Left Termination of the query pattern minimum_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
    ↳3 PrologToPiTRSProof (⇐)
        ↳4 PiTRS
        ↳5 DependencyPairsProof (⇔)
        ↳6 PiDP
        ↳7 DependencyGraphProof (⇔)
        ↳8 PiDP
        ↳9 UsableRulesProof (⇔)
        ↳10 PiDP
        ↳11 PiDPToQDPProof (⇐)
        ↳12 QDP
        ↳13 NonTerminationProof (⇔)
        ↳14 NO
    ↳15 PrologToPiTRSProof (⇐)
        ↳16 PiTRS
        ↳17 DependencyPairsProof (⇔)
        ↳18 PiDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 PiDP
        ↳21 UsableRulesProof (⇔)
        ↳22 PiDP
        ↳23 PiDPToQDPProof (⇐)
        ↳24 QDP
        ↳25 NonTerminationProof (⇔)
        ↳26 NO

(0) Obligation:

Clauses:

minimum(tree(X, void, X1), X).
minimum(tree(X2, Left, X3), X) :- minimum(Left, X).

Queries:

minimum(a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

minimum1(tree(T5, void, T6), T5).
minimum1(tree(T11, tree(T24, void, T25), T13), T24).
minimum1(tree(T11, tree(T34, T38, T36), T13), T37) :- minimum1(T38, T37).
minimum1(tree(T47, tree(T60, void, T61), T49), T60).
minimum1(tree(T47, tree(T70, T74, T72), T49), T73) :- minimum1(T74, T73).

Queries:

minimum1(a,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minimum1_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x7)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → U1_AG(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)
MINIMUM1_IN_AG(tree(T47, tree(T70, T74, T72), T49), T73) → U2_AG(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x7)
U2_AG(x1, x2, x3, x4, x5, x6, x7)  =  U2_AG(x7)

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

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → U1_AG(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)
MINIMUM1_IN_AG(tree(T47, tree(T70, T74, T72), T49), T73) → U2_AG(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x7)
U2_AG(x1, x2, x3, x4, x5, x6, x7)  =  U2_AG(x7)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x2)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(T37) → MINIMUM1_IN_AG(T37)

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

(13) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MINIMUM1_IN_AG(T37) evaluates to t =MINIMUM1_IN_AG(T37)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MINIMUM1_IN_AG(T37) to MINIMUM1_IN_AG(T37).



(14) NO

(15) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minimum1_in: (f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1, x2)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x6, x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x6, x7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(16) Obligation:

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

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1, x2)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x6, x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x6, x7)

(17) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → U1_AG(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)
MINIMUM1_IN_AG(tree(T47, tree(T70, T74, T72), T49), T73) → U2_AG(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1, x2)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x6, x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x6, x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x6, x7)
U2_AG(x1, x2, x3, x4, x5, x6, x7)  =  U2_AG(x6, x7)

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

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → U1_AG(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)
MINIMUM1_IN_AG(tree(T47, tree(T70, T74, T72), T49), T73) → U2_AG(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1, x2)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x6, x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x6, x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)
U1_AG(x1, x2, x3, x4, x5, x6, x7)  =  U1_AG(x6, x7)
U2_AG(x1, x2, x3, x4, x5, x6, x7)  =  U2_AG(x6, x7)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(20) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)

The TRS R consists of the following rules:

minimum1_in_ag(tree(T5, void, T6), T5) → minimum1_out_ag(tree(T5, void, T6), T5)
minimum1_in_ag(tree(T11, tree(T24, void, T25), T13), T24) → minimum1_out_ag(tree(T11, tree(T24, void, T25), T13), T24)
minimum1_in_ag(tree(T11, tree(T34, T38, T36), T13), T37) → U1_ag(T11, T34, T38, T36, T13, T37, minimum1_in_ag(T38, T37))
minimum1_in_ag(tree(T47, tree(T70, T74, T72), T49), T73) → U2_ag(T47, T70, T74, T72, T49, T73, minimum1_in_ag(T74, T73))
U2_ag(T47, T70, T74, T72, T49, T73, minimum1_out_ag(T74, T73)) → minimum1_out_ag(tree(T47, tree(T70, T74, T72), T49), T73)
U1_ag(T11, T34, T38, T36, T13, T37, minimum1_out_ag(T38, T37)) → minimum1_out_ag(tree(T11, tree(T34, T38, T36), T13), T37)

The argument filtering Pi contains the following mapping:
minimum1_in_ag(x1, x2)  =  minimum1_in_ag(x2)
minimum1_out_ag(x1, x2)  =  minimum1_out_ag(x1, x2)
tree(x1, x2, x3)  =  tree(x2)
U1_ag(x1, x2, x3, x4, x5, x6, x7)  =  U1_ag(x6, x7)
U2_ag(x1, x2, x3, x4, x5, x6, x7)  =  U2_ag(x6, x7)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(tree(T11, tree(T34, T38, T36), T13), T37) → MINIMUM1_IN_AG(T38, T37)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x2)
MINIMUM1_IN_AG(x1, x2)  =  MINIMUM1_IN_AG(x2)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_AG(T37) → MINIMUM1_IN_AG(T37)

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

(25) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = MINIMUM1_IN_AG(T37) evaluates to t =MINIMUM1_IN_AG(T37)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from MINIMUM1_IN_AG(T37) to MINIMUM1_IN_AG(T37).



(26) NO