(0) Obligation:
Clauses:
append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
append([], Y, Y).
Query: append(g,g,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
appendA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) :- appendA(T30, T31, T33).
appendA(.(T8, []), T42, .([], T42)).
appendA([], T44, T44).
Query: appendA(g,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:
appendA_in: (b,b,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
appendA_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
appendA_in_gga(.(T8, []), T42, .([], T42)) → appendA_out_gga(.(T8, []), T42, .([], T42))
appendA_in_gga([], T44, T44) → appendA_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, appendA_out_gga(T30, T31, T33)) → appendA_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))
The argument filtering Pi contains the following mapping:
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x2,
x3,
x6)
[] =
[]
appendA_out_gga(
x1,
x2,
x3) =
appendA_out_gga(
x3)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
appendA_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
appendA_in_gga(.(T8, []), T42, .([], T42)) → appendA_out_gga(.(T8, []), T42, .([], T42))
appendA_in_gga([], T44, T44) → appendA_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, appendA_out_gga(T30, T31, T33)) → appendA_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))
The argument filtering Pi contains the following mapping:
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x2,
x3,
x6)
[] =
[]
appendA_out_gga(
x1,
x2,
x3) =
appendA_out_gga(
x3)
(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:
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPENDA_IN_GGA(T30, T31, T33)
The TRS R consists of the following rules:
appendA_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
appendA_in_gga(.(T8, []), T42, .([], T42)) → appendA_out_gga(.(T8, []), T42, .([], T42))
appendA_in_gga([], T44, T44) → appendA_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, appendA_out_gga(T30, T31, T33)) → appendA_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))
The argument filtering Pi contains the following mapping:
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x2,
x3,
x6)
[] =
[]
appendA_out_gga(
x1,
x2,
x3) =
appendA_out_gga(
x3)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGA(
x2,
x3,
x6)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_GGA(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPENDA_IN_GGA(T30, T31, T33)
The TRS R consists of the following rules:
appendA_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
appendA_in_gga(.(T8, []), T42, .([], T42)) → appendA_out_gga(.(T8, []), T42, .([], T42))
appendA_in_gga([], T44, T44) → appendA_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, appendA_out_gga(T30, T31, T33)) → appendA_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))
The argument filtering Pi contains the following mapping:
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x2,
x3,
x6)
[] =
[]
appendA_out_gga(
x1,
x2,
x3) =
appendA_out_gga(
x3)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGA(
x2,
x3,
x6)
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:
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPENDA_IN_GGA(T30, T31, T33)
The TRS R consists of the following rules:
appendA_in_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → U1_gga(T8, T29, T30, T31, T33, appendA_in_gga(T30, T31, T33))
appendA_in_gga(.(T8, []), T42, .([], T42)) → appendA_out_gga(.(T8, []), T42, .([], T42))
appendA_in_gga([], T44, T44) → appendA_out_gga([], T44, T44)
U1_gga(T8, T29, T30, T31, T33, appendA_out_gga(T30, T31, T33)) → appendA_out_gga(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33)))
The argument filtering Pi contains the following mapping:
appendA_in_gga(
x1,
x2,
x3) =
appendA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x2,
x3,
x6)
[] =
[]
appendA_out_gga(
x1,
x2,
x3) =
appendA_out_gga(
x3)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
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:
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31, .(.(T29, T30), .(T30, T33))) → APPENDA_IN_GGA(T30, T31, T33)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GGA(
x1,
x2,
x3) =
APPENDA_IN_GGA(
x1,
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:
APPENDA_IN_GGA(.(T8, .(T29, T30)), T31) → APPENDA_IN_GGA(T30, T31)
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:
- APPENDA_IN_GGA(.(T8, .(T29, T30)), T31) → APPENDA_IN_GGA(T30, T31)
The graph contains the following edges 1 > 1, 2 >= 2
(14) YES