/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) CutEliminatorProof [SOUND, 0 ms] (2) Prolog (3) PrologToPiTRSProof [SOUND, 0 ms] (4) PiTRS (5) DependencyPairsProof [EQUIVALENT, 23 ms] (6) PiDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) PiDP (9) UsableRulesProof [EQUIVALENT, 0 ms] (10) PiDP (11) PiDPToQDPProof [SOUND, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES ---------------------------------------- (0) Obligation: Clauses: goal(Name, Code) :- ','(eq(Name, .(First, Others)), ','(reduce(Others, s(0), First, Reduced), eq(Code, Reduced))). reduce(X1, s(s(s(s(0)))), X2, []) :- !. reduce([], X3, X4, []) :- !. reduce(.(Current, Others), Count, Current, Code) :- ','(vowel_h_w_y(Current), ','(!, reduce(Others, Count, Current, Code))). reduce(.(Letter, Others), Count, Letter, Code) :- ','(!, reduce(Others, Count, Letter, Code)). reduce(.(Current, Others), Count, X5, .(Current, Code)) :- reduce(Others, s(Count), Current, Code). vowel_h_w_y(97). vowel_h_w_y(101). vowel_h_w_y(105). vowel_h_w_y(111). vowel_h_w_y(117). vowel_h_w_y(104). vowel_h_w_y(119). vowel_h_w_y(121). eq(X, X). Query: goal(g,a) ---------------------------------------- (1) CutEliminatorProof (SOUND) Eliminated all cuts by simply ignoring them[PROLOG]. ---------------------------------------- (2) Obligation: Clauses: goal(Name, Code) :- ','(eq(Name, .(First, Others)), ','(reduce(Others, s(0), First, Reduced), eq(Code, Reduced))). reduce(X1, s(s(s(s(0)))), X2, []). reduce([], X3, X4, []). reduce(.(Current, Others), Count, Current, Code) :- ','(vowel_h_w_y(Current), reduce(Others, Count, Current, Code)). reduce(.(Letter, Others), Count, Letter, Code) :- reduce(Others, Count, Letter, Code). reduce(.(Current, Others), Count, X5, .(Current, Code)) :- reduce(Others, s(Count), Current, Code). vowel_h_w_y(97). vowel_h_w_y(101). vowel_h_w_y(105). vowel_h_w_y(111). vowel_h_w_y(117). vowel_h_w_y(104). vowel_h_w_y(119). vowel_h_w_y(121). eq(X, X). Query: goal(g,a) ---------------------------------------- (3) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_2: (b,f) reduce_in_4: (b,b,f,f) (b,b,b,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_ga(Name, Code) -> U1_ga(Name, Code, eq_in_ga(Name, .(First, Others))) eq_in_ga(X, X) -> eq_out_ga(X, X) U1_ga(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_ga(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) reduce_in_ggaa(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggaa(X1, s(s(s(s(0)))), X2, []) reduce_in_ggaa([], X3, X4, []) -> reduce_out_ggaa([], X3, X4, []) reduce_in_ggaa(.(Current, Others), Count, Current, Code) -> U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_ggaa(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggga(X1, s(s(s(s(0)))), X2, []) reduce_in_ggga([], X3, X4, []) -> reduce_out_ggga([], X3, X4, []) reduce_in_ggga(.(Current, Others), Count, Current, Code) -> U4_ggga(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) U4_ggga(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_ggga(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(.(Letter, Others), Count, Letter, Code) -> U6_ggga(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) reduce_in_ggga(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggga(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) reduce_in_ggaa(.(Letter, Others), Count, Letter, Code) -> U6_ggaa(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) reduce_in_ggaa(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggaa(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) U7_ggaa(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, X5, .(Current, Code)) U6_ggaa(Letter, Others, Count, Code, reduce_out_ggaa(Others, Count, Letter, Code)) -> reduce_out_ggaa(.(Letter, Others), Count, Letter, Code) U7_ggga(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, X5, .(Current, Code)) U6_ggga(Letter, Others, Count, Code, reduce_out_ggga(Others, Count, Letter, Code)) -> reduce_out_ggga(.(Letter, Others), Count, Letter, Code) U5_ggga(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, Current, Code) U5_ggaa(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, Current, Code) U2_ga(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_ga(Name, Code, eq_in_ag(Code, Reduced)) eq_in_ag(X, X) -> eq_out_ag(X, X) U3_ga(Name, Code, eq_out_ag(Code, Reduced)) -> goal_out_ga(Name, Code) The argument filtering Pi contains the following mapping: goal_in_ga(x1, x2) = goal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) eq_in_ga(x1, x2) = eq_in_ga(x1) .(x1, x2) = .(x2) eq_out_ga(x1, x2) = eq_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) reduce_in_ggaa(x1, x2, x3, x4) = reduce_in_ggaa(x1, x2) s(x1) = s(x1) 0 = 0 reduce_out_ggaa(x1, x2, x3, x4) = reduce_out_ggaa(x4) [] = [] U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x2, x3, x5) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) U5_ggaa(x1, x2, x3, x4, x5) = U5_ggaa(x5) reduce_in_ggga(x1, x2, x3, x4) = reduce_in_ggga(x1, x2, x3) reduce_out_ggga(x1, x2, x3, x4) = reduce_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) U6_ggga(x1, x2, x3, x4, x5) = U6_ggga(x5) U7_ggga(x1, x2, x3, x4, x5, x6) = U7_ggga(x6) U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x5) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x6) U3_ga(x1, x2, x3) = U3_ga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) goal_out_ga(x1, x2) = goal_out_ga(x2) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (4) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_ga(Name, Code) -> U1_ga(Name, Code, eq_in_ga(Name, .(First, Others))) eq_in_ga(X, X) -> eq_out_ga(X, X) U1_ga(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_ga(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) reduce_in_ggaa(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggaa(X1, s(s(s(s(0)))), X2, []) reduce_in_ggaa([], X3, X4, []) -> reduce_out_ggaa([], X3, X4, []) reduce_in_ggaa(.(Current, Others), Count, Current, Code) -> U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_ggaa(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggga(X1, s(s(s(s(0)))), X2, []) reduce_in_ggga([], X3, X4, []) -> reduce_out_ggga([], X3, X4, []) reduce_in_ggga(.(Current, Others), Count, Current, Code) -> U4_ggga(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) U4_ggga(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_ggga(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(.(Letter, Others), Count, Letter, Code) -> U6_ggga(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) reduce_in_ggga(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggga(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) reduce_in_ggaa(.(Letter, Others), Count, Letter, Code) -> U6_ggaa(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) reduce_in_ggaa(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggaa(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) U7_ggaa(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, X5, .(Current, Code)) U6_ggaa(Letter, Others, Count, Code, reduce_out_ggaa(Others, Count, Letter, Code)) -> reduce_out_ggaa(.(Letter, Others), Count, Letter, Code) U7_ggga(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, X5, .(Current, Code)) U6_ggga(Letter, Others, Count, Code, reduce_out_ggga(Others, Count, Letter, Code)) -> reduce_out_ggga(.(Letter, Others), Count, Letter, Code) U5_ggga(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, Current, Code) U5_ggaa(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, Current, Code) U2_ga(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_ga(Name, Code, eq_in_ag(Code, Reduced)) eq_in_ag(X, X) -> eq_out_ag(X, X) U3_ga(Name, Code, eq_out_ag(Code, Reduced)) -> goal_out_ga(Name, Code) The argument filtering Pi contains the following mapping: goal_in_ga(x1, x2) = goal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) eq_in_ga(x1, x2) = eq_in_ga(x1) .(x1, x2) = .(x2) eq_out_ga(x1, x2) = eq_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) reduce_in_ggaa(x1, x2, x3, x4) = reduce_in_ggaa(x1, x2) s(x1) = s(x1) 0 = 0 reduce_out_ggaa(x1, x2, x3, x4) = reduce_out_ggaa(x4) [] = [] U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x2, x3, x5) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) U5_ggaa(x1, x2, x3, x4, x5) = U5_ggaa(x5) reduce_in_ggga(x1, x2, x3, x4) = reduce_in_ggga(x1, x2, x3) reduce_out_ggga(x1, x2, x3, x4) = reduce_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) U6_ggga(x1, x2, x3, x4, x5) = U6_ggga(x5) U7_ggga(x1, x2, x3, x4, x5, x6) = U7_ggga(x6) U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x5) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x6) U3_ga(x1, x2, x3) = U3_ga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) goal_out_ga(x1, x2) = goal_out_ga(x2) ---------------------------------------- (5) 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: GOAL_IN_GA(Name, Code) -> U1_GA(Name, Code, eq_in_ga(Name, .(First, Others))) GOAL_IN_GA(Name, Code) -> EQ_IN_GA(Name, .(First, Others)) U1_GA(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_GA(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) U1_GA(Name, Code, eq_out_ga(Name, .(First, Others))) -> REDUCE_IN_GGAA(Others, s(0), First, Reduced) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> VOWEL_H_W_Y_IN_A(Current) U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_GGAA(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> VOWEL_H_W_Y_IN_G(Current) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_GGGA(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> U6_GGGA(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGGA(Others, Count, Letter, Code) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> U7_GGGA(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> U6_GGAA(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGAA(Others, Count, Letter, Code) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> U7_GGAA(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) U2_GA(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_GA(Name, Code, eq_in_ag(Code, Reduced)) U2_GA(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> EQ_IN_AG(Code, Reduced) The TRS R consists of the following rules: goal_in_ga(Name, Code) -> U1_ga(Name, Code, eq_in_ga(Name, .(First, Others))) eq_in_ga(X, X) -> eq_out_ga(X, X) U1_ga(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_ga(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) reduce_in_ggaa(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggaa(X1, s(s(s(s(0)))), X2, []) reduce_in_ggaa([], X3, X4, []) -> reduce_out_ggaa([], X3, X4, []) reduce_in_ggaa(.(Current, Others), Count, Current, Code) -> U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_ggaa(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggga(X1, s(s(s(s(0)))), X2, []) reduce_in_ggga([], X3, X4, []) -> reduce_out_ggga([], X3, X4, []) reduce_in_ggga(.(Current, Others), Count, Current, Code) -> U4_ggga(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) U4_ggga(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_ggga(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(.(Letter, Others), Count, Letter, Code) -> U6_ggga(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) reduce_in_ggga(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggga(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) reduce_in_ggaa(.(Letter, Others), Count, Letter, Code) -> U6_ggaa(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) reduce_in_ggaa(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggaa(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) U7_ggaa(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, X5, .(Current, Code)) U6_ggaa(Letter, Others, Count, Code, reduce_out_ggaa(Others, Count, Letter, Code)) -> reduce_out_ggaa(.(Letter, Others), Count, Letter, Code) U7_ggga(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, X5, .(Current, Code)) U6_ggga(Letter, Others, Count, Code, reduce_out_ggga(Others, Count, Letter, Code)) -> reduce_out_ggga(.(Letter, Others), Count, Letter, Code) U5_ggga(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, Current, Code) U5_ggaa(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, Current, Code) U2_ga(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_ga(Name, Code, eq_in_ag(Code, Reduced)) eq_in_ag(X, X) -> eq_out_ag(X, X) U3_ga(Name, Code, eq_out_ag(Code, Reduced)) -> goal_out_ga(Name, Code) The argument filtering Pi contains the following mapping: goal_in_ga(x1, x2) = goal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) eq_in_ga(x1, x2) = eq_in_ga(x1) .(x1, x2) = .(x2) eq_out_ga(x1, x2) = eq_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) reduce_in_ggaa(x1, x2, x3, x4) = reduce_in_ggaa(x1, x2) s(x1) = s(x1) 0 = 0 reduce_out_ggaa(x1, x2, x3, x4) = reduce_out_ggaa(x4) [] = [] U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x2, x3, x5) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) U5_ggaa(x1, x2, x3, x4, x5) = U5_ggaa(x5) reduce_in_ggga(x1, x2, x3, x4) = reduce_in_ggga(x1, x2, x3) reduce_out_ggga(x1, x2, x3, x4) = reduce_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) U6_ggga(x1, x2, x3, x4, x5) = U6_ggga(x5) U7_ggga(x1, x2, x3, x4, x5, x6) = U7_ggga(x6) U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x5) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x6) U3_ga(x1, x2, x3) = U3_ga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) goal_out_ga(x1, x2) = goal_out_ga(x2) GOAL_IN_GA(x1, x2) = GOAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) EQ_IN_GA(x1, x2) = EQ_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) REDUCE_IN_GGAA(x1, x2, x3, x4) = REDUCE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x2, x3, x5) VOWEL_H_W_Y_IN_A(x1) = VOWEL_H_W_Y_IN_A U5_GGAA(x1, x2, x3, x4, x5) = U5_GGAA(x5) REDUCE_IN_GGGA(x1, x2, x3, x4) = REDUCE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) VOWEL_H_W_Y_IN_G(x1) = VOWEL_H_W_Y_IN_G(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x5) U6_GGGA(x1, x2, x3, x4, x5) = U6_GGGA(x5) U7_GGGA(x1, x2, x3, x4, x5, x6) = U7_GGGA(x6) U6_GGAA(x1, x2, x3, x4, x5) = U6_GGAA(x5) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x6) U3_GA(x1, x2, x3) = U3_GA(x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (6) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_GA(Name, Code) -> U1_GA(Name, Code, eq_in_ga(Name, .(First, Others))) GOAL_IN_GA(Name, Code) -> EQ_IN_GA(Name, .(First, Others)) U1_GA(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_GA(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) U1_GA(Name, Code, eq_out_ga(Name, .(First, Others))) -> REDUCE_IN_GGAA(Others, s(0), First, Reduced) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> VOWEL_H_W_Y_IN_A(Current) U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_GGAA(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> VOWEL_H_W_Y_IN_G(Current) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_GGGA(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> U6_GGGA(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGGA(Others, Count, Letter, Code) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> U7_GGGA(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> U6_GGAA(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGAA(Others, Count, Letter, Code) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> U7_GGAA(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) U2_GA(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_GA(Name, Code, eq_in_ag(Code, Reduced)) U2_GA(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> EQ_IN_AG(Code, Reduced) The TRS R consists of the following rules: goal_in_ga(Name, Code) -> U1_ga(Name, Code, eq_in_ga(Name, .(First, Others))) eq_in_ga(X, X) -> eq_out_ga(X, X) U1_ga(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_ga(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) reduce_in_ggaa(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggaa(X1, s(s(s(s(0)))), X2, []) reduce_in_ggaa([], X3, X4, []) -> reduce_out_ggaa([], X3, X4, []) reduce_in_ggaa(.(Current, Others), Count, Current, Code) -> U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_ggaa(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggga(X1, s(s(s(s(0)))), X2, []) reduce_in_ggga([], X3, X4, []) -> reduce_out_ggga([], X3, X4, []) reduce_in_ggga(.(Current, Others), Count, Current, Code) -> U4_ggga(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) U4_ggga(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_ggga(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(.(Letter, Others), Count, Letter, Code) -> U6_ggga(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) reduce_in_ggga(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggga(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) reduce_in_ggaa(.(Letter, Others), Count, Letter, Code) -> U6_ggaa(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) reduce_in_ggaa(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggaa(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) U7_ggaa(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, X5, .(Current, Code)) U6_ggaa(Letter, Others, Count, Code, reduce_out_ggaa(Others, Count, Letter, Code)) -> reduce_out_ggaa(.(Letter, Others), Count, Letter, Code) U7_ggga(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, X5, .(Current, Code)) U6_ggga(Letter, Others, Count, Code, reduce_out_ggga(Others, Count, Letter, Code)) -> reduce_out_ggga(.(Letter, Others), Count, Letter, Code) U5_ggga(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, Current, Code) U5_ggaa(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, Current, Code) U2_ga(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_ga(Name, Code, eq_in_ag(Code, Reduced)) eq_in_ag(X, X) -> eq_out_ag(X, X) U3_ga(Name, Code, eq_out_ag(Code, Reduced)) -> goal_out_ga(Name, Code) The argument filtering Pi contains the following mapping: goal_in_ga(x1, x2) = goal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) eq_in_ga(x1, x2) = eq_in_ga(x1) .(x1, x2) = .(x2) eq_out_ga(x1, x2) = eq_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) reduce_in_ggaa(x1, x2, x3, x4) = reduce_in_ggaa(x1, x2) s(x1) = s(x1) 0 = 0 reduce_out_ggaa(x1, x2, x3, x4) = reduce_out_ggaa(x4) [] = [] U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x2, x3, x5) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) U5_ggaa(x1, x2, x3, x4, x5) = U5_ggaa(x5) reduce_in_ggga(x1, x2, x3, x4) = reduce_in_ggga(x1, x2, x3) reduce_out_ggga(x1, x2, x3, x4) = reduce_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) U6_ggga(x1, x2, x3, x4, x5) = U6_ggga(x5) U7_ggga(x1, x2, x3, x4, x5, x6) = U7_ggga(x6) U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x5) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x6) U3_ga(x1, x2, x3) = U3_ga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) goal_out_ga(x1, x2) = goal_out_ga(x2) GOAL_IN_GA(x1, x2) = GOAL_IN_GA(x1) U1_GA(x1, x2, x3) = U1_GA(x3) EQ_IN_GA(x1, x2) = EQ_IN_GA(x1) U2_GA(x1, x2, x3) = U2_GA(x3) REDUCE_IN_GGAA(x1, x2, x3, x4) = REDUCE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x2, x3, x5) VOWEL_H_W_Y_IN_A(x1) = VOWEL_H_W_Y_IN_A U5_GGAA(x1, x2, x3, x4, x5) = U5_GGAA(x5) REDUCE_IN_GGGA(x1, x2, x3, x4) = REDUCE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) VOWEL_H_W_Y_IN_G(x1) = VOWEL_H_W_Y_IN_G(x1) U5_GGGA(x1, x2, x3, x4, x5) = U5_GGGA(x5) U6_GGGA(x1, x2, x3, x4, x5) = U6_GGGA(x5) U7_GGGA(x1, x2, x3, x4, x5, x6) = U7_GGGA(x6) U6_GGAA(x1, x2, x3, x4, x5) = U6_GGAA(x5) U7_GGAA(x1, x2, x3, x4, x5, x6) = U7_GGAA(x6) U3_GA(x1, x2, x3) = U3_GA(x3) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 14 less nodes. ---------------------------------------- (8) Obligation: Pi DP problem: The TRS P consists of the following rules: U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGGA(Others, Count, Letter, Code) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGAA(Others, Count, Letter, Code) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) The TRS R consists of the following rules: goal_in_ga(Name, Code) -> U1_ga(Name, Code, eq_in_ga(Name, .(First, Others))) eq_in_ga(X, X) -> eq_out_ga(X, X) U1_ga(Name, Code, eq_out_ga(Name, .(First, Others))) -> U2_ga(Name, Code, reduce_in_ggaa(Others, s(0), First, Reduced)) reduce_in_ggaa(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggaa(X1, s(s(s(s(0)))), X2, []) reduce_in_ggaa([], X3, X4, []) -> reduce_out_ggaa([], X3, X4, []) reduce_in_ggaa(.(Current, Others), Count, Current, Code) -> U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) U4_ggaa(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> U5_ggaa(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(X1, s(s(s(s(0)))), X2, []) -> reduce_out_ggga(X1, s(s(s(s(0)))), X2, []) reduce_in_ggga([], X3, X4, []) -> reduce_out_ggga([], X3, X4, []) reduce_in_ggga(.(Current, Others), Count, Current, Code) -> U4_ggga(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) U4_ggga(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> U5_ggga(Current, Others, Count, Code, reduce_in_ggga(Others, Count, Current, Code)) reduce_in_ggga(.(Letter, Others), Count, Letter, Code) -> U6_ggga(Letter, Others, Count, Code, reduce_in_ggga(Others, Count, Letter, Code)) reduce_in_ggga(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggga(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) reduce_in_ggaa(.(Letter, Others), Count, Letter, Code) -> U6_ggaa(Letter, Others, Count, Code, reduce_in_ggaa(Others, Count, Letter, Code)) reduce_in_ggaa(.(Current, Others), Count, X5, .(Current, Code)) -> U7_ggaa(Current, Others, Count, X5, Code, reduce_in_ggaa(Others, s(Count), Current, Code)) U7_ggaa(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, X5, .(Current, Code)) U6_ggaa(Letter, Others, Count, Code, reduce_out_ggaa(Others, Count, Letter, Code)) -> reduce_out_ggaa(.(Letter, Others), Count, Letter, Code) U7_ggga(Current, Others, Count, X5, Code, reduce_out_ggaa(Others, s(Count), Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, X5, .(Current, Code)) U6_ggga(Letter, Others, Count, Code, reduce_out_ggga(Others, Count, Letter, Code)) -> reduce_out_ggga(.(Letter, Others), Count, Letter, Code) U5_ggga(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggga(.(Current, Others), Count, Current, Code) U5_ggaa(Current, Others, Count, Code, reduce_out_ggga(Others, Count, Current, Code)) -> reduce_out_ggaa(.(Current, Others), Count, Current, Code) U2_ga(Name, Code, reduce_out_ggaa(Others, s(0), First, Reduced)) -> U3_ga(Name, Code, eq_in_ag(Code, Reduced)) eq_in_ag(X, X) -> eq_out_ag(X, X) U3_ga(Name, Code, eq_out_ag(Code, Reduced)) -> goal_out_ga(Name, Code) The argument filtering Pi contains the following mapping: goal_in_ga(x1, x2) = goal_in_ga(x1) U1_ga(x1, x2, x3) = U1_ga(x3) eq_in_ga(x1, x2) = eq_in_ga(x1) .(x1, x2) = .(x2) eq_out_ga(x1, x2) = eq_out_ga(x2) U2_ga(x1, x2, x3) = U2_ga(x3) reduce_in_ggaa(x1, x2, x3, x4) = reduce_in_ggaa(x1, x2) s(x1) = s(x1) 0 = 0 reduce_out_ggaa(x1, x2, x3, x4) = reduce_out_ggaa(x4) [] = [] U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x2, x3, x5) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) U5_ggaa(x1, x2, x3, x4, x5) = U5_ggaa(x5) reduce_in_ggga(x1, x2, x3, x4) = reduce_in_ggga(x1, x2, x3) reduce_out_ggga(x1, x2, x3, x4) = reduce_out_ggga(x4) U4_ggga(x1, x2, x3, x4, x5) = U4_ggga(x1, x2, x3, x5) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 U5_ggga(x1, x2, x3, x4, x5) = U5_ggga(x5) U6_ggga(x1, x2, x3, x4, x5) = U6_ggga(x5) U7_ggga(x1, x2, x3, x4, x5, x6) = U7_ggga(x6) U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x5) U7_ggaa(x1, x2, x3, x4, x5, x6) = U7_ggaa(x6) U3_ga(x1, x2, x3) = U3_ga(x3) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) goal_out_ga(x1, x2) = goal_out_ga(x2) REDUCE_IN_GGAA(x1, x2, x3, x4) = REDUCE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x2, x3, x5) REDUCE_IN_GGGA(x1, x2, x3, x4) = REDUCE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (9) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (10) Obligation: Pi DP problem: The TRS P consists of the following rules: U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Current, Others), Count, Current, Code) -> U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_in_g(Current)) U4_GGGA(Current, Others, Count, Code, vowel_h_w_y_out_g(Current)) -> REDUCE_IN_GGGA(Others, Count, Current, Code) REDUCE_IN_GGGA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGGA(Others, Count, Letter, Code) REDUCE_IN_GGGA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) REDUCE_IN_GGAA(.(Current, Others), Count, Current, Code) -> U4_GGAA(Current, Others, Count, Code, vowel_h_w_y_in_a(Current)) REDUCE_IN_GGAA(.(Letter, Others), Count, Letter, Code) -> REDUCE_IN_GGAA(Others, Count, Letter, Code) REDUCE_IN_GGAA(.(Current, Others), Count, X5, .(Current, Code)) -> REDUCE_IN_GGAA(Others, s(Count), Current, Code) The TRS R consists of the following rules: vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g(97) vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g(101) vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g(105) vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g(111) vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g(117) vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g(104) vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g(119) vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g(121) vowel_h_w_y_in_a(97) -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a(101) -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a(105) -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a(111) -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a(117) -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a(104) -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a(119) -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a(121) -> vowel_h_w_y_out_a(121) The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) s(x1) = s(x1) vowel_h_w_y_in_a(x1) = vowel_h_w_y_in_a vowel_h_w_y_out_a(x1) = vowel_h_w_y_out_a(x1) vowel_h_w_y_in_g(x1) = vowel_h_w_y_in_g(x1) 97 = 97 vowel_h_w_y_out_g(x1) = vowel_h_w_y_out_g 101 = 101 105 = 105 111 = 111 117 = 117 104 = 104 119 = 119 121 = 121 REDUCE_IN_GGAA(x1, x2, x3, x4) = REDUCE_IN_GGAA(x1, x2) U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x2, x3, x5) REDUCE_IN_GGGA(x1, x2, x3, x4) = REDUCE_IN_GGGA(x1, x2, x3) U4_GGGA(x1, x2, x3, x4, x5) = U4_GGGA(x1, x2, x3, x5) We have to consider all (P,R,Pi)-chains ---------------------------------------- (11) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U4_GGAA(Others, Count, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current) REDUCE_IN_GGGA(.(Others), Count, Current) -> U4_GGGA(Current, Others, Count, vowel_h_w_y_in_g(Current)) U4_GGGA(Current, Others, Count, vowel_h_w_y_out_g) -> REDUCE_IN_GGGA(Others, Count, Current) REDUCE_IN_GGGA(.(Others), Count, Letter) -> REDUCE_IN_GGGA(Others, Count, Letter) REDUCE_IN_GGGA(.(Others), Count, X5) -> REDUCE_IN_GGAA(Others, s(Count)) REDUCE_IN_GGAA(.(Others), Count) -> U4_GGAA(Others, Count, vowel_h_w_y_in_a) REDUCE_IN_GGAA(.(Others), Count) -> REDUCE_IN_GGAA(Others, Count) REDUCE_IN_GGAA(.(Others), Count) -> REDUCE_IN_GGAA(Others, s(Count)) The TRS R consists of the following rules: vowel_h_w_y_in_g(97) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(101) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(105) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(111) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(117) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(104) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(119) -> vowel_h_w_y_out_g vowel_h_w_y_in_g(121) -> vowel_h_w_y_out_g vowel_h_w_y_in_a -> vowel_h_w_y_out_a(97) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(101) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(105) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(111) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(117) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(104) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(119) vowel_h_w_y_in_a -> vowel_h_w_y_out_a(121) The set Q consists of the following terms: vowel_h_w_y_in_g(x0) vowel_h_w_y_in_a We have to consider all (P,Q,R)-chains. ---------------------------------------- (13) 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: *REDUCE_IN_GGAA(.(Others), Count) -> U4_GGAA(Others, Count, vowel_h_w_y_in_a) The graph contains the following edges 1 > 1, 2 >= 2 *REDUCE_IN_GGGA(.(Others), Count, Letter) -> REDUCE_IN_GGGA(Others, Count, Letter) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U4_GGGA(Current, Others, Count, vowel_h_w_y_out_g) -> REDUCE_IN_GGGA(Others, Count, Current) The graph contains the following edges 2 >= 1, 3 >= 2, 1 >= 3 *U4_GGAA(Others, Count, vowel_h_w_y_out_a(Current)) -> REDUCE_IN_GGGA(Others, Count, Current) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *REDUCE_IN_GGGA(.(Others), Count, Current) -> U4_GGGA(Current, Others, Count, vowel_h_w_y_in_g(Current)) The graph contains the following edges 3 >= 1, 1 > 2, 2 >= 3 *REDUCE_IN_GGGA(.(Others), Count, X5) -> REDUCE_IN_GGAA(Others, s(Count)) The graph contains the following edges 1 > 1 *REDUCE_IN_GGAA(.(Others), Count) -> REDUCE_IN_GGAA(Others, Count) The graph contains the following edges 1 > 1, 2 >= 2 *REDUCE_IN_GGAA(.(Others), Count) -> REDUCE_IN_GGAA(Others, s(Count)) The graph contains the following edges 1 > 1 ---------------------------------------- (14) YES