/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern convert(g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (27) QDP (28) PisEmptyProof [EQUIVALENT, 0 ms] (29) YES ---------------------------------------- (0) Obligation: Clauses: convert([], B, 0). convert(.(0, XS), B, X) :- ','(convert(XS, B, Y), times(Y, B, X)). convert(.(s(Y), XS), B, s(X)) :- convert(.(Y, XS), B, X). plus(0, Y, Y). plus(s(X), Y, s(Z)) :- plus(X, Y, Z). times(0, Y, 0). times(s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)). Query: convert(g,g,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: convert_in_3: (b,b,f) times_in_3: (b,b,f) plus_in_3: (b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: CONVERT_IN_GGA(.(0, XS), B, X) -> U1_GGA(XS, B, X, convert_in_gga(XS, B, Y)) CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> U3_GGA(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_GGA(XS, B, X, times_in_gga(Y, B, X)) U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> TIMES_IN_GGA(Y, B, X) TIMES_IN_GGA(s(X), Y, Z) -> U5_GGA(X, Y, Z, times_in_gga(X, Y, U)) TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U6_GGA(X, Y, Z, plus_in_gga(Y, U, Z)) U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) PLUS_IN_GGA(s(X), Y, s(Z)) -> U4_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: CONVERT_IN_GGA(.(0, XS), B, X) -> U1_GGA(XS, B, X, convert_in_gga(XS, B, Y)) CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> U3_GGA(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_GGA(XS, B, X, times_in_gga(Y, B, X)) U1_GGA(XS, B, X, convert_out_gga(XS, B, Y)) -> TIMES_IN_GGA(Y, B, X) TIMES_IN_GGA(s(X), Y, Z) -> U5_GGA(X, Y, Z, times_in_gga(X, Y, U)) TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> U6_GGA(X, Y, Z, plus_in_gga(Y, U, Z)) U5_GGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z) PLUS_IN_GGA(s(X), Y, s(Z)) -> U4_GGA(X, Y, Z, plus_in_gga(X, Y, Z)) PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4) U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x3, x5) U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4) U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) PLUS_IN_GGA(x1, x2, x3) = PLUS_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *PLUS_IN_GGA(s(X), Y) -> PLUS_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) TIMES_IN_GGA(x1, x2, x3) = TIMES_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *TIMES_IN_GGA(s(X), Y) -> TIMES_IN_GGA(X, Y) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) The TRS R consists of the following rules: convert_in_gga([], B, 0) -> convert_out_gga([], B, 0) convert_in_gga(.(0, XS), B, X) -> U1_gga(XS, B, X, convert_in_gga(XS, B, Y)) convert_in_gga(.(s(Y), XS), B, s(X)) -> U3_gga(Y, XS, B, X, convert_in_gga(.(Y, XS), B, X)) U3_gga(Y, XS, B, X, convert_out_gga(.(Y, XS), B, X)) -> convert_out_gga(.(s(Y), XS), B, s(X)) U1_gga(XS, B, X, convert_out_gga(XS, B, Y)) -> U2_gga(XS, B, X, times_in_gga(Y, B, X)) times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0) times_in_gga(s(X), Y, Z) -> U5_gga(X, Y, Z, times_in_gga(X, Y, U)) U5_gga(X, Y, Z, times_out_gga(X, Y, U)) -> U6_gga(X, Y, Z, plus_in_gga(Y, U, Z)) plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y) plus_in_gga(s(X), Y, s(Z)) -> U4_gga(X, Y, Z, plus_in_gga(X, Y, Z)) U4_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z)) U6_gga(X, Y, Z, plus_out_gga(Y, U, Z)) -> times_out_gga(s(X), Y, Z) U2_gga(XS, B, X, times_out_gga(Y, B, X)) -> convert_out_gga(.(0, XS), B, X) The argument filtering Pi contains the following mapping: convert_in_gga(x1, x2, x3) = convert_in_gga(x1, x2) [] = [] convert_out_gga(x1, x2, x3) = convert_out_gga(x1, x2, x3) .(x1, x2) = .(x1, x2) 0 = 0 U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4) s(x1) = s(x1) U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x3, x5) U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4) times_in_gga(x1, x2, x3) = times_in_gga(x1, x2) times_out_gga(x1, x2, x3) = times_out_gga(x1, x2, x3) U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4) U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4) plus_in_gga(x1, x2, x3) = plus_in_gga(x1, x2) plus_out_gga(x1, x2, x3) = plus_out_gga(x1, x2, x3) U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4) CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: CONVERT_IN_GGA(.(s(Y), XS), B, s(X)) -> CONVERT_IN_GGA(.(Y, XS), B, X) CONVERT_IN_GGA(.(0, XS), B, X) -> CONVERT_IN_GGA(XS, B, Y) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) 0 = 0 s(x1) = s(x1) CONVERT_IN_GGA(x1, x2, x3) = CONVERT_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: CONVERT_IN_GGA(.(s(Y), XS), B) -> CONVERT_IN_GGA(.(Y, XS), B) CONVERT_IN_GGA(.(0, XS), B) -> CONVERT_IN_GGA(XS, B) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesReductionPairsProof (EQUIVALENT) 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. The following dependency pairs can be deleted: CONVERT_IN_GGA(.(s(Y), XS), B) -> CONVERT_IN_GGA(.(Y, XS), B) CONVERT_IN_GGA(.(0, XS), B) -> CONVERT_IN_GGA(XS, B) No rules are removed from R. Used ordering: POLO with Polynomial interpretation [POLO]: POL(.(x_1, x_2)) = x_1 + x_2 POL(0) = 0 POL(CONVERT_IN_GGA(x_1, x_2)) = 2*x_1 + x_2 POL(s(x_1)) = 2*x_1 ---------------------------------------- (27) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (28) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (29) YES