3.98/1.83 YES 3.98/1.85 proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl 3.98/1.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.98/1.85 3.98/1.85 3.98/1.85 Left Termination of the query pattern 3.98/1.85 3.98/1.85 goal(g) 3.98/1.85 3.98/1.85 w.r.t. the given Prolog program could successfully be proven: 3.98/1.85 3.98/1.85 (0) Prolog 3.98/1.85 (1) PrologToPiTRSProof [SOUND, 0 ms] 3.98/1.85 (2) PiTRS 3.98/1.85 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 3.98/1.85 (4) PiDP 3.98/1.85 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 3.98/1.85 (6) AND 3.98/1.85 (7) PiDP 3.98/1.85 (8) UsableRulesProof [EQUIVALENT, 0 ms] 3.98/1.85 (9) PiDP 3.98/1.85 (10) PiDPToQDPProof [SOUND, 0 ms] 3.98/1.85 (11) QDP 3.98/1.85 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.98/1.85 (13) YES 3.98/1.85 (14) PiDP 3.98/1.85 (15) UsableRulesProof [EQUIVALENT, 0 ms] 3.98/1.85 (16) PiDP 3.98/1.85 (17) PiDPToQDPProof [SOUND, 0 ms] 3.98/1.85 (18) QDP 3.98/1.85 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 3.98/1.85 (20) YES 3.98/1.85 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (0) 3.98/1.85 Obligation: 3.98/1.85 Clauses: 3.98/1.85 3.98/1.85 append([], XS, XS). 3.98/1.85 append(.(X, XS), YS, .(X, ZS)) :- append(XS, YS, ZS). 3.98/1.85 s2l(s(X), .(Y, Xs)) :- s2l(X, Xs). 3.98/1.85 s2l(0, []). 3.98/1.85 goal(X) :- ','(s2l(X, XS), append(XS, YS, ZS)). 3.98/1.85 3.98/1.85 3.98/1.85 Query: goal(g) 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (1) PrologToPiTRSProof (SOUND) 3.98/1.85 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 3.98/1.85 3.98/1.85 goal_in_1: (b) 3.98/1.85 3.98/1.85 s2l_in_2: (b,f) 3.98/1.85 3.98/1.85 append_in_3: (b,f,f) 3.98/1.85 3.98/1.85 Transforming Prolog into the following Term Rewriting System: 3.98/1.85 3.98/1.85 Pi-finite rewrite system: 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 3.98/1.85 3.98/1.85 3.98/1.85 3.98/1.85 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 3.98/1.85 3.98/1.85 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (2) 3.98/1.85 Obligation: 3.98/1.85 Pi-finite rewrite system: 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (3) DependencyPairsProof (EQUIVALENT) 3.98/1.85 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 GOAL_IN_G(X) -> U3_G(X, s2l_in_ga(X, XS)) 3.98/1.85 GOAL_IN_G(X) -> S2L_IN_GA(X, XS) 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) 3.98/1.85 U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U3_G(X, s2l_out_ga(X, XS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> U1_GAA(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 GOAL_IN_G(x1) = GOAL_IN_G(x1) 3.98/1.85 3.98/1.85 U3_G(x1, x2) = U3_G(x2) 3.98/1.85 3.98/1.85 S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 3.98/1.85 3.98/1.85 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 3.98/1.85 3.98/1.85 U4_G(x1, x2) = U4_G(x2) 3.98/1.85 3.98/1.85 APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 3.98/1.85 3.98/1.85 U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (4) 3.98/1.85 Obligation: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 GOAL_IN_G(X) -> U3_G(X, s2l_in_ga(X, XS)) 3.98/1.85 GOAL_IN_G(X) -> S2L_IN_GA(X, XS) 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> U2_GA(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) 3.98/1.85 U3_G(X, s2l_out_ga(X, XS)) -> U4_G(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U3_G(X, s2l_out_ga(X, XS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> U1_GAA(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 GOAL_IN_G(x1) = GOAL_IN_G(x1) 3.98/1.85 3.98/1.85 U3_G(x1, x2) = U3_G(x2) 3.98/1.85 3.98/1.85 S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 3.98/1.85 3.98/1.85 U2_GA(x1, x2, x3, x4) = U2_GA(x4) 3.98/1.85 3.98/1.85 U4_G(x1, x2) = U4_G(x2) 3.98/1.85 3.98/1.85 APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 3.98/1.85 3.98/1.85 U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x5) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (5) DependencyGraphProof (EQUIVALENT) 3.98/1.85 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (6) 3.98/1.85 Complex Obligation (AND) 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (7) 3.98/1.85 Obligation: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (8) UsableRulesProof (EQUIVALENT) 3.98/1.85 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (9) 3.98/1.85 Obligation: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 APPEND_IN_GAA(.(X, XS), YS, .(X, ZS)) -> APPEND_IN_GAA(XS, YS, ZS) 3.98/1.85 3.98/1.85 R is empty. 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 APPEND_IN_GAA(x1, x2, x3) = APPEND_IN_GAA(x1) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (10) PiDPToQDPProof (SOUND) 3.98/1.85 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (11) 3.98/1.85 Obligation: 3.98/1.85 Q DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 APPEND_IN_GAA(.(XS)) -> APPEND_IN_GAA(XS) 3.98/1.85 3.98/1.85 R is empty. 3.98/1.85 Q is empty. 3.98/1.85 We have to consider all (P,Q,R)-chains. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (12) QDPSizeChangeProof (EQUIVALENT) 3.98/1.85 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. 3.98/1.85 3.98/1.85 From the DPs we obtained the following set of size-change graphs: 3.98/1.85 *APPEND_IN_GAA(.(XS)) -> APPEND_IN_GAA(XS) 3.98/1.85 The graph contains the following edges 1 > 1 3.98/1.85 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (13) 3.98/1.85 YES 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (14) 3.98/1.85 Obligation: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) 3.98/1.85 3.98/1.85 The TRS R consists of the following rules: 3.98/1.85 3.98/1.85 goal_in_g(X) -> U3_g(X, s2l_in_ga(X, XS)) 3.98/1.85 s2l_in_ga(s(X), .(Y, Xs)) -> U2_ga(X, Y, Xs, s2l_in_ga(X, Xs)) 3.98/1.85 s2l_in_ga(0, []) -> s2l_out_ga(0, []) 3.98/1.85 U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) 3.98/1.85 U3_g(X, s2l_out_ga(X, XS)) -> U4_g(X, append_in_gaa(XS, YS, ZS)) 3.98/1.85 append_in_gaa([], XS, XS) -> append_out_gaa([], XS, XS) 3.98/1.85 append_in_gaa(.(X, XS), YS, .(X, ZS)) -> U1_gaa(X, XS, YS, ZS, append_in_gaa(XS, YS, ZS)) 3.98/1.85 U1_gaa(X, XS, YS, ZS, append_out_gaa(XS, YS, ZS)) -> append_out_gaa(.(X, XS), YS, .(X, ZS)) 3.98/1.85 U4_g(X, append_out_gaa(XS, YS, ZS)) -> goal_out_g(X) 3.98/1.85 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 goal_in_g(x1) = goal_in_g(x1) 3.98/1.85 3.98/1.85 U3_g(x1, x2) = U3_g(x2) 3.98/1.85 3.98/1.85 s2l_in_ga(x1, x2) = s2l_in_ga(x1) 3.98/1.85 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 U2_ga(x1, x2, x3, x4) = U2_ga(x4) 3.98/1.85 3.98/1.85 0 = 0 3.98/1.85 3.98/1.85 s2l_out_ga(x1, x2) = s2l_out_ga(x2) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 U4_g(x1, x2) = U4_g(x2) 3.98/1.85 3.98/1.85 append_in_gaa(x1, x2, x3) = append_in_gaa(x1) 3.98/1.85 3.98/1.85 [] = [] 3.98/1.85 3.98/1.85 append_out_gaa(x1, x2, x3) = append_out_gaa 3.98/1.85 3.98/1.85 U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x5) 3.98/1.85 3.98/1.85 goal_out_g(x1) = goal_out_g 3.98/1.85 3.98/1.85 S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (15) UsableRulesProof (EQUIVALENT) 3.98/1.85 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (16) 3.98/1.85 Obligation: 3.98/1.85 Pi DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) 3.98/1.85 3.98/1.85 R is empty. 3.98/1.85 The argument filtering Pi contains the following mapping: 3.98/1.85 s(x1) = s(x1) 3.98/1.85 3.98/1.85 .(x1, x2) = .(x2) 3.98/1.85 3.98/1.85 S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) 3.98/1.85 3.98/1.85 3.98/1.85 We have to consider all (P,R,Pi)-chains 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (17) PiDPToQDPProof (SOUND) 3.98/1.85 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (18) 3.98/1.85 Obligation: 3.98/1.85 Q DP problem: 3.98/1.85 The TRS P consists of the following rules: 3.98/1.85 3.98/1.85 S2L_IN_GA(s(X)) -> S2L_IN_GA(X) 3.98/1.85 3.98/1.85 R is empty. 3.98/1.85 Q is empty. 3.98/1.85 We have to consider all (P,Q,R)-chains. 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (19) QDPSizeChangeProof (EQUIVALENT) 3.98/1.85 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. 3.98/1.85 3.98/1.85 From the DPs we obtained the following set of size-change graphs: 3.98/1.85 *S2L_IN_GA(s(X)) -> S2L_IN_GA(X) 3.98/1.85 The graph contains the following edges 1 > 1 3.98/1.85 3.98/1.85 3.98/1.85 ---------------------------------------- 3.98/1.85 3.98/1.85 (20) 3.98/1.85 YES 3.98/1.87 EOF