(0) Obligation:
Clauses:append([], L, L).
append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).
Queries:
append3(a,a,a,g).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:append7(.(T38, T42), T43, .(T38, T41)) :- append7(T42, T43, T41).
append19(.(T77, T80), T81, .(T77, X85)) :- append19(T80, T81, X85).
append31([], T21, T22, T12) :- append7(T21, T22, T12).
append31(.(T58, T55), T56, T57, T12) :- append19(T55, T56, X56).
append31(.(T62, T55), T56, T63, T12) :- ','(appendc19(T55, T56, T61), append7(.(T62, T61), T63, T12)).
Clauses:
appendc7([], T29, T29).
appendc7(.(T38, T42), T43, .(T38, T41)) :- appendc7(T42, T43, T41).
appendc19([], T70, T70).
appendc19(.(T77, T80), T81, .(T77, X85)) :- appendc19(T80, T81, X85).
Afs:
append31(x1, x2, x3, x4) = append31(x4)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:append31_in: (f,f,f,b)
append7_in: (f,f,b)
append19_in: (f,f,f)
appendc19_in: (f,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, appendc19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, appendc19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, appendc19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)
The TRS R consists of the following rules:
appendc19_in_aaa([], T70, T70) → appendc19_out_aaa([], T70, T70)
appendc19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U9_aaa(T77, T80, T81, X85, appendc19_in_aaa(T80, T81, X85))
U9_aaa(T77, T80, T81, X85, appendc19_out_aaa(T80, T81, X85)) → appendc19_out_aaa(.(T77, T80), T81, .(T77, X85))
The argument filtering Pi contains the following mapping:
append7_in_aag(x1, x2, x3) = append7_in_aag(x3)
.(x1, x2) = .(x2)
append19_in_aaa(x1, x2, x3) = append19_in_aaa
appendc19_in_aaa(x1, x2, x3) = appendc19_in_aaa
appendc19_out_aaa(x1, x2, x3) = appendc19_out_aaa(x1)
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
[] = []
APPEND31_IN_AAAG(x1, x2, x3, x4) = APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4) = U3_AAAG(x3, x4)
APPEND7_IN_AAG(x1, x2, x3) = APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U4_AAAG(x1, x2, x3, x4, x5, x6) = U4_AAAG(x5, x6)
APPEND19_IN_AAA(x1, x2, x3) = APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6) = U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6) = U6_AAAG(x2, x5, x6)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND31_IN_AAAG([], T21, T22, T12) → U3_AAAG(T21, T22, T12, append7_in_aag(T21, T22, T12))
APPEND31_IN_AAAG([], T21, T22, T12) → APPEND7_IN_AAG(T21, T22, T12)
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → U1_AAG(T38, T42, T43, T41, append7_in_aag(T42, T43, T41))
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → U4_AAAG(T58, T55, T56, T57, T12, append19_in_aaa(T55, T56, X56))
APPEND31_IN_AAAG(.(T58, T55), T56, T57, T12) → APPEND19_IN_AAA(T55, T56, X56)
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → U2_AAA(T77, T80, T81, X85, append19_in_aaa(T80, T81, X85))
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
APPEND31_IN_AAAG(.(T62, T55), T56, T63, T12) → U5_AAAG(T62, T55, T56, T63, T12, appendc19_in_aaa(T55, T56, T61))
U5_AAAG(T62, T55, T56, T63, T12, appendc19_out_aaa(T55, T56, T61)) → U6_AAAG(T62, T55, T56, T63, T12, append7_in_aag(.(T62, T61), T63, T12))
U5_AAAG(T62, T55, T56, T63, T12, appendc19_out_aaa(T55, T56, T61)) → APPEND7_IN_AAG(.(T62, T61), T63, T12)
The TRS R consists of the following rules:
appendc19_in_aaa([], T70, T70) → appendc19_out_aaa([], T70, T70)
appendc19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U9_aaa(T77, T80, T81, X85, appendc19_in_aaa(T80, T81, X85))
U9_aaa(T77, T80, T81, X85, appendc19_out_aaa(T80, T81, X85)) → appendc19_out_aaa(.(T77, T80), T81, .(T77, X85))
The argument filtering Pi contains the following mapping:
append7_in_aag(x1, x2, x3) = append7_in_aag(x3)
.(x1, x2) = .(x2)
append19_in_aaa(x1, x2, x3) = append19_in_aaa
appendc19_in_aaa(x1, x2, x3) = appendc19_in_aaa
appendc19_out_aaa(x1, x2, x3) = appendc19_out_aaa(x1)
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
[] = []
APPEND31_IN_AAAG(x1, x2, x3, x4) = APPEND31_IN_AAAG(x4)
U3_AAAG(x1, x2, x3, x4) = U3_AAAG(x3, x4)
APPEND7_IN_AAG(x1, x2, x3) = APPEND7_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5) = U1_AAG(x4, x5)
U4_AAAG(x1, x2, x3, x4, x5, x6) = U4_AAAG(x5, x6)
APPEND19_IN_AAA(x1, x2, x3) = APPEND19_IN_AAA
U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5)
U5_AAAG(x1, x2, x3, x4, x5, x6) = U5_AAAG(x5, x6)
U6_AAAG(x1, x2, x3, x4, x5, x6) = U6_AAAG(x2, x5, x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
The TRS R consists of the following rules:
appendc19_in_aaa([], T70, T70) → appendc19_out_aaa([], T70, T70)
appendc19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U9_aaa(T77, T80, T81, X85, appendc19_in_aaa(T80, T81, X85))
U9_aaa(T77, T80, T81, X85, appendc19_out_aaa(T80, T81, X85)) → appendc19_out_aaa(.(T77, T80), T81, .(T77, X85))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
appendc19_in_aaa(x1, x2, x3) = appendc19_in_aaa
appendc19_out_aaa(x1, x2, x3) = appendc19_out_aaa(x1)
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
[] = []
APPEND19_IN_AAA(x1, x2, x3) = APPEND19_IN_AAA
We have to consider all (P,R,Pi)-chains
(8) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(9) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND19_IN_AAA(.(T77, T80), T81, .(T77, X85)) → APPEND19_IN_AAA(T80, T81, X85)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APPEND19_IN_AAA(x1, x2, x3) = APPEND19_IN_AAA
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(11) Obligation:
Q DP problem:The TRS P consists of the following rules:
APPEND19_IN_AAA → APPEND19_IN_AAA
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(12) 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 = APPEND19_IN_AAA evaluates to t =APPEND19_IN_AAA
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 APPEND19_IN_AAA to APPEND19_IN_AAA.
(13) NO
(14) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
The TRS R consists of the following rules:
appendc19_in_aaa([], T70, T70) → appendc19_out_aaa([], T70, T70)
appendc19_in_aaa(.(T77, T80), T81, .(T77, X85)) → U9_aaa(T77, T80, T81, X85, appendc19_in_aaa(T80, T81, X85))
U9_aaa(T77, T80, T81, X85, appendc19_out_aaa(T80, T81, X85)) → appendc19_out_aaa(.(T77, T80), T81, .(T77, X85))
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
appendc19_in_aaa(x1, x2, x3) = appendc19_in_aaa
appendc19_out_aaa(x1, x2, x3) = appendc19_out_aaa(x1)
U9_aaa(x1, x2, x3, x4, x5) = U9_aaa(x5)
[] = []
APPEND7_IN_AAG(x1, x2, x3) = APPEND7_IN_AAG(x3)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(16) Obligation:
Pi DP problem:The TRS P consists of the following rules:
APPEND7_IN_AAG(.(T38, T42), T43, .(T38, T41)) → APPEND7_IN_AAG(T42, T43, T41)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APPEND7_IN_AAG(x1, x2, x3) = APPEND7_IN_AAG(x3)
We have to consider all (P,R,Pi)-chains
(17) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(18) Obligation:
Q DP problem:The TRS P consists of the following rules:
APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(19) 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:
- APPEND7_IN_AAG(.(T41)) → APPEND7_IN_AAG(T41)
The graph contains the following edges 1 > 1