(0) Obligation:
Clauses:
minimum(tree(X, void, X1), X).
minimum(tree(X2, Left, X3), X) :- minimum(Left, X).
Query: minimum(g,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
minimumA(tree(T5, void, T6), T5).
minimumA(tree(T18, tree(T31, void, T32), T20), T31).
minimumA(tree(T18, tree(T41, T42, T43), T20), T45) :- minimumA(T42, T45).
Query: minimumA(g,a)
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
minimumA_in: (b,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)
The argument filtering Pi contains the following mapping:
minimumA_in_ga(
x1,
x2) =
minimumA_in_ga(
x1)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
void =
void
minimumA_out_ga(
x1,
x2) =
minimumA_out_ga(
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_ga(
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:
minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)
The argument filtering Pi contains the following mapping:
minimumA_in_ga(
x1,
x2) =
minimumA_in_ga(
x1)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
void =
void
minimumA_out_ga(
x1,
x2) =
minimumA_out_ga(
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_ga(
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:
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)
The TRS R consists of the following rules:
minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)
The argument filtering Pi contains the following mapping:
minimumA_in_ga(
x1,
x2) =
minimumA_in_ga(
x1)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
void =
void
minimumA_out_ga(
x1,
x2) =
minimumA_out_ga(
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_ga(
x7)
MINIMUMA_IN_GA(
x1,
x2) =
MINIMUMA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_GA(
x7)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)
The TRS R consists of the following rules:
minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)
The argument filtering Pi contains the following mapping:
minimumA_in_ga(
x1,
x2) =
minimumA_in_ga(
x1)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
void =
void
minimumA_out_ga(
x1,
x2) =
minimumA_out_ga(
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_ga(
x7)
MINIMUMA_IN_GA(
x1,
x2) =
MINIMUMA_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_GA(
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 1 less node.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)
The TRS R consists of the following rules:
minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)
The argument filtering Pi contains the following mapping:
minimumA_in_ga(
x1,
x2) =
minimumA_in_ga(
x1)
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
void =
void
minimumA_out_ga(
x1,
x2) =
minimumA_out_ga(
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U1_ga(
x7)
MINIMUMA_IN_GA(
x1,
x2) =
MINIMUMA_IN_GA(
x1)
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:
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)
R is empty.
The argument filtering Pi contains the following mapping:
tree(
x1,
x2,
x3) =
tree(
x1,
x2,
x3)
MINIMUMA_IN_GA(
x1,
x2) =
MINIMUMA_IN_GA(
x1)
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:
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUMA_IN_GA(T42)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) QDPSizeChangeProof (EQUIVALENT transformation)
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:
- MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUMA_IN_GA(T42)
The graph contains the following edges 1 > 1
(14) YES