YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/45nHG.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

fold(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) PiDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) PiDP
(9) PiDPToQDPProof [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Clauses:

fold(X, .(Y, Ys), Z) :- ','(myop(X, Y, V), fold(V, Ys, Z)).
fold(X, [], X).
myop(a, b, c).


Query: fold(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:

fold_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:

   fold_in_gga(X, .(Y, Ys), Z) -> U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)
   U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   fold_in_gga(X, [], X) -> fold_out_gga(X, [], X)
   U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) -> fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)

[]  =  []

fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   fold_in_gga(X, .(Y, Ys), Z) -> U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)
   U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   fold_in_gga(X, [], X) -> fold_out_gga(X, [], X)
   U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) -> fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)

[]  =  []

fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)



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

(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:

   FOLD_IN_GGA(X, .(Y, Ys), Z) -> U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   FOLD_IN_GGA(X, .(Y, Ys), Z) -> MYOP_IN_GGA(X, Y, V)
   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_GGA(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> FOLD_IN_GGA(V, Ys, Z)

The TRS R consists of the following rules:

   fold_in_gga(X, .(Y, Ys), Z) -> U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)
   U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   fold_in_gga(X, [], X) -> fold_out_gga(X, [], X)
   U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) -> fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)

[]  =  []

fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)

FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)

MYOP_IN_GGA(x1, x2, x3)  =  MYOP_IN_GGA(x1, x2)

U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)


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

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

   FOLD_IN_GGA(X, .(Y, Ys), Z) -> U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   FOLD_IN_GGA(X, .(Y, Ys), Z) -> MYOP_IN_GGA(X, Y, V)
   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_GGA(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> FOLD_IN_GGA(V, Ys, Z)

The TRS R consists of the following rules:

   fold_in_gga(X, .(Y, Ys), Z) -> U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)
   U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   fold_in_gga(X, [], X) -> fold_out_gga(X, [], X)
   U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) -> fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)

[]  =  []

fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)

FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)

MYOP_IN_GGA(x1, x2, x3)  =  MYOP_IN_GGA(x1, x2)

U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x5)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> FOLD_IN_GGA(V, Ys, Z)
   FOLD_IN_GGA(X, .(Y, Ys), Z) -> U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))

The TRS R consists of the following rules:

   fold_in_gga(X, .(Y, Ys), Z) -> U1_gga(X, Y, Ys, Z, myop_in_gga(X, Y, V))
   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)
   U1_gga(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> U2_gga(X, Y, Ys, Z, fold_in_gga(V, Ys, Z))
   fold_in_gga(X, [], X) -> fold_out_gga(X, [], X)
   U2_gga(X, Y, Ys, Z, fold_out_gga(V, Ys, Z)) -> fold_out_gga(X, .(Y, Ys), Z)

The argument filtering Pi contains the following mapping:
fold_in_gga(x1, x2, x3)  =  fold_in_gga(x1, x2)

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

U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x3, x5)

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x5)

[]  =  []

fold_out_gga(x1, x2, x3)  =  fold_out_gga(x3)

FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)


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

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

(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GGA(X, Y, Ys, Z, myop_out_gga(X, Y, V)) -> FOLD_IN_GGA(V, Ys, Z)
   FOLD_IN_GGA(X, .(Y, Ys), Z) -> U1_GGA(X, Y, Ys, Z, myop_in_gga(X, Y, V))

The TRS R consists of the following rules:

   myop_in_gga(a, b, c) -> myop_out_gga(a, b, c)

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

myop_in_gga(x1, x2, x3)  =  myop_in_gga(x1, x2)

a  =  a

b  =  b

myop_out_gga(x1, x2, x3)  =  myop_out_gga(x3)

FOLD_IN_GGA(x1, x2, x3)  =  FOLD_IN_GGA(x1, x2)

U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x3, x5)


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

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GGA(Ys, myop_out_gga(V)) -> FOLD_IN_GGA(V, Ys)
   FOLD_IN_GGA(X, .(Y, Ys)) -> U1_GGA(Ys, myop_in_gga(X, Y))

The TRS R consists of the following rules:

   myop_in_gga(a, b) -> myop_out_gga(c)

The set Q consists of the following terms:

   myop_in_gga(x0, x1)

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

(11) 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:
*FOLD_IN_GGA(X, .(Y, Ys)) -> U1_GGA(Ys, myop_in_gga(X, Y))
The graph contains the following edges 2 > 1


*U1_GGA(Ys, myop_out_gga(V)) -> FOLD_IN_GGA(V, Ys)
The graph contains the following edges 2 > 1, 1 >= 2


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

(12)
YES
