↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PrologToPiTRSProof
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → U7_AAA(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → U8_AAA(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, H, R)) → U5_AA(X, L, H, R, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(L, H, R)) → U6_AA(X, L, H, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → U7_AAA(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → U8_AAA(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, H, R)) → U5_AA(X, L, H, R, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(L, H, R)) → U6_AA(X, L, H, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
TLAST_IN_AA → TLAST_IN_AA
TLAST_IN_AA → TLAST_IN_AA
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PrologToPiTRSProof
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PrologToPiTRSProof
TAPPEND_IN_AAA → TAPPEND_IN_AAA
TAPPEND_IN_AAA → TAPPEND_IN_AAA
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PrologToPiTRSProof
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PrologToPiTRSProof
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PrologToPiTRSProof
S2T_IN_GA(s(X)) → S2T_IN_GA(X)
From the DPs we obtained the following set of size-change graphs:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → U7_AAA(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → U8_AAA(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, H, R)) → U5_AA(X, L, H, R, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(L, H, R)) → U6_AA(X, L, H, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2t_in_ga(A, T))
GOAL_IN_GAA(A, B, C) → S2T_IN_GA(A, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U9_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U10_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U11_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U1_GAA(A, B, C, s2t_out_ga(A, T)) → U2_GAA(A, B, C, tapplast_in_aaa(T, B, C))
U1_GAA(A, B, C, s2t_out_ga(A, T)) → TAPPLAST_IN_AAA(T, B, C)
TAPPLAST_IN_AAA(L, X, Last) → U3_AAA(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
TAPPLAST_IN_AAA(L, X, Last) → TAPPEND_IN_AAA(L, node(nil, X, nil), LX)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → U7_AAA(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → U8_AAA(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_AAA(L, X, Last, tlast_in_aa(Last, LX))
U3_AAA(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → TLAST_IN_AA(Last, LX)
TLAST_IN_AA(X, node(L, H, R)) → U5_AA(X, L, H, R, tlast_in_aa(X, L))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
TLAST_IN_AA(X, node(L, H, R)) → U6_AA(X, L, H, R, tlast_in_aa(X, R))
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, R)
TLAST_IN_AA(X, node(L, H, R)) → TLAST_IN_AA(X, L)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
↳ PiDP
TLAST_IN_AA → TLAST_IN_AA
TLAST_IN_AA → TLAST_IN_AA
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(T1, X, U)) → TAPPEND_IN_AAA(T2, T3, U)
TAPPEND_IN_AAA(node(T1, X, T2), T3, node(U, X, T2)) → TAPPEND_IN_AAA(T1, T3, U)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ NonTerminationProof
↳ PiDP
TAPPEND_IN_AAA → TAPPEND_IN_AAA
TAPPEND_IN_AAA → TAPPEND_IN_AAA
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2t_in_ga(A, T))
s2t_in_ga(s(X), node(T, Y, T)) → U9_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U10_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U11_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U11_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U10_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U9_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U1_gaa(A, B, C, s2t_out_ga(A, T)) → U2_gaa(A, B, C, tapplast_in_aaa(T, B, C))
tapplast_in_aaa(L, X, Last) → U3_aaa(L, X, Last, tappend_in_aaa(L, node(nil, X, nil), LX))
tappend_in_aaa(nil, T, T) → tappend_out_aaa(nil, T, T)
tappend_in_aaa(node(nil, X, T2), T1, node(T1, X, T2)) → tappend_out_aaa(node(nil, X, T2), T1, node(T1, X, T2))
tappend_in_aaa(node(T1, X, nil), T2, node(T1, X, T2)) → tappend_out_aaa(node(T1, X, nil), T2, node(T1, X, T2))
tappend_in_aaa(node(T1, X, T2), T3, node(U, X, T2)) → U7_aaa(T1, X, T2, T3, U, tappend_in_aaa(T1, T3, U))
tappend_in_aaa(node(T1, X, T2), T3, node(T1, X, U)) → U8_aaa(T1, X, T2, T3, U, tappend_in_aaa(T2, T3, U))
U8_aaa(T1, X, T2, T3, U, tappend_out_aaa(T2, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(T1, X, U))
U7_aaa(T1, X, T2, T3, U, tappend_out_aaa(T1, T3, U)) → tappend_out_aaa(node(T1, X, T2), T3, node(U, X, T2))
U3_aaa(L, X, Last, tappend_out_aaa(L, node(nil, X, nil), LX)) → U4_aaa(L, X, Last, tlast_in_aa(Last, LX))
tlast_in_aa(X, node(nil, X, nil)) → tlast_out_aa(X, node(nil, X, nil))
tlast_in_aa(X, node(L, H, R)) → U5_aa(X, L, H, R, tlast_in_aa(X, L))
tlast_in_aa(X, node(L, H, R)) → U6_aa(X, L, H, R, tlast_in_aa(X, R))
U6_aa(X, L, H, R, tlast_out_aa(X, R)) → tlast_out_aa(X, node(L, H, R))
U5_aa(X, L, H, R, tlast_out_aa(X, L)) → tlast_out_aa(X, node(L, H, R))
U4_aaa(L, X, Last, tlast_out_aa(Last, LX)) → tapplast_out_aaa(L, X, Last)
U2_gaa(A, B, C, tapplast_out_aaa(T, B, C)) → goal_out_gaa(A, B, C)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
↳ Prolog
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
S2T_IN_GA(s(X)) → S2T_IN_GA(X)
From the DPs we obtained the following set of size-change graphs: