/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Left Termination of the query pattern shanoi(g,g,g,g,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof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, 3 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) PiDPToQDPProof [SOUND, 1 ms] (16) QDP (17) QDPOrderProof [EQUIVALENT, 90 ms] (18) QDP (19) DependencyGraphProof [EQUIVALENT, 0 ms] (20) TRUE ---------------------------------------- (0) Obligation: Clauses: shanoi(s(0), A, B, C, .(mv(A, C), [])). shanoi(s(s(X)), A, B, C, M) :- ','(eq(N1, s(X)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M))))). append([], L, L). append(.(H, L), L1, .(H, R)) :- append(L, L1, R). eq(X, X). Query: shanoi(g,g,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: shanoi_in_5: (b,b,b,b,f) append_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: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) ---------------------------------------- (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: SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X))) SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> EQ_IN_AG(N1, s(X)) U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1) U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_GGGGA(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2) U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_GGGGA(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> APPEND_IN_GGA(M1, .(mv(A, C), []), T) APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> U6_GGA(H, L, L1, R, append_in_gga(L, L1, R)) APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R) U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_GGGGA(X, A, B, C, M, append_in_gga(T, M2, M)) U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> APPEND_IN_GGA(T, M2, M) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) SHANOI_IN_GGGGA(x1, x2, x3, x4, x5) = SHANOI_IN_GGGGA(x1, x2, x3, x4) U1_GGGGA(x1, x2, x3, x4, x5, x6) = U1_GGGGA(x2, x3, x4, x6) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U2_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U2_GGGGA(x2, x3, x4, x6, x7) U3_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U3_GGGGA(x2, x4, x6, x7) U4_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U4_GGGGA(x6, x7) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x5) U5_GGGGA(x1, x2, x3, x4, x5, x6) = U5_GGGGA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X))) SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> EQ_IN_AG(N1, s(X)) U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1) U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_GGGGA(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2) U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_GGGGA(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) U3_GGGGA(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> APPEND_IN_GGA(M1, .(mv(A, C), []), T) APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> U6_GGA(H, L, L1, R, append_in_gga(L, L1, R)) APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R) U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_GGGGA(X, A, B, C, M, append_in_gga(T, M2, M)) U4_GGGGA(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> APPEND_IN_GGA(T, M2, M) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) SHANOI_IN_GGGGA(x1, x2, x3, x4, x5) = SHANOI_IN_GGGGA(x1, x2, x3, x4) U1_GGGGA(x1, x2, x3, x4, x5, x6) = U1_GGGGA(x2, x3, x4, x6) EQ_IN_AG(x1, x2) = EQ_IN_AG(x2) U2_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U2_GGGGA(x2, x3, x4, x6, x7) U3_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U3_GGGGA(x2, x4, x6, x7) U4_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U4_GGGGA(x6, x7) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x1, x5) U5_GGGGA(x1, x2, x3, x4, x5, x6) = U5_GGGGA(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: APPEND_IN_GGA(.(H, L), L1, .(H, R)) -> APPEND_IN_GGA(L, L1, R) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x1, x2) APPEND_IN_GGA(x1, x2, x3) = APPEND_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: APPEND_IN_GGA(.(H, L), L1) -> APPEND_IN_GGA(L, L1) 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: *APPEND_IN_GGA(.(H, L), L1) -> APPEND_IN_GGA(L, L1) 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: U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_GGGGA(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_GGGGA(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> SHANOI_IN_GGGGA(N1, B, A, C, M2) SHANOI_IN_GGGGA(s(s(X)), A, B, C, M) -> U1_GGGGA(X, A, B, C, M, eq_in_ag(N1, s(X))) U1_GGGGA(X, A, B, C, M, eq_out_ag(N1, s(X))) -> SHANOI_IN_GGGGA(N1, A, C, B, M1) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C, .(mv(A, C), [])) -> shanoi_out_gggga(s(0), A, B, C, .(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C, M) -> U1_gggga(X, A, B, C, M, eq_in_ag(N1, s(X))) eq_in_ag(X, X) -> eq_out_ag(X, X) U1_gggga(X, A, B, C, M, eq_out_ag(N1, s(X))) -> U2_gggga(X, A, B, C, M, N1, shanoi_in_gggga(N1, A, C, B, M1)) U2_gggga(X, A, B, C, M, N1, shanoi_out_gggga(N1, A, C, B, M1)) -> U3_gggga(X, A, B, C, M, M1, shanoi_in_gggga(N1, B, A, C, M2)) U3_gggga(X, A, B, C, M, M1, shanoi_out_gggga(N1, B, A, C, M2)) -> U4_gggga(X, A, B, C, M, M2, append_in_gga(M1, .(mv(A, C), []), T)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L), L1, .(H, R)) -> U6_gga(H, L, L1, R, append_in_gga(L, L1, R)) U6_gga(H, L, L1, R, append_out_gga(L, L1, R)) -> append_out_gga(.(H, L), L1, .(H, R)) U4_gggga(X, A, B, C, M, M2, append_out_gga(M1, .(mv(A, C), []), T)) -> U5_gggga(X, A, B, C, M, append_in_gga(T, M2, M)) U5_gggga(X, A, B, C, M, append_out_gga(T, M2, M)) -> shanoi_out_gggga(s(s(X)), A, B, C, M) The argument filtering Pi contains the following mapping: shanoi_in_gggga(x1, x2, x3, x4, x5) = shanoi_in_gggga(x1, x2, x3, x4) s(x1) = s(x1) 0 = 0 shanoi_out_gggga(x1, x2, x3, x4, x5) = shanoi_out_gggga(x5) U1_gggga(x1, x2, x3, x4, x5, x6) = U1_gggga(x2, x3, x4, x6) eq_in_ag(x1, x2) = eq_in_ag(x2) eq_out_ag(x1, x2) = eq_out_ag(x1) U2_gggga(x1, x2, x3, x4, x5, x6, x7) = U2_gggga(x2, x3, x4, x6, x7) U3_gggga(x1, x2, x3, x4, x5, x6, x7) = U3_gggga(x2, x4, x6, x7) U4_gggga(x1, x2, x3, x4, x5, x6, x7) = U4_gggga(x6, x7) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) .(x1, x2) = .(x1, x2) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x1, x5) mv(x1, x2) = mv(x1, x2) U5_gggga(x1, x2, x3, x4, x5, x6) = U5_gggga(x6) SHANOI_IN_GGGGA(x1, x2, x3, x4, x5) = SHANOI_IN_GGGGA(x1, x2, x3, x4) U1_GGGGA(x1, x2, x3, x4, x5, x6) = U1_GGGGA(x2, x3, x4, x6) U2_GGGGA(x1, x2, x3, x4, x5, x6, x7) = U2_GGGGA(x2, x3, x4, x6, x7) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GGGGA(A, B, C, eq_out_ag(N1)) -> U2_GGGGA(A, B, C, N1, shanoi_in_gggga(N1, A, C, B)) U2_GGGGA(A, B, C, N1, shanoi_out_gggga(M1)) -> SHANOI_IN_GGGGA(N1, B, A, C) SHANOI_IN_GGGGA(s(s(X)), A, B, C) -> U1_GGGGA(A, B, C, eq_in_ag(s(X))) U1_GGGGA(A, B, C, eq_out_ag(N1)) -> SHANOI_IN_GGGGA(N1, A, C, B) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C) -> shanoi_out_gggga(.(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C) -> U1_gggga(A, B, C, eq_in_ag(s(X))) eq_in_ag(X) -> eq_out_ag(X) U1_gggga(A, B, C, eq_out_ag(N1)) -> U2_gggga(A, B, C, N1, shanoi_in_gggga(N1, A, C, B)) U2_gggga(A, B, C, N1, shanoi_out_gggga(M1)) -> U3_gggga(A, C, M1, shanoi_in_gggga(N1, B, A, C)) U3_gggga(A, C, M1, shanoi_out_gggga(M2)) -> U4_gggga(M2, append_in_gga(M1, .(mv(A, C), []))) append_in_gga([], L) -> append_out_gga(L) append_in_gga(.(H, L), L1) -> U6_gga(H, append_in_gga(L, L1)) U6_gga(H, append_out_gga(R)) -> append_out_gga(.(H, R)) U4_gggga(M2, append_out_gga(T)) -> U5_gggga(append_in_gga(T, M2)) U5_gggga(append_out_gga(M)) -> shanoi_out_gggga(M) The set Q consists of the following terms: shanoi_in_gggga(x0, x1, x2, x3) eq_in_ag(x0) U1_gggga(x0, x1, x2, x3) U2_gggga(x0, x1, x2, x3, x4) U3_gggga(x0, x1, x2, x3) append_in_gga(x0, x1) U6_gga(x0, x1) U4_gggga(x0, x1) U5_gggga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. SHANOI_IN_GGGGA(s(s(X)), A, B, C) -> U1_GGGGA(A, B, C, eq_in_ag(s(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U2_GGGGA_5(x_1, ..., x_5) ) = x_4 POL( shanoi_in_gggga_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + x_4 POL( s_1(x_1) ) = x_1 + 2 POL( 0 ) = 0 POL( shanoi_out_gggga_1(x_1) ) = max{0, 2x_1 - 2} POL( ._2(x_1, x_2) ) = 2x_2 + 2 POL( mv_2(x_1, x_2) ) = 2 POL( [] ) = 0 POL( U1_gggga_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + 2x_3 - 2} POL( eq_in_ag_1(x_1) ) = x_1 + 1 POL( U1_GGGGA_4(x_1, ..., x_4) ) = x_4 POL( eq_out_ag_1(x_1) ) = x_1 POL( U2_gggga_5(x_1, ..., x_5) ) = max{0, 2x_1 + 2x_2 + 2x_4 + 2x_5 - 2} POL( U3_gggga_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2} POL( U4_gggga_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( append_in_gga_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2} POL( U5_gggga_1(x_1) ) = 2 POL( append_out_gga_1(x_1) ) = max{0, -2} POL( U6_gga_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( SHANOI_IN_GGGGA_4(x_1, ..., x_4) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: eq_in_ag(X) -> eq_out_ag(X) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U1_GGGGA(A, B, C, eq_out_ag(N1)) -> U2_GGGGA(A, B, C, N1, shanoi_in_gggga(N1, A, C, B)) U2_GGGGA(A, B, C, N1, shanoi_out_gggga(M1)) -> SHANOI_IN_GGGGA(N1, B, A, C) U1_GGGGA(A, B, C, eq_out_ag(N1)) -> SHANOI_IN_GGGGA(N1, A, C, B) The TRS R consists of the following rules: shanoi_in_gggga(s(0), A, B, C) -> shanoi_out_gggga(.(mv(A, C), [])) shanoi_in_gggga(s(s(X)), A, B, C) -> U1_gggga(A, B, C, eq_in_ag(s(X))) eq_in_ag(X) -> eq_out_ag(X) U1_gggga(A, B, C, eq_out_ag(N1)) -> U2_gggga(A, B, C, N1, shanoi_in_gggga(N1, A, C, B)) U2_gggga(A, B, C, N1, shanoi_out_gggga(M1)) -> U3_gggga(A, C, M1, shanoi_in_gggga(N1, B, A, C)) U3_gggga(A, C, M1, shanoi_out_gggga(M2)) -> U4_gggga(M2, append_in_gga(M1, .(mv(A, C), []))) append_in_gga([], L) -> append_out_gga(L) append_in_gga(.(H, L), L1) -> U6_gga(H, append_in_gga(L, L1)) U6_gga(H, append_out_gga(R)) -> append_out_gga(.(H, R)) U4_gggga(M2, append_out_gga(T)) -> U5_gggga(append_in_gga(T, M2)) U5_gggga(append_out_gga(M)) -> shanoi_out_gggga(M) The set Q consists of the following terms: shanoi_in_gggga(x0, x1, x2, x3) eq_in_ag(x0) U1_gggga(x0, x1, x2, x3) U2_gggga(x0, x1, x2, x3, x4) U3_gggga(x0, x1, x2, x3) append_in_gga(x0, x1) U6_gga(x0, x1) U4_gggga(x0, x1) U5_gggga(x0) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (20) TRUE