↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
With regard to the inferred argument filtering the predicates were used in the following modes:
reach4: (b,b,b,b)
member2: (b,b)
member12: (f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PrologToPiTRSProof
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PrologToPiTRSProof
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PiDP
↳ PrologToPiTRSProof
MEMBER1_2_IN_AG1(._22(H, L)) -> MEMBER1_2_IN_AG1(L)
From the DPs we obtained the following set of size-change graphs:
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PrologToPiTRSProof
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
↳ PrologToPiTRSProof
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
↳ PrologToPiTRSProof
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
From the DPs we obtained the following set of size-change graphs:
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PrologToPiTRSProof
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PrologToPiTRSProof
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
↳ PROLOG
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ PrologToPiTRSProof
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG4(Z, Edges, Visited, member1_2_in_ag1(Edges))
IF_REACH_4_IN_3_GGGG5(Z, Edges, Visited, Y, member_2_out_gg) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
IF_REACH_4_IN_2_GGGG4(Z, Edges, Visited, member1_2_out_ag1(._22(X, ._22(Y, []_0)))) -> IF_REACH_4_IN_3_GGGG5(Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
member1_2_in_ag1(._22(H, L)) -> member1_2_out_ag1(H)
member1_2_in_ag1(._22(H, L)) -> if_member1_2_in_1_ag1(member1_2_in_ag1(L))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg1(member_2_in_gg2(X, L))
if_member1_2_in_1_ag1(member1_2_out_ag1(X)) -> member1_2_out_ag1(X)
if_member_2_in_1_gg1(member_2_out_gg) -> member_2_out_gg
member1_2_in_ag1(x0)
member_2_in_gg2(x0, x1)
if_member1_2_in_1_ag1(x0)
if_member_2_in_1_gg1(x0)
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG
↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> IF_REACH_4_IN_1_GGGG5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Y, Edges, Visited) -> MEMBER_2_IN_GG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER_2_IN_GG2(X, ._22(H, L)) -> IF_MEMBER_2_IN_1_GG4(X, H, L, member_2_in_gg2(X, L))
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> MEMBER1_2_IN_AG2(._22(X, ._22(Y, []_0)), Edges)
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> IF_MEMBER1_2_IN_1_AG4(X, H, L, member1_2_in_ag2(X, L))
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> MEMBER_2_IN_GG2(Y, Visited)
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> IF_REACH_4_IN_4_GGGG6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
MEMBER1_2_IN_AG2(X, ._22(H, L)) -> MEMBER1_2_IN_AG2(X, L)
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
MEMBER_2_IN_GG2(X, ._22(H, L)) -> MEMBER_2_IN_GG2(X, L)
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)
↳ PROLOG
↳ PrologToPiTRSProof
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ PiDP
REACH_4_IN_GGGG4(X, Z, Edges, Visited) -> IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> REACH_4_IN_GGGG4(Y, Z, Edges, ._22(Y, Visited))
IF_REACH_4_IN_2_GGGG5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> IF_REACH_4_IN_3_GGGG6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
reach_4_in_gggg4(X, Y, Edges, Visited) -> if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_in_gg2(._22(X, ._22(Y, []_0)), Edges))
member_2_in_gg2(H, ._22(H, L)) -> member_2_out_gg2(H, ._22(H, L))
member_2_in_gg2(X, ._22(H, L)) -> if_member_2_in_1_gg4(X, H, L, member_2_in_gg2(X, L))
if_member_2_in_1_gg4(X, H, L, member_2_out_gg2(X, L)) -> member_2_out_gg2(X, ._22(H, L))
if_reach_4_in_1_gggg5(X, Y, Edges, Visited, member_2_out_gg2(._22(X, ._22(Y, []_0)), Edges)) -> reach_4_out_gggg4(X, Y, Edges, Visited)
reach_4_in_gggg4(X, Z, Edges, Visited) -> if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_in_ag2(._22(X, ._22(Y, []_0)), Edges))
member1_2_in_ag2(H, ._22(H, L)) -> member1_2_out_ag2(H, ._22(H, L))
member1_2_in_ag2(X, ._22(H, L)) -> if_member1_2_in_1_ag4(X, H, L, member1_2_in_ag2(X, L))
if_member1_2_in_1_ag4(X, H, L, member1_2_out_ag2(X, L)) -> member1_2_out_ag2(X, ._22(H, L))
if_reach_4_in_2_gggg5(X, Z, Edges, Visited, member1_2_out_ag2(._22(X, ._22(Y, []_0)), Edges)) -> if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_in_gg2(Y, Visited))
if_reach_4_in_3_gggg6(X, Z, Edges, Visited, Y, member_2_out_gg2(Y, Visited)) -> if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_in_gggg4(Y, Z, Edges, ._22(Y, Visited)))
if_reach_4_in_4_gggg6(X, Z, Edges, Visited, Y, reach_4_out_gggg4(Y, Z, Edges, ._22(Y, Visited))) -> reach_4_out_gggg4(X, Z, Edges, Visited)