YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/MHVpZ.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

sameleaves(g,g)

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) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES
    (21) PiDP
        (22) UsableRulesProof [EQUIVALENT, 0 ms]
        (23) PiDP
        (24) PiDPToQDPProof [SOUND, 0 ms]
        (25) QDP
        (26) UsableRulesReductionPairsProof [EQUIVALENT, 15 ms]
        (27) QDP
        (28) DependencyGraphProof [EQUIVALENT, 0 ms]
        (29) TRUE


----------------------------------------

(0)
Obligation:
Clauses:

sameleaves(leaf(L), leaf(L)).
sameleaves(tree(T1, T2), tree(S1, S2)) :- ','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S))).
getleave(leaf(A), C, A, C).
getleave(tree(A, B), C, L, O) :- getleave(A, tree(B, C), L, O).


Query: sameleaves(g,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

sameleaves_in_2: (b,b)

getleave_in_4: (b,b,f,f) (b,b,b,f)

Transforming Prolog into the following Term Rewriting System:

Pi-finite rewrite system:
The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)



----------------------------------------

(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:

   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> GETLEAVE_IN_GGAA(T1, T2, L, T)
   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> U4_GGAA(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O)
   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> GETLEAVE_IN_GGGA(S1, S2, L, S)
   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> U4_GGGA(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O)
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_GG(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S)

The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

SAMELEAVES_IN_GG(x1, x2)  =  SAMELEAVES_IN_GG(x1, x2)

U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)

GETLEAVE_IN_GGAA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGAA(x1, x2)

U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)

U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x5, x6)

GETLEAVE_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGGA(x1, x2, x3)

U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x6)

U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> GETLEAVE_IN_GGAA(T1, T2, L, T)
   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> U4_GGAA(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O)
   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> GETLEAVE_IN_GGGA(S1, S2, L, S)
   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> U4_GGGA(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O)
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_GG(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S)

The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

SAMELEAVES_IN_GG(x1, x2)  =  SAMELEAVES_IN_GG(x1, x2)

U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)

GETLEAVE_IN_GGAA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGAA(x1, x2)

U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x6)

U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x5, x6)

GETLEAVE_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGGA(x1, x2, x3)

U4_GGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGGA(x6)

U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 5 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O)

The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

GETLEAVE_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGGA(x1, x2, x3)


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:

   GETLEAVE_IN_GGGA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGGA(A, tree(B, C), L, O)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)

GETLEAVE_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGGA(x1, x2, x3)


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:

   GETLEAVE_IN_GGGA(tree(A, B), C, L) -> GETLEAVE_IN_GGGA(A, tree(B, C), L)

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:
*GETLEAVE_IN_GGGA(tree(A, B), C, L) -> GETLEAVE_IN_GGGA(A, tree(B, C), L)
The graph contains the following edges 1 > 1, 3 >= 3


----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O)

The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

GETLEAVE_IN_GGAA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGAA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(16)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GETLEAVE_IN_GGAA(tree(A, B), C, L, O) -> GETLEAVE_IN_GGAA(A, tree(B, C), L, O)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)

GETLEAVE_IN_GGAA(x1, x2, x3, x4)  =  GETLEAVE_IN_GGAA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(17) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GETLEAVE_IN_GGAA(tree(A, B), C) -> GETLEAVE_IN_GGAA(A, tree(B, C))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(19) 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:
*GETLEAVE_IN_GGAA(tree(A, B), C) -> GETLEAVE_IN_GGAA(A, tree(B, C))
The graph contains the following edges 1 > 1


----------------------------------------

(20)
YES

----------------------------------------

(21)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S)
   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))

The TRS R consists of the following rules:

   sameleaves_in_gg(leaf(L), leaf(L)) -> sameleaves_out_gg(leaf(L), leaf(L))
   sameleaves_in_gg(tree(T1, T2), tree(S1, S2)) -> U1_gg(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)
   U1_gg(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_gg(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U2_gg(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> U3_gg(T1, T2, S1, S2, sameleaves_in_gg(T, S))
   U3_gg(T1, T2, S1, S2, sameleaves_out_gg(T, S)) -> sameleaves_out_gg(tree(T1, T2), tree(S1, S2))

The argument filtering Pi contains the following mapping:
sameleaves_in_gg(x1, x2)  =  sameleaves_in_gg(x1, x2)

leaf(x1)  =  leaf(x1)

sameleaves_out_gg(x1, x2)  =  sameleaves_out_gg

tree(x1, x2)  =  tree(x1, x2)

U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

U2_gg(x1, x2, x3, x4, x5, x6)  =  U2_gg(x5, x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x5)

SAMELEAVES_IN_GG(x1, x2)  =  SAMELEAVES_IN_GG(x1, x2)

U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)

U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x5, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(22) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(23)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GG(T1, T2, S1, S2, getleave_out_ggaa(T1, T2, L, T)) -> U2_GG(T1, T2, S1, S2, T, getleave_in_ggga(S1, S2, L, S))
   U2_GG(T1, T2, S1, S2, T, getleave_out_ggga(S1, S2, L, S)) -> SAMELEAVES_IN_GG(T, S)
   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(T1, T2, S1, S2, getleave_in_ggaa(T1, T2, L, T))

The TRS R consists of the following rules:

   getleave_in_ggga(leaf(A), C, A, C) -> getleave_out_ggga(leaf(A), C, A, C)
   getleave_in_ggga(tree(A, B), C, L, O) -> U4_ggga(A, B, C, L, O, getleave_in_ggga(A, tree(B, C), L, O))
   getleave_in_ggaa(leaf(A), C, A, C) -> getleave_out_ggaa(leaf(A), C, A, C)
   getleave_in_ggaa(tree(A, B), C, L, O) -> U4_ggaa(A, B, C, L, O, getleave_in_ggaa(A, tree(B, C), L, O))
   U4_ggga(A, B, C, L, O, getleave_out_ggga(A, tree(B, C), L, O)) -> getleave_out_ggga(tree(A, B), C, L, O)
   U4_ggaa(A, B, C, L, O, getleave_out_ggaa(A, tree(B, C), L, O)) -> getleave_out_ggaa(tree(A, B), C, L, O)

The argument filtering Pi contains the following mapping:
leaf(x1)  =  leaf(x1)

tree(x1, x2)  =  tree(x1, x2)

getleave_in_ggaa(x1, x2, x3, x4)  =  getleave_in_ggaa(x1, x2)

getleave_out_ggaa(x1, x2, x3, x4)  =  getleave_out_ggaa(x3, x4)

U4_ggaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaa(x6)

getleave_in_ggga(x1, x2, x3, x4)  =  getleave_in_ggga(x1, x2, x3)

getleave_out_ggga(x1, x2, x3, x4)  =  getleave_out_ggga(x4)

U4_ggga(x1, x2, x3, x4, x5, x6)  =  U4_ggga(x6)

SAMELEAVES_IN_GG(x1, x2)  =  SAMELEAVES_IN_GG(x1, x2)

U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)

U2_GG(x1, x2, x3, x4, x5, x6)  =  U2_GG(x5, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(24) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GG(S1, S2, getleave_out_ggaa(L, T)) -> U2_GG(T, getleave_in_ggga(S1, S2, L))
   U2_GG(T, getleave_out_ggga(S)) -> SAMELEAVES_IN_GG(T, S)
   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(S1, S2, getleave_in_ggaa(T1, T2))

The TRS R consists of the following rules:

   getleave_in_ggga(leaf(A), C, A) -> getleave_out_ggga(C)
   getleave_in_ggga(tree(A, B), C, L) -> U4_ggga(getleave_in_ggga(A, tree(B, C), L))
   getleave_in_ggaa(leaf(A), C) -> getleave_out_ggaa(A, C)
   getleave_in_ggaa(tree(A, B), C) -> U4_ggaa(getleave_in_ggaa(A, tree(B, C)))
   U4_ggga(getleave_out_ggga(O)) -> getleave_out_ggga(O)
   U4_ggaa(getleave_out_ggaa(L, O)) -> getleave_out_ggaa(L, O)

The set Q consists of the following terms:

   getleave_in_ggga(x0, x1, x2)
   getleave_in_ggaa(x0, x1)
   U4_ggga(x0)
   U4_ggaa(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(26) UsableRulesReductionPairsProof (EQUIVALENT)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

   SAMELEAVES_IN_GG(tree(T1, T2), tree(S1, S2)) -> U1_GG(S1, S2, getleave_in_ggaa(T1, T2))
The following rules are removed from R:

   getleave_in_ggga(leaf(A), C, A) -> getleave_out_ggga(C)
   getleave_in_ggaa(leaf(A), C) -> getleave_out_ggaa(A, C)
Used ordering: POLO with Polynomial interpretation [POLO]:

   POL(SAMELEAVES_IN_GG(x_1, x_2)) = 2*x_1 + x_2
   POL(U1_GG(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(U2_GG(x_1, x_2)) = 2*x_1 + x_2
   POL(U4_ggaa(x_1)) = x_1
   POL(U4_ggga(x_1)) = x_1
   POL(getleave_in_ggaa(x_1, x_2)) = x_1 + x_2
   POL(getleave_in_ggga(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(getleave_out_ggaa(x_1, x_2)) = 2*x_1 + x_2
   POL(getleave_out_ggga(x_1)) = x_1
   POL(leaf(x_1)) = 2*x_1
   POL(tree(x_1, x_2)) = 1 + x_1 + x_2


----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GG(S1, S2, getleave_out_ggaa(L, T)) -> U2_GG(T, getleave_in_ggga(S1, S2, L))
   U2_GG(T, getleave_out_ggga(S)) -> SAMELEAVES_IN_GG(T, S)

The TRS R consists of the following rules:

   getleave_in_ggaa(tree(A, B), C) -> U4_ggaa(getleave_in_ggaa(A, tree(B, C)))
   U4_ggaa(getleave_out_ggaa(L, O)) -> getleave_out_ggaa(L, O)
   getleave_in_ggga(tree(A, B), C, L) -> U4_ggga(getleave_in_ggga(A, tree(B, C), L))
   U4_ggga(getleave_out_ggga(O)) -> getleave_out_ggga(O)

The set Q consists of the following terms:

   getleave_in_ggga(x0, x1, x2)
   getleave_in_ggaa(x0, x1)
   U4_ggga(x0)
   U4_ggaa(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(28) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(29)
TRUE
