YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/TRfqX.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty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, 0 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, 0 ms]
        (16) QDP
        (17) QDPOrderProof [EQUIVALENT, 31 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
