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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

rotate(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) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES


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

(0)
Obligation:
Clauses:

rotate(X, Y) :- ','(append(A, B, X), append(B, A, Y)).
append(.(X, Xs), Ys, .(X, Zs)) :- append(Xs, Ys, Zs).
append([], Ys, Ys).


Query: rotate(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:

rotate_in_2: (b,f)

append_in_3: (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:

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)



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

(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:

   ROTATE_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aag(A, B, X))
   ROTATE_IN_GA(X, Y) -> APPEND_IN_AAG(A, B, X)
   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U3_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
   U1_GA(X, Y, append_out_aag(A, B, X)) -> U2_GA(X, Y, append_in_gga(B, A, Y))
   U1_GA(X, Y, append_out_aag(A, B, X)) -> APPEND_IN_GGA(B, A, Y)
   APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)

ROTATE_IN_GA(x1, x2)  =  ROTATE_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)

U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, x5)

U2_GA(x1, x2, x3)  =  U2_GA(x3)

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)

U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)


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

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

   ROTATE_IN_GA(X, Y) -> U1_GA(X, Y, append_in_aag(A, B, X))
   ROTATE_IN_GA(X, Y) -> APPEND_IN_AAG(A, B, X)
   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> U3_AAG(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)
   U1_GA(X, Y, append_out_aag(A, B, X)) -> U2_GA(X, Y, append_in_gga(B, A, Y))
   U1_GA(X, Y, append_out_aag(A, B, X)) -> APPEND_IN_GGA(B, A, Y)
   APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U3_GGA(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)

ROTATE_IN_GA(x1, x2)  =  ROTATE_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x3)

APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)

U3_AAG(x1, x2, x3, x4, x5)  =  U3_AAG(x1, x5)

U2_GA(x1, x2, x3)  =  U2_GA(x3)

APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)

U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x5)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   APPEND_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)

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(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_GGA(Xs, Ys, Zs)

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(.(X, Xs), Ys) -> APPEND_IN_GGA(Xs, Ys)

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(.(X, Xs), Ys) -> APPEND_IN_GGA(Xs, Ys)
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:

   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   rotate_in_ga(X, Y) -> U1_ga(X, Y, append_in_aag(A, B, X))
   append_in_aag(.(X, Xs), Ys, .(X, Zs)) -> U3_aag(X, Xs, Ys, Zs, append_in_aag(Xs, Ys, Zs))
   append_in_aag([], Ys, Ys) -> append_out_aag([], Ys, Ys)
   U3_aag(X, Xs, Ys, Zs, append_out_aag(Xs, Ys, Zs)) -> append_out_aag(.(X, Xs), Ys, .(X, Zs))
   U1_ga(X, Y, append_out_aag(A, B, X)) -> U2_ga(X, Y, append_in_gga(B, A, Y))
   append_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U3_gga(X, Xs, Ys, Zs, append_in_gga(Xs, Ys, Zs))
   append_in_gga([], Ys, Ys) -> append_out_gga([], Ys, Ys)
   U3_gga(X, Xs, Ys, Zs, append_out_gga(Xs, Ys, Zs)) -> append_out_gga(.(X, Xs), Ys, .(X, Zs))
   U2_ga(X, Y, append_out_gga(B, A, Y)) -> rotate_out_ga(X, Y)

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

U1_ga(x1, x2, x3)  =  U1_ga(x3)

append_in_aag(x1, x2, x3)  =  append_in_aag(x3)

.(x1, x2)  =  .(x1, x2)

U3_aag(x1, x2, x3, x4, x5)  =  U3_aag(x1, x5)

append_out_aag(x1, x2, x3)  =  append_out_aag(x1, x2)

U2_ga(x1, x2, x3)  =  U2_ga(x3)

append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)

U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x5)

[]  =  []

append_out_gga(x1, x2, x3)  =  append_out_gga(x3)

rotate_out_ga(x1, x2)  =  rotate_out_ga(x2)

APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)


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:

   APPEND_IN_AAG(.(X, Xs), Ys, .(X, Zs)) -> APPEND_IN_AAG(Xs, Ys, Zs)

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

APPEND_IN_AAG(x1, x2, x3)  =  APPEND_IN_AAG(x3)


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:

   APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs)

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:
*APPEND_IN_AAG(.(X, Zs)) -> APPEND_IN_AAG(Zs)
The graph contains the following edges 1 > 1


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

(20)
YES
