES 5.23/2.16 (14) PiDP 5.23/2.16 (15) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.16 (16) PiDP 5.23/2.16 (17) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.16 (18) QDP 5.23/2.16 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 5.23/2.16 (20) YES 5.23/2.16 (21) PiDP 5.23/2.16 (22) UsableRulesProof [EQUIVALENT, 0 ms] 5.23/2.16 (23) PiDP 5.23/2.16 (24) PiDPToQDPProof [SOUND, 0 ms] 5.23/2.16 (25) QDP 5.23/2.16 (26) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 5.23/2.16 (27) QDP 5.23/2.16 (28) PisEmptyProof [EQUIVALENT, 0 ms] 5.23/2.16 (29) YES 5.23/2.16 5.23/2.16 5.23/2.16 ---------------------------------------- 5.23/2.16 5.23/2.16 (0) 5.23/2.16 Obligation: 5.23/2.16 Clauses: 5.23/2.16 5.23/2.16 convert([], B, 0). 5.23/2.16 convert(.(0, XS), B, X) :- ','(convert(XS, B, Y), times(Y, B, X)). 5.23/2.16 convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X). 5.23/2.16 plus(0, Y, Y). 5.23/2.16 plus(s(X), Y, s(Z)) :- plus(X, Y, Z). 5.23/2.16 times(0, Y, 0). 5.23/2.16 times(s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)). 5.23/2.16 5.23/2.16 5.23/2.16 Query: convert(g,g,a) 5.23/2.16 ---------------------------------------- 5.23/2.16 5.23/2.16 (1) PrologToPiTRSProof (SOUND) 5.23/2.16 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.23/2.16 5.23/2.16 convert_in_3: (b,b,f) 5.23/2.16 5.23/2.16 times_in_3: (b,b,f) 5.23/2.16 5.23/2.16 plus_in_3: (b,b,f) 5.23/2.16 5.23/2.16 Transforming Prolog into the following Term Rewriting System: 5.23/2.16 5.23/2.16 Pi-finite rewrite system: 5.23/2.16 The TRS R consists of the following rules: 5.23/2.16 5.23/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.23/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.23/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.23/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.23/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.23/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.23/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.23/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.23/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.23/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.23/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.23/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.23/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.23/2.16 5.23/2.16 The argument filtering Pi contains the following mapping: 5.23/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.23/2.16 5.23/2.16 [] = [] 5.23/2.16 5.23/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 .(x1, x2) = .(x1, x2) 5.23/2.16 5.23/2.16 0 = 0 5.23/2.16 5.23/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 s(x1) = s(x1) 5.23/2.16 5.23/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.23/2.16 5.23/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.23/2.16 5.23/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.23/2.16 5.23/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 5.23/2.16 5.23/2.16 5.23/2.16 5.23/2.16 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.23/2.16 5.23/2.16 5.23/2.16 5.23/2.16 ---------------------------------------- 5.23/2.16 5.23/2.16 (2) 5.23/2.16 Obligation: 5.23/2.16 Pi-finite rewrite system: 5.23/2.16 The TRS R consists of the following rules: 5.23/2.16 5.23/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.23/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.23/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.23/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.23/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.23/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.23/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.23/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.23/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.23/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.23/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.23/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.23/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.23/2.16 5.23/2.16 The argument filtering Pi contains the following mapping: 5.23/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.23/2.16 5.23/2.16 [] = [] 5.23/2.16 5.23/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 .(x1, x2) = .(x1, x2) 5.23/2.16 5.23/2.16 0 = 0 5.23/2.16 5.23/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 s(x1) = s(x1) 5.23/2.16 5.23/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.23/2.16 5.23/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.23/2.16 5.23/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.23/2.16 5.23/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 5.23/2.16 5.23/2.16 ---------------------------------------- 5.23/2.16 5.23/2.16 (3) DependencyPairsProof (EQUIVALENT) 5.23/2.16 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.23/2.16 Pi DP problem: 5.23/2.16 The TRS P consists of the following rules: 5.23/2.16 5.23/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> U1_GGA(XS, B, X, convert_in_gga(XS, B, Y)) 5.23/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) 5.23/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> U3_GGA(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.23/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) 5.23/2.16 U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_GGA(XS, B, X, times_in_gga(Y, B, X)) 5.23/2.16 U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> TIMES_IN_GGA(Y, B, X) 5.23/2.16 TIMES_IN_GGA(s(X), Y, Z) -> U5_GGA(X, Y, Z, times_in_gga(X, Y, U)) 5.23/2.16 TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) 5.23/2.16 U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U6_GGA(X, Y, Z, plus_in_gga(Y, U, Z)) 5.23/2.16 U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) 5.23/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> U4_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) 5.23/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) 5.23/2.16 5.23/2.16 The TRS R consists of the following rules: 5.23/2.16 5.23/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.23/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.23/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.23/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.23/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.23/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.23/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.23/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.23/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.23/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.23/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.23/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.23/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.23/2.16 5.23/2.16 The argument filtering Pi contains the following mapping: 5.23/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.23/2.16 5.23/2.16 [] = [] 5.23/2.16 5.23/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 .(x1, x2) = .(x1, x2) 5.23/2.16 5.23/2.16 0 = 0 5.23/2.16 5.23/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 s(x1) = s(x1) 5.23/2.16 5.23/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.23/2.16 5.23/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.23/2.16 5.23/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.23/2.16 5.23/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.23/2.16 5.23/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.56/2.16 5.56/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (4) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> U1_GGA(XS, B, X, convert_in_gga(XS, B, Y)) 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> U3_GGA(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) 5.56/2.16 U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_GGA(XS, B, X, times_in_gga(Y, B, X)) 5.56/2.16 U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> TIMES_IN_GGA(Y, B, X) 5.56/2.16 TIMES_IN_GGA(s(X), Y, Z) -> U5_GGA(X, Y, Z, times_in_gga(X, Y, U)) 5.56/2.16 TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) 5.56/2.16 U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U6_GGA(X, Y, Z, plus_in_gga(Y, U, Z)) 5.56/2.16 U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) 5.56/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> U4_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) 5.56/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) 5.56/2.16 5.56/2.16 The TRS R consists of the following rules: 5.56/2.16 5.56/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.56/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.56/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.56/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.56/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.56/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.56/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.56/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.56/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.56/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.56/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.56/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.56/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.56/2.16 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.56/2.16 5.56/2.16 [] = [] 5.56/2.16 5.56/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 .(x1, x2) = .(x1, x2) 5.56/2.16 5.56/2.16 0 = 0 5.56/2.16 5.56/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.56/2.16 5.56/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.56/2.16 5.56/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (5) DependencyGraphProof (EQUIVALENT) 5.56/2.16 The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (6) 5.56/2.16 Complex Obligation (AND) 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (7) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) 5.56/2.16 5.56/2.16 The TRS R consists of the following rules: 5.56/2.16 5.56/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.56/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.56/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.56/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.56/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.56/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.56/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.56/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.56/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.56/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.56/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.56/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.56/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.56/2.16 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.56/2.16 5.56/2.16 [] = [] 5.56/2.16 5.56/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 .(x1, x2) = .(x1, x2) 5.56/2.16 5.56/2.16 0 = 0 5.56/2.16 5.56/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.56/2.16 5.56/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.56/2.16 5.56/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (8) UsableRulesProof (EQUIVALENT) 5.56/2.16 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (9) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (10) PiDPToQDPProof (SOUND) 5.56/2.16 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (11) 5.56/2.16 Obligation: 5.56/2.16 Q DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 Q is empty. 5.56/2.16 We have to consider all (P,Q,R)-chains. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (12) QDPSizeChangeProof (EQUIVALENT) 5.56/2.16 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. 5.56/2.16 5.56/2.16 From the DPs we obtained the following set of size-change graphs: 5.56/2.16 *PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y) 5.56/2.16 The graph contains the following edges 1 > 1, 2 >= 2 5.56/2.16 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (13) 5.56/2.16 YES 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (14) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) 5.56/2.16 5.56/2.16 The TRS R consists of the following rules: 5.56/2.16 5.56/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.56/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.56/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.56/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.56/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.56/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.56/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.56/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.56/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.56/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.56/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.56/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.56/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.56/2.16 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.56/2.16 5.56/2.16 [] = [] 5.56/2.16 5.56/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 .(x1, x2) = .(x1, x2) 5.56/2.16 5.56/2.16 0 = 0 5.56/2.16 5.56/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.56/2.16 5.56/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.56/2.16 5.56/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (15) UsableRulesProof (EQUIVALENT) 5.56/2.16 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (16) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (17) PiDPToQDPProof (SOUND) 5.56/2.16 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (18) 5.56/2.16 Obligation: 5.56/2.16 Q DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 Q is empty. 5.56/2.16 We have to consider all (P,Q,R)-chains. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (19) QDPSizeChangeProof (EQUIVALENT) 5.56/2.16 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. 5.56/2.16 5.56/2.16 From the DPs we obtained the following set of size-change graphs: 5.56/2.16 *TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) 5.56/2.16 The graph contains the following edges 1 > 1, 2 >= 2 5.56/2.16 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (20) 5.56/2.16 YES 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (21) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) 5.56/2.16 5.56/2.16 The TRS R consists of the following rules: 5.56/2.16 5.56/2.16 convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) 5.56/2.16 convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) 5.56/2.16 convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) 5.56/2.16 U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) 5.56/2.16 U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) 5.56/2.16 times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) 5.56/2.16 times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) 5.56/2.16 U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) 5.56/2.16 plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) 5.56/2.16 plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) 5.56/2.16 U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) 5.56/2.16 U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) 5.56/2.16 U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) 5.56/2.16 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) 5.56/2.16 5.56/2.16 [] = [] 5.56/2.16 5.56/2.16 convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 .(x1, x2) = .(x1, x2) 5.56/2.16 5.56/2.16 0 = 0 5.56/2.16 5.56/2.16 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) 5.56/2.16 5.56/2.16 U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) 5.56/2.16 5.56/2.16 times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) 5.56/2.16 5.56/2.16 plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) 5.56/2.16 5.56/2.16 U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (22) UsableRulesProof (EQUIVALENT) 5.56/2.16 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (23) 5.56/2.16 Obligation: 5.56/2.16 Pi DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 The argument filtering Pi contains the following mapping: 5.56/2.16 .(x1, x2) = .(x1, x2) 5.56/2.16 5.56/2.16 0 = 0 5.56/2.16 5.56/2.16 s(x1) = s(x1) 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) 5.56/2.16 5.56/2.16 5.56/2.16 We have to consider all (P,R,Pi)-chains 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (24) PiDPToQDPProof (SOUND) 5.56/2.16 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (25) 5.56/2.16 Obligation: 5.56/2.16 Q DP problem: 5.56/2.16 The TRS P consists of the following rules: 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B) -> CONVERT_IN_GGA(.(Y, XS), B) 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B) -> CONVERT_IN_GGA(XS, B) 5.56/2.16 5.56/2.16 R is empty. 5.56/2.16 Q is empty. 5.56/2.16 We have to consider all (P,Q,R)-chains. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (26) UsableRulesReductionPairsProof (EQUIVALENT) 5.56/2.16 By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. 5.56/2.16 5.56/2.16 The following dependency pairs can be deleted: 5.56/2.16 5.56/2.16 CONVERT_IN_GGA(.(s(Y), XS), B) -> CONVERT_IN_GGA(.(Y, XS), B) 5.56/2.16 CONVERT_IN_GGA(.(0, XS), B) -> CONVERT_IN_GGA(XS, B) 5.56/2.16 No rules are removed from R. 5.56/2.16 5.56/2.16 Used ordering: POLO with Polynomial interpretation [POLO]: 5.56/2.16 5.56/2.16 POL(.(x_1, x_2)) = x_1 + x_2 5.56/2.16 POL(0) = 0 5.56/2.16 POL(CONVERT_IN_GGA(x_1, x_2)) = 2*x_1 + x_2 5.56/2.16 POL(s(x_1)) = 2*x_1 5.56/2.16 5.56/2.16 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (27) 5.56/2.16 Obligation: 5.56/2.16 Q DP problem: 5.56/2.16 P is empty. 5.56/2.16 R is empty. 5.56/2.16 Q is empty. 5.56/2.16 We have to consider all (P,Q,R)-chains. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (28) PisEmptyProof (EQUIVALENT) 5.56/2.16 The TRS P is empty. Hence, there is no (P,Q,R) chain. 5.56/2.16 ---------------------------------------- 5.56/2.16 5.56/2.16 (29) 5.56/2.16 YES 5.76/2.68 EOF