5.61/2.20 YES 5.61/2.21 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 5.61/2.21 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.61/2.21 5.61/2.21 5.61/2.21 Left Termination of the query pattern 5.61/2.21 5.61/2.21 e(g,a) 5.61/2.21 5.61/2.21 w.r.t. the given Prolog program could successfully be proven: 5.61/2.21 5.61/2.21 (0) Prolog 5.61/2.21 (1) PrologToPiTRSProof [SOUND, 0 ms] 5.61/2.21 (2) PiTRS 5.61/2.21 (3) DependencyPairsProof [EQUIVALENT, 12 ms] 5.61/2.21 (4) PiDP 5.61/2.21 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.21 (6) PiDP 5.61/2.21 (7) PiDPToQDPProof [SOUND, 15 ms] 5.61/2.21 (8) QDP 5.61/2.21 (9) UsableRulesReductionPairsProof [EQUIVALENT, 55 ms] 5.61/2.21 (10) QDP 5.61/2.21 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 5.61/2.21 (12) TRUE 5.61/2.21 5.61/2.21 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (0) 5.61/2.21 Obligation: 5.61/2.21 Clauses: 5.61/2.21 5.61/2.21 e(L, T) :- t(L, T). 5.61/2.21 e(L, T) :- ','(t(L, .(plus, C)), e(C, T)). 5.61/2.21 t(L, T) :- n(L, T). 5.61/2.21 t(L, T) :- ','(n(L, .(star, C)), t(C, T)). 5.61/2.21 n(.(L, T), T) :- z(L). 5.61/2.21 n(.(lbrace, A), B) :- e(A, .(rbrace, B)). 5.61/2.21 z(a). 5.61/2.21 z(b). 5.61/2.21 z(c). 5.61/2.21 5.61/2.21 5.61/2.21 Query: e(g,a) 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (1) PrologToPiTRSProof (SOUND) 5.61/2.21 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 5.61/2.21 5.61/2.21 e_in_2: (b,f) 5.61/2.21 5.61/2.21 t_in_2: (b,f) 5.61/2.21 5.61/2.21 n_in_2: (b,f) 5.61/2.21 5.61/2.21 Transforming Prolog into the following Term Rewriting System: 5.61/2.21 5.61/2.21 Pi-finite rewrite system: 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T)) 5.61/2.21 t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T)) 5.61/2.21 n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g(a) 5.61/2.21 z_in_g(b) -> z_out_g(b) 5.61/2.21 z_in_g(c) -> z_out_g(c) 5.61/2.21 U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T) 5.61/2.21 n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T)) 5.61/2.21 U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T) 5.61/2.21 U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T)) 5.61/2.21 U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T) 5.61/2.21 U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B) 5.61/2.21 U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T) 5.61/2.21 U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T) 5.61/2.21 5.61/2.21 The argument filtering Pi contains the following mapping: 5.61/2.21 e_in_ga(x1, x2) = e_in_ga(x1) 5.61/2.21 5.61/2.21 U1_ga(x1, x2, x3) = U1_ga(x3) 5.61/2.21 5.61/2.21 t_in_ga(x1, x2) = t_in_ga(x1) 5.61/2.21 5.61/2.21 U4_ga(x1, x2, x3) = U4_ga(x3) 5.61/2.21 5.61/2.21 n_in_ga(x1, x2) = n_in_ga(x1) 5.61/2.21 5.61/2.21 .(x1, x2) = .(x1, x2) 5.61/2.21 5.61/2.21 U7_ga(x1, x2, x3) = U7_ga(x2, x3) 5.61/2.21 5.61/2.21 z_in_g(x1) = z_in_g(x1) 5.61/2.21 5.61/2.21 a = a 5.61/2.21 5.61/2.21 z_out_g(x1) = z_out_g 5.61/2.21 5.61/2.21 b = b 5.61/2.21 5.61/2.21 c = c 5.61/2.21 5.61/2.21 n_out_ga(x1, x2) = n_out_ga(x2) 5.61/2.21 5.61/2.21 lbrace = lbrace 5.61/2.21 5.61/2.21 U8_ga(x1, x2, x3) = U8_ga(x3) 5.61/2.21 5.61/2.21 U2_ga(x1, x2, x3) = U2_ga(x3) 5.61/2.21 5.61/2.21 U5_ga(x1, x2, x3) = U5_ga(x3) 5.61/2.21 5.61/2.21 star = star 5.61/2.21 5.61/2.21 U6_ga(x1, x2, x3) = U6_ga(x3) 5.61/2.21 5.61/2.21 t_out_ga(x1, x2) = t_out_ga(x2) 5.61/2.21 5.61/2.21 plus = plus 5.61/2.21 5.61/2.21 U3_ga(x1, x2, x3) = U3_ga(x3) 5.61/2.21 5.61/2.21 e_out_ga(x1, x2) = e_out_ga(x2) 5.61/2.21 5.61/2.21 rbrace = rbrace 5.61/2.21 5.61/2.21 5.61/2.21 5.61/2.21 5.61/2.21 5.61/2.21 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 5.61/2.21 5.61/2.21 5.61/2.21 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (2) 5.61/2.21 Obligation: 5.61/2.21 Pi-finite rewrite system: 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T)) 5.61/2.21 t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T)) 5.61/2.21 n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g(a) 5.61/2.21 z_in_g(b) -> z_out_g(b) 5.61/2.21 z_in_g(c) -> z_out_g(c) 5.61/2.21 U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T) 5.61/2.21 n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T)) 5.61/2.21 U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T) 5.61/2.21 U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T)) 5.61/2.21 U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T) 5.61/2.21 U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B) 5.61/2.21 U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T) 5.61/2.21 U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T) 5.61/2.21 5.61/2.21 The argument filtering Pi contains the following mapping: 5.61/2.21 e_in_ga(x1, x2) = e_in_ga(x1) 5.61/2.21 5.61/2.21 U1_ga(x1, x2, x3) = U1_ga(x3) 5.61/2.21 5.61/2.21 t_in_ga(x1, x2) = t_in_ga(x1) 5.61/2.21 5.61/2.21 U4_ga(x1, x2, x3) = U4_ga(x3) 5.61/2.21 5.61/2.21 n_in_ga(x1, x2) = n_in_ga(x1) 5.61/2.21 5.61/2.21 .(x1, x2) = .(x1, x2) 5.61/2.21 5.61/2.21 U7_ga(x1, x2, x3) = U7_ga(x2, x3) 5.61/2.21 5.61/2.21 z_in_g(x1) = z_in_g(x1) 5.61/2.21 5.61/2.21 a = a 5.61/2.21 5.61/2.21 z_out_g(x1) = z_out_g 5.61/2.21 5.61/2.21 b = b 5.61/2.21 5.61/2.21 c = c 5.61/2.21 5.61/2.21 n_out_ga(x1, x2) = n_out_ga(x2) 5.61/2.21 5.61/2.21 lbrace = lbrace 5.61/2.21 5.61/2.21 U8_ga(x1, x2, x3) = U8_ga(x3) 5.61/2.21 5.61/2.21 U2_ga(x1, x2, x3) = U2_ga(x3) 5.61/2.21 5.61/2.21 U5_ga(x1, x2, x3) = U5_ga(x3) 5.61/2.21 5.61/2.21 star = star 5.61/2.21 5.61/2.21 U6_ga(x1, x2, x3) = U6_ga(x3) 5.61/2.21 5.61/2.21 t_out_ga(x1, x2) = t_out_ga(x2) 5.61/2.21 5.61/2.21 plus = plus 5.61/2.21 5.61/2.21 U3_ga(x1, x2, x3) = U3_ga(x3) 5.61/2.21 5.61/2.21 e_out_ga(x1, x2) = e_out_ga(x2) 5.61/2.21 5.61/2.21 rbrace = rbrace 5.61/2.21 5.61/2.21 5.61/2.21 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (3) DependencyPairsProof (EQUIVALENT) 5.61/2.21 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 5.61/2.21 Pi DP problem: 5.61/2.21 The TRS P consists of the following rules: 5.61/2.21 5.61/2.21 E_IN_GA(L, T) -> U1_GA(L, T, t_in_ga(L, T)) 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, T) 5.61/2.21 T_IN_GA(L, T) -> U4_GA(L, T, n_in_ga(L, T)) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, T) 5.61/2.21 N_IN_GA(.(L, T), T) -> U7_GA(L, T, z_in_g(L)) 5.61/2.21 N_IN_GA(.(L, T), T) -> Z_IN_G(L) 5.61/2.21 N_IN_GA(.(lbrace, A), B) -> U8_GA(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B)) 5.61/2.21 E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C)) 5.61/2.21 T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, .(star, C)) 5.61/2.21 U5_GA(L, T, n_out_ga(L, .(star, C))) -> U6_GA(L, T, t_in_ga(C, T)) 5.61/2.21 U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T) 5.61/2.21 U2_GA(L, T, t_out_ga(L, .(plus, C))) -> U3_GA(L, T, e_in_ga(C, T)) 5.61/2.21 U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T) 5.61/2.21 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T)) 5.61/2.21 t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T)) 5.61/2.21 n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g(a) 5.61/2.21 z_in_g(b) -> z_out_g(b) 5.61/2.21 z_in_g(c) -> z_out_g(c) 5.61/2.21 U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T) 5.61/2.21 n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T)) 5.61/2.21 U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T) 5.61/2.21 U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T)) 5.61/2.21 U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T) 5.61/2.21 U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B) 5.61/2.21 U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T) 5.61/2.21 U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T) 5.61/2.21 5.61/2.21 The argument filtering Pi contains the following mapping: 5.61/2.21 e_in_ga(x1, x2) = e_in_ga(x1) 5.61/2.21 5.61/2.21 U1_ga(x1, x2, x3) = U1_ga(x3) 5.61/2.21 5.61/2.21 t_in_ga(x1, x2) = t_in_ga(x1) 5.61/2.21 5.61/2.21 U4_ga(x1, x2, x3) = U4_ga(x3) 5.61/2.21 5.61/2.21 n_in_ga(x1, x2) = n_in_ga(x1) 5.61/2.21 5.61/2.21 .(x1, x2) = .(x1, x2) 5.61/2.21 5.61/2.21 U7_ga(x1, x2, x3) = U7_ga(x2, x3) 5.61/2.21 5.61/2.21 z_in_g(x1) = z_in_g(x1) 5.61/2.21 5.61/2.21 a = a 5.61/2.21 5.61/2.21 z_out_g(x1) = z_out_g 5.61/2.21 5.61/2.21 b = b 5.61/2.21 5.61/2.21 c = c 5.61/2.21 5.61/2.21 n_out_ga(x1, x2) = n_out_ga(x2) 5.61/2.21 5.61/2.21 lbrace = lbrace 5.61/2.21 5.61/2.21 U8_ga(x1, x2, x3) = U8_ga(x3) 5.61/2.21 5.61/2.21 U2_ga(x1, x2, x3) = U2_ga(x3) 5.61/2.21 5.61/2.21 U5_ga(x1, x2, x3) = U5_ga(x3) 5.61/2.21 5.61/2.21 star = star 5.61/2.21 5.61/2.21 U6_ga(x1, x2, x3) = U6_ga(x3) 5.61/2.21 5.61/2.21 t_out_ga(x1, x2) = t_out_ga(x2) 5.61/2.21 5.61/2.21 plus = plus 5.61/2.21 5.61/2.21 U3_ga(x1, x2, x3) = U3_ga(x3) 5.61/2.21 5.61/2.21 e_out_ga(x1, x2) = e_out_ga(x2) 5.61/2.21 5.61/2.21 rbrace = rbrace 5.61/2.21 5.61/2.21 E_IN_GA(x1, x2) = E_IN_GA(x1) 5.61/2.21 5.61/2.21 U1_GA(x1, x2, x3) = U1_GA(x3) 5.61/2.21 5.61/2.21 T_IN_GA(x1, x2) = T_IN_GA(x1) 5.61/2.21 5.61/2.21 U4_GA(x1, x2, x3) = U4_GA(x3) 5.61/2.21 5.61/2.21 N_IN_GA(x1, x2) = N_IN_GA(x1) 5.61/2.21 5.61/2.21 U7_GA(x1, x2, x3) = U7_GA(x2, x3) 5.61/2.21 5.61/2.21 Z_IN_G(x1) = Z_IN_G(x1) 5.61/2.21 5.61/2.21 U8_GA(x1, x2, x3) = U8_GA(x3) 5.61/2.21 5.61/2.21 U2_GA(x1, x2, x3) = U2_GA(x3) 5.61/2.21 5.61/2.21 U5_GA(x1, x2, x3) = U5_GA(x3) 5.61/2.21 5.61/2.21 U6_GA(x1, x2, x3) = U6_GA(x3) 5.61/2.21 5.61/2.21 U3_GA(x1, x2, x3) = U3_GA(x3) 5.61/2.21 5.61/2.21 5.61/2.21 We have to consider all (P,R,Pi)-chains 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (4) 5.61/2.21 Obligation: 5.61/2.21 Pi DP problem: 5.61/2.21 The TRS P consists of the following rules: 5.61/2.21 5.61/2.21 E_IN_GA(L, T) -> U1_GA(L, T, t_in_ga(L, T)) 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, T) 5.61/2.21 T_IN_GA(L, T) -> U4_GA(L, T, n_in_ga(L, T)) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, T) 5.61/2.21 N_IN_GA(.(L, T), T) -> U7_GA(L, T, z_in_g(L)) 5.61/2.21 N_IN_GA(.(L, T), T) -> Z_IN_G(L) 5.61/2.21 N_IN_GA(.(lbrace, A), B) -> U8_GA(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B)) 5.61/2.21 E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C)) 5.61/2.21 T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, .(star, C)) 5.61/2.21 U5_GA(L, T, n_out_ga(L, .(star, C))) -> U6_GA(L, T, t_in_ga(C, T)) 5.61/2.21 U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T) 5.61/2.21 U2_GA(L, T, t_out_ga(L, .(plus, C))) -> U3_GA(L, T, e_in_ga(C, T)) 5.61/2.21 U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T) 5.61/2.21 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T)) 5.61/2.21 t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T)) 5.61/2.21 n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g(a) 5.61/2.21 z_in_g(b) -> z_out_g(b) 5.61/2.21 z_in_g(c) -> z_out_g(c) 5.61/2.21 U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T) 5.61/2.21 n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T)) 5.61/2.21 U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T) 5.61/2.21 U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T)) 5.61/2.21 U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T) 5.61/2.21 U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B) 5.61/2.21 U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T) 5.61/2.21 U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T) 5.61/2.21 5.61/2.21 The argument filtering Pi contains the following mapping: 5.61/2.21 e_in_ga(x1, x2) = e_in_ga(x1) 5.61/2.21 5.61/2.21 U1_ga(x1, x2, x3) = U1_ga(x3) 5.61/2.21 5.61/2.21 t_in_ga(x1, x2) = t_in_ga(x1) 5.61/2.21 5.61/2.21 U4_ga(x1, x2, x3) = U4_ga(x3) 5.61/2.21 5.61/2.21 n_in_ga(x1, x2) = n_in_ga(x1) 5.61/2.21 5.61/2.21 .(x1, x2) = .(x1, x2) 5.61/2.21 5.61/2.21 U7_ga(x1, x2, x3) = U7_ga(x2, x3) 5.61/2.21 5.61/2.21 z_in_g(x1) = z_in_g(x1) 5.61/2.21 5.61/2.21 a = a 5.61/2.21 5.61/2.21 z_out_g(x1) = z_out_g 5.61/2.21 5.61/2.21 b = b 5.61/2.21 5.61/2.21 c = c 5.61/2.21 5.61/2.21 n_out_ga(x1, x2) = n_out_ga(x2) 5.61/2.21 5.61/2.21 lbrace = lbrace 5.61/2.21 5.61/2.21 U8_ga(x1, x2, x3) = U8_ga(x3) 5.61/2.21 5.61/2.21 U2_ga(x1, x2, x3) = U2_ga(x3) 5.61/2.21 5.61/2.21 U5_ga(x1, x2, x3) = U5_ga(x3) 5.61/2.21 5.61/2.21 star = star 5.61/2.21 5.61/2.21 U6_ga(x1, x2, x3) = U6_ga(x3) 5.61/2.21 5.61/2.21 t_out_ga(x1, x2) = t_out_ga(x2) 5.61/2.21 5.61/2.21 plus = plus 5.61/2.21 5.61/2.21 U3_ga(x1, x2, x3) = U3_ga(x3) 5.61/2.21 5.61/2.21 e_out_ga(x1, x2) = e_out_ga(x2) 5.61/2.21 5.61/2.21 rbrace = rbrace 5.61/2.21 5.61/2.21 E_IN_GA(x1, x2) = E_IN_GA(x1) 5.61/2.21 5.61/2.21 U1_GA(x1, x2, x3) = U1_GA(x3) 5.61/2.21 5.61/2.21 T_IN_GA(x1, x2) = T_IN_GA(x1) 5.61/2.21 5.61/2.21 U4_GA(x1, x2, x3) = U4_GA(x3) 5.61/2.21 5.61/2.21 N_IN_GA(x1, x2) = N_IN_GA(x1) 5.61/2.21 5.61/2.21 U7_GA(x1, x2, x3) = U7_GA(x2, x3) 5.61/2.21 5.61/2.21 Z_IN_G(x1) = Z_IN_G(x1) 5.61/2.21 5.61/2.21 U8_GA(x1, x2, x3) = U8_GA(x3) 5.61/2.21 5.61/2.21 U2_GA(x1, x2, x3) = U2_GA(x3) 5.61/2.21 5.61/2.21 U5_GA(x1, x2, x3) = U5_GA(x3) 5.61/2.21 5.61/2.21 U6_GA(x1, x2, x3) = U6_GA(x3) 5.61/2.21 5.61/2.21 U3_GA(x1, x2, x3) = U3_GA(x3) 5.61/2.21 5.61/2.21 5.61/2.21 We have to consider all (P,R,Pi)-chains 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (5) DependencyGraphProof (EQUIVALENT) 5.61/2.21 The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes. 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (6) 5.61/2.21 Obligation: 5.61/2.21 Pi DP problem: 5.61/2.21 The TRS P consists of the following rules: 5.61/2.21 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, T) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, T) 5.61/2.21 N_IN_GA(.(lbrace, A), B) -> E_IN_GA(A, .(rbrace, B)) 5.61/2.21 E_IN_GA(L, T) -> U2_GA(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 U2_GA(L, T, t_out_ga(L, .(plus, C))) -> E_IN_GA(C, T) 5.61/2.21 E_IN_GA(L, T) -> T_IN_GA(L, .(plus, C)) 5.61/2.21 T_IN_GA(L, T) -> U5_GA(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_GA(L, T, n_out_ga(L, .(star, C))) -> T_IN_GA(C, T) 5.61/2.21 T_IN_GA(L, T) -> N_IN_GA(L, .(star, C)) 5.61/2.21 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L, T) -> U1_ga(L, T, t_in_ga(L, T)) 5.61/2.21 t_in_ga(L, T) -> U4_ga(L, T, n_in_ga(L, T)) 5.61/2.21 n_in_ga(.(L, T), T) -> U7_ga(L, T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g(a) 5.61/2.21 z_in_g(b) -> z_out_g(b) 5.61/2.21 z_in_g(c) -> z_out_g(c) 5.61/2.21 U7_ga(L, T, z_out_g(L)) -> n_out_ga(.(L, T), T) 5.61/2.21 n_in_ga(.(lbrace, A), B) -> U8_ga(A, B, e_in_ga(A, .(rbrace, B))) 5.61/2.21 e_in_ga(L, T) -> U2_ga(L, T, t_in_ga(L, .(plus, C))) 5.61/2.21 t_in_ga(L, T) -> U5_ga(L, T, n_in_ga(L, .(star, C))) 5.61/2.21 U5_ga(L, T, n_out_ga(L, .(star, C))) -> U6_ga(L, T, t_in_ga(C, T)) 5.61/2.21 U6_ga(L, T, t_out_ga(C, T)) -> t_out_ga(L, T) 5.61/2.21 U2_ga(L, T, t_out_ga(L, .(plus, C))) -> U3_ga(L, T, e_in_ga(C, T)) 5.61/2.21 U3_ga(L, T, e_out_ga(C, T)) -> e_out_ga(L, T) 5.61/2.21 U8_ga(A, B, e_out_ga(A, .(rbrace, B))) -> n_out_ga(.(lbrace, A), B) 5.61/2.21 U4_ga(L, T, n_out_ga(L, T)) -> t_out_ga(L, T) 5.61/2.21 U1_ga(L, T, t_out_ga(L, T)) -> e_out_ga(L, T) 5.61/2.21 5.61/2.21 The argument filtering Pi contains the following mapping: 5.61/2.21 e_in_ga(x1, x2) = e_in_ga(x1) 5.61/2.21 5.61/2.21 U1_ga(x1, x2, x3) = U1_ga(x3) 5.61/2.21 5.61/2.21 t_in_ga(x1, x2) = t_in_ga(x1) 5.61/2.21 5.61/2.21 U4_ga(x1, x2, x3) = U4_ga(x3) 5.61/2.21 5.61/2.21 n_in_ga(x1, x2) = n_in_ga(x1) 5.61/2.21 5.61/2.21 .(x1, x2) = .(x1, x2) 5.61/2.21 5.61/2.21 U7_ga(x1, x2, x3) = U7_ga(x2, x3) 5.61/2.21 5.61/2.21 z_in_g(x1) = z_in_g(x1) 5.61/2.21 5.61/2.21 a = a 5.61/2.21 5.61/2.21 z_out_g(x1) = z_out_g 5.61/2.21 5.61/2.21 b = b 5.61/2.21 5.61/2.21 c = c 5.61/2.21 5.61/2.21 n_out_ga(x1, x2) = n_out_ga(x2) 5.61/2.21 5.61/2.21 lbrace = lbrace 5.61/2.21 5.61/2.21 U8_ga(x1, x2, x3) = U8_ga(x3) 5.61/2.21 5.61/2.21 U2_ga(x1, x2, x3) = U2_ga(x3) 5.61/2.21 5.61/2.21 U5_ga(x1, x2, x3) = U5_ga(x3) 5.61/2.21 5.61/2.21 star = star 5.61/2.21 5.61/2.21 U6_ga(x1, x2, x3) = U6_ga(x3) 5.61/2.21 5.61/2.21 t_out_ga(x1, x2) = t_out_ga(x2) 5.61/2.21 5.61/2.21 plus = plus 5.61/2.21 5.61/2.21 U3_ga(x1, x2, x3) = U3_ga(x3) 5.61/2.21 5.61/2.21 e_out_ga(x1, x2) = e_out_ga(x2) 5.61/2.21 5.61/2.21 rbrace = rbrace 5.61/2.21 5.61/2.21 E_IN_GA(x1, x2) = E_IN_GA(x1) 5.61/2.21 5.61/2.21 T_IN_GA(x1, x2) = T_IN_GA(x1) 5.61/2.21 5.61/2.21 N_IN_GA(x1, x2) = N_IN_GA(x1) 5.61/2.21 5.61/2.21 U2_GA(x1, x2, x3) = U2_GA(x3) 5.61/2.21 5.61/2.21 U5_GA(x1, x2, x3) = U5_GA(x3) 5.61/2.21 5.61/2.21 5.61/2.21 We have to consider all (P,R,Pi)-chains 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (7) PiDPToQDPProof (SOUND) 5.61/2.21 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (8) 5.61/2.21 Obligation: 5.61/2.21 Q DP problem: 5.61/2.21 The TRS P consists of the following rules: 5.61/2.21 5.61/2.21 E_IN_GA(L) -> T_IN_GA(L) 5.61/2.21 T_IN_GA(L) -> N_IN_GA(L) 5.61/2.21 N_IN_GA(.(lbrace, A)) -> E_IN_GA(A) 5.61/2.21 E_IN_GA(L) -> U2_GA(t_in_ga(L)) 5.61/2.21 U2_GA(t_out_ga(.(plus, C))) -> E_IN_GA(C) 5.61/2.21 T_IN_GA(L) -> U5_GA(n_in_ga(L)) 5.61/2.21 U5_GA(n_out_ga(.(star, C))) -> T_IN_GA(C) 5.61/2.21 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L) -> U1_ga(t_in_ga(L)) 5.61/2.21 t_in_ga(L) -> U4_ga(n_in_ga(L)) 5.61/2.21 n_in_ga(.(L, T)) -> U7_ga(T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g 5.61/2.21 z_in_g(b) -> z_out_g 5.61/2.21 z_in_g(c) -> z_out_g 5.61/2.21 U7_ga(T, z_out_g) -> n_out_ga(T) 5.61/2.21 n_in_ga(.(lbrace, A)) -> U8_ga(e_in_ga(A)) 5.61/2.21 e_in_ga(L) -> U2_ga(t_in_ga(L)) 5.61/2.21 t_in_ga(L) -> U5_ga(n_in_ga(L)) 5.61/2.21 U5_ga(n_out_ga(.(star, C))) -> U6_ga(t_in_ga(C)) 5.61/2.21 U6_ga(t_out_ga(T)) -> t_out_ga(T) 5.61/2.21 U2_ga(t_out_ga(.(plus, C))) -> U3_ga(e_in_ga(C)) 5.61/2.21 U3_ga(e_out_ga(T)) -> e_out_ga(T) 5.61/2.21 U8_ga(e_out_ga(.(rbrace, B))) -> n_out_ga(B) 5.61/2.21 U4_ga(n_out_ga(T)) -> t_out_ga(T) 5.61/2.21 U1_ga(t_out_ga(T)) -> e_out_ga(T) 5.61/2.21 5.61/2.21 The set Q consists of the following terms: 5.61/2.21 5.61/2.21 e_in_ga(x0) 5.61/2.21 t_in_ga(x0) 5.61/2.21 n_in_ga(x0) 5.61/2.21 z_in_g(x0) 5.61/2.21 U7_ga(x0, x1) 5.61/2.21 U5_ga(x0) 5.61/2.21 U6_ga(x0) 5.61/2.21 U2_ga(x0) 5.61/2.21 U3_ga(x0) 5.61/2.21 U8_ga(x0) 5.61/2.21 U4_ga(x0) 5.61/2.21 U1_ga(x0) 5.61/2.21 5.61/2.21 We have to consider all (P,Q,R)-chains. 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (9) UsableRulesReductionPairsProof (EQUIVALENT) 5.61/2.21 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. 5.61/2.21 5.61/2.21 The following dependency pairs can be deleted: 5.61/2.21 5.61/2.21 N_IN_GA(.(lbrace, A)) -> E_IN_GA(A) 5.61/2.21 U2_GA(t_out_ga(.(plus, C))) -> E_IN_GA(C) 5.61/2.21 U5_GA(n_out_ga(.(star, C))) -> T_IN_GA(C) 5.61/2.21 The following rules are removed from R: 5.61/2.21 5.61/2.21 n_in_ga(.(L, T)) -> U7_ga(T, z_in_g(L)) 5.61/2.21 z_in_g(a) -> z_out_g 5.61/2.21 z_in_g(b) -> z_out_g 5.61/2.21 z_in_g(c) -> z_out_g 5.61/2.21 n_in_ga(.(lbrace, A)) -> U8_ga(e_in_ga(A)) 5.61/2.21 U5_ga(n_out_ga(.(star, C))) -> U6_ga(t_in_ga(C)) 5.61/2.21 U6_ga(t_out_ga(T)) -> t_out_ga(T) 5.61/2.21 U2_ga(t_out_ga(.(plus, C))) -> U3_ga(e_in_ga(C)) 5.61/2.21 U8_ga(e_out_ga(.(rbrace, B))) -> n_out_ga(B) 5.61/2.21 Used ordering: POLO with Polynomial interpretation [POLO]: 5.61/2.21 5.61/2.21 POL(.(x_1, x_2)) = 2*x_1 + x_2 5.61/2.21 POL(E_IN_GA(x_1)) = x_1 5.61/2.21 POL(N_IN_GA(x_1)) = x_1 5.61/2.21 POL(T_IN_GA(x_1)) = x_1 5.61/2.21 POL(U1_ga(x_1)) = x_1 5.61/2.21 POL(U2_GA(x_1)) = x_1 5.61/2.21 POL(U2_ga(x_1)) = x_1 5.61/2.21 POL(U3_ga(x_1)) = x_1 5.61/2.21 POL(U4_ga(x_1)) = x_1 5.61/2.21 POL(U5_GA(x_1)) = x_1 5.61/2.21 POL(U5_ga(x_1)) = x_1 5.61/2.21 POL(U6_ga(x_1)) = 2 + x_1 5.61/2.21 POL(U7_ga(x_1, x_2)) = x_1 + x_2 5.61/2.21 POL(U8_ga(x_1)) = x_1 5.61/2.21 POL(a) = 0 5.61/2.21 POL(b) = 0 5.61/2.21 POL(c) = 0 5.61/2.21 POL(e_in_ga(x_1)) = x_1 5.61/2.21 POL(e_out_ga(x_1)) = x_1 5.61/2.21 POL(lbrace) = 0 5.61/2.21 POL(n_in_ga(x_1)) = x_1 5.61/2.21 POL(n_out_ga(x_1)) = x_1 5.61/2.21 POL(plus) = 1 5.61/2.21 POL(rbrace) = 0 5.61/2.21 POL(star) = 2 5.61/2.21 POL(t_in_ga(x_1)) = x_1 5.61/2.21 POL(t_out_ga(x_1)) = x_1 5.61/2.21 POL(z_in_g(x_1)) = x_1 5.61/2.21 POL(z_out_g) = 0 5.61/2.21 5.61/2.21 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (10) 5.61/2.21 Obligation: 5.61/2.21 Q DP problem: 5.61/2.21 The TRS P consists of the following rules: 5.61/2.21 5.61/2.21 E_IN_GA(L) -> T_IN_GA(L) 5.61/2.21 T_IN_GA(L) -> N_IN_GA(L) 5.61/2.21 E_IN_GA(L) -> U2_GA(t_in_ga(L)) 5.61/2.21 T_IN_GA(L) -> U5_GA(n_in_ga(L)) 5.61/2.21 5.61/2.21 The TRS R consists of the following rules: 5.61/2.21 5.61/2.21 e_in_ga(L) -> U1_ga(t_in_ga(L)) 5.61/2.21 e_in_ga(L) -> U2_ga(t_in_ga(L)) 5.61/2.21 t_in_ga(L) -> U4_ga(n_in_ga(L)) 5.61/2.21 t_in_ga(L) -> U5_ga(n_in_ga(L)) 5.61/2.21 U3_ga(e_out_ga(T)) -> e_out_ga(T) 5.61/2.21 U4_ga(n_out_ga(T)) -> t_out_ga(T) 5.61/2.21 U1_ga(t_out_ga(T)) -> e_out_ga(T) 5.61/2.21 U7_ga(T, z_out_g) -> n_out_ga(T) 5.61/2.21 5.61/2.21 The set Q consists of the following terms: 5.61/2.21 5.61/2.21 e_in_ga(x0) 5.61/2.21 t_in_ga(x0) 5.61/2.21 n_in_ga(x0) 5.61/2.21 z_in_g(x0) 5.61/2.21 U7_ga(x0, x1) 5.61/2.21 U5_ga(x0) 5.61/2.21 U6_ga(x0) 5.61/2.21 U2_ga(x0) 5.61/2.21 U3_ga(x0) 5.61/2.21 U8_ga(x0) 5.61/2.21 U4_ga(x0) 5.61/2.21 U1_ga(x0) 5.61/2.21 5.61/2.21 We have to consider all (P,Q,R)-chains. 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (11) DependencyGraphProof (EQUIVALENT) 5.61/2.21 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 5.61/2.21 ---------------------------------------- 5.61/2.21 5.61/2.21 (12) 5.61/2.21 TRUE 5.61/2.24 EOF