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# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

max(a,a,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) 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:

max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).


Query: max(a,a,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:

max_in_3: (f,f,b)

less_in_2: (f,b)

Transforming Prolog into the following Term Rewriting System:

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

   max_in_aag(X, Y, X) -> U1_aag(X, Y, less_in_ag(Y, X))
   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
   U1_aag(X, Y, less_out_ag(Y, X)) -> max_out_aag(X, Y, X)
   max_in_aag(X, Y, Y) -> U2_aag(X, Y, less_in_ag(X, s(Y)))
   U2_aag(X, Y, less_out_ag(X, s(Y))) -> max_out_aag(X, Y, Y)

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

U1_aag(x1, x2, x3)  =  U1_aag(x1, x3)

less_in_ag(x1, x2)  =  less_in_ag(x2)

s(x1)  =  s(x1)

less_out_ag(x1, x2)  =  less_out_ag(x1)

U3_ag(x1, x2, x3)  =  U3_ag(x3)

max_out_aag(x1, x2, x3)  =  max_out_aag(x1, x2)

U2_aag(x1, x2, x3)  =  U2_aag(x2, 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:

   max_in_aag(X, Y, X) -> U1_aag(X, Y, less_in_ag(Y, X))
   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
   U1_aag(X, Y, less_out_ag(Y, X)) -> max_out_aag(X, Y, X)
   max_in_aag(X, Y, Y) -> U2_aag(X, Y, less_in_ag(X, s(Y)))
   U2_aag(X, Y, less_out_ag(X, s(Y))) -> max_out_aag(X, Y, Y)

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

U1_aag(x1, x2, x3)  =  U1_aag(x1, x3)

less_in_ag(x1, x2)  =  less_in_ag(x2)

s(x1)  =  s(x1)

less_out_ag(x1, x2)  =  less_out_ag(x1)

U3_ag(x1, x2, x3)  =  U3_ag(x3)

max_out_aag(x1, x2, x3)  =  max_out_aag(x1, x2)

U2_aag(x1, x2, x3)  =  U2_aag(x2, 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:

   MAX_IN_AAG(X, Y, X) -> U1_AAG(X, Y, less_in_ag(Y, X))
   MAX_IN_AAG(X, Y, X) -> LESS_IN_AG(Y, X)
   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
   MAX_IN_AAG(X, Y, Y) -> U2_AAG(X, Y, less_in_ag(X, s(Y)))
   MAX_IN_AAG(X, Y, Y) -> LESS_IN_AG(X, s(Y))

The TRS R consists of the following rules:

   max_in_aag(X, Y, X) -> U1_aag(X, Y, less_in_ag(Y, X))
   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
   U1_aag(X, Y, less_out_ag(Y, X)) -> max_out_aag(X, Y, X)
   max_in_aag(X, Y, Y) -> U2_aag(X, Y, less_in_ag(X, s(Y)))
   U2_aag(X, Y, less_out_ag(X, s(Y))) -> max_out_aag(X, Y, Y)

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

U1_aag(x1, x2, x3)  =  U1_aag(x1, x3)

less_in_ag(x1, x2)  =  less_in_ag(x2)

s(x1)  =  s(x1)

less_out_ag(x1, x2)  =  less_out_ag(x1)

U3_ag(x1, x2, x3)  =  U3_ag(x3)

max_out_aag(x1, x2, x3)  =  max_out_aag(x1, x2)

U2_aag(x1, x2, x3)  =  U2_aag(x2, x3)

MAX_IN_AAG(x1, x2, x3)  =  MAX_IN_AAG(x3)

U1_AAG(x1, x2, x3)  =  U1_AAG(x1, x3)

LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

U3_AG(x1, x2, x3)  =  U3_AG(x3)

U2_AAG(x1, x2, x3)  =  U2_AAG(x2, x3)


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

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

   MAX_IN_AAG(X, Y, X) -> U1_AAG(X, Y, less_in_ag(Y, X))
   MAX_IN_AAG(X, Y, X) -> LESS_IN_AG(Y, X)
   LESS_IN_AG(s(X), s(Y)) -> U3_AG(X, Y, less_in_ag(X, Y))
   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)
   MAX_IN_AAG(X, Y, Y) -> U2_AAG(X, Y, less_in_ag(X, s(Y)))
   MAX_IN_AAG(X, Y, Y) -> LESS_IN_AG(X, s(Y))

The TRS R consists of the following rules:

   max_in_aag(X, Y, X) -> U1_aag(X, Y, less_in_ag(Y, X))
   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
   U1_aag(X, Y, less_out_ag(Y, X)) -> max_out_aag(X, Y, X)
   max_in_aag(X, Y, Y) -> U2_aag(X, Y, less_in_ag(X, s(Y)))
   U2_aag(X, Y, less_out_ag(X, s(Y))) -> max_out_aag(X, Y, Y)

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

U1_aag(x1, x2, x3)  =  U1_aag(x1, x3)

less_in_ag(x1, x2)  =  less_in_ag(x2)

s(x1)  =  s(x1)

less_out_ag(x1, x2)  =  less_out_ag(x1)

U3_ag(x1, x2, x3)  =  U3_ag(x3)

max_out_aag(x1, x2, x3)  =  max_out_aag(x1, x2)

U2_aag(x1, x2, x3)  =  U2_aag(x2, x3)

MAX_IN_AAG(x1, x2, x3)  =  MAX_IN_AAG(x3)

U1_AAG(x1, x2, x3)  =  U1_AAG(x1, x3)

LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)

U3_AG(x1, x2, x3)  =  U3_AG(x3)

U2_AAG(x1, x2, x3)  =  U2_AAG(x2, x3)


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

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

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

   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)

The TRS R consists of the following rules:

   max_in_aag(X, Y, X) -> U1_aag(X, Y, less_in_ag(Y, X))
   less_in_ag(0, s(X1)) -> less_out_ag(0, s(X1))
   less_in_ag(s(X), s(Y)) -> U3_ag(X, Y, less_in_ag(X, Y))
   U3_ag(X, Y, less_out_ag(X, Y)) -> less_out_ag(s(X), s(Y))
   U1_aag(X, Y, less_out_ag(Y, X)) -> max_out_aag(X, Y, X)
   max_in_aag(X, Y, Y) -> U2_aag(X, Y, less_in_ag(X, s(Y)))
   U2_aag(X, Y, less_out_ag(X, s(Y))) -> max_out_aag(X, Y, Y)

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

U1_aag(x1, x2, x3)  =  U1_aag(x1, x3)

less_in_ag(x1, x2)  =  less_in_ag(x2)

s(x1)  =  s(x1)

less_out_ag(x1, x2)  =  less_out_ag(x1)

U3_ag(x1, x2, x3)  =  U3_ag(x3)

max_out_aag(x1, x2, x3)  =  max_out_aag(x1, x2)

U2_aag(x1, x2, x3)  =  U2_aag(x2, x3)

LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)


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:

   LESS_IN_AG(s(X), s(Y)) -> LESS_IN_AG(X, Y)

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

LESS_IN_AG(x1, x2)  =  LESS_IN_AG(x2)


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:

   LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)

R is empty.
Q is empty.
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:
*LESS_IN_AG(s(Y)) -> LESS_IN_AG(Y)
The graph contains the following edges 1 > 1


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(12)
YES
