(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), list(Xs)).
list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Query: goal(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

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

goalA_in_g(0) → U1_g(listB_in_)
listB_in_listB_out_
U1_g(listB_out_) → goalA_out_g(0)
goalA_in_g(s(T9)) → U2_g(T9, pC_in_gaa(T9, X36, X35))
pC_in_gaa(T9, T10, X35) → U7_gaa(T9, T10, X35, s2lD_in_ga(T9, T10))
s2lD_in_ga(0, []) → s2lD_out_ga(0, [])
s2lD_in_ga(s(T16), .(X75, X76)) → U3_ga(T16, X75, X76, s2lD_in_ga(T16, X76))
U3_ga(T16, X75, X76, s2lD_out_ga(T16, X76)) → s2lD_out_ga(s(T16), .(X75, X76))
U7_gaa(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_gaa(T9, T10, X35, listF_in_ag(X35, T10))
listF_in_ag(X100, T26) → U6_ag(X100, T26, listE_in_g(T26))
listE_in_g([]) → listE_out_g([])
listE_in_g([]) → U4_g(listB_in_)
U4_g(listB_out_) → listE_out_g([])
listE_in_g(.(T36, T38)) → U5_g(T36, T38, listE_in_g(T38))
U5_g(T36, T38, listE_out_g(T38)) → listE_out_g(.(T36, T38))
U6_ag(X100, T26, listE_out_g(T26)) → listF_out_ag(X100, T26)
U8_gaa(T9, T10, X35, listF_out_ag(X35, T10)) → pC_out_gaa(T9, T10, X35)
U2_g(T9, pC_out_gaa(T9, X36, X35)) → goalA_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalA_in_g(x1)  =  goalA_in_g(x1)
0  =  0
U1_g(x1)  =  U1_g(x1)
listB_in_  =  listB_in_
listB_out_  =  listB_out_
goalA_out_g(x1)  =  goalA_out_g(x1)
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
s2lD_in_ga(x1, x2)  =  s2lD_in_ga(x1)
s2lD_out_ga(x1, x2)  =  s2lD_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
.(x1, x2)  =  .(x2)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x2, x4)
listF_in_ag(x1, x2)  =  listF_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
listE_in_g(x1)  =  listE_in_g(x1)
[]  =  []
listE_out_g(x1)  =  listE_out_g(x1)
U4_g(x1)  =  U4_g(x1)
U5_g(x1, x2, x3)  =  U5_g(x2, x3)
listF_out_ag(x1, x2)  =  listF_out_ag(x2)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2)

(3) 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:

GOALA_IN_G(0) → U1_G(listB_in_)
GOALA_IN_G(0) → LISTB_IN_
GOALA_IN_G(s(T9)) → U2_G(T9, pC_in_gaa(T9, X36, X35))
GOALA_IN_G(s(T9)) → PC_IN_GAA(T9, X36, X35)
PC_IN_GAA(T9, T10, X35) → U7_GAA(T9, T10, X35, s2lD_in_ga(T9, T10))
PC_IN_GAA(T9, T10, X35) → S2LD_IN_GA(T9, T10)
S2LD_IN_GA(s(T16), .(X75, X76)) → U3_GA(T16, X75, X76, s2lD_in_ga(T16, X76))
S2LD_IN_GA(s(T16), .(X75, X76)) → S2LD_IN_GA(T16, X76)
U7_GAA(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_GAA(T9, T10, X35, listF_in_ag(X35, T10))
U7_GAA(T9, T10, X35, s2lD_out_ga(T9, T10)) → LISTF_IN_AG(X35, T10)
LISTF_IN_AG(X100, T26) → U6_AG(X100, T26, listE_in_g(T26))
LISTF_IN_AG(X100, T26) → LISTE_IN_G(T26)
LISTE_IN_G([]) → U4_G(listB_in_)
LISTE_IN_G([]) → LISTB_IN_
LISTE_IN_G(.(T36, T38)) → U5_G(T36, T38, listE_in_g(T38))
LISTE_IN_G(.(T36, T38)) → LISTE_IN_G(T38)

The TRS R consists of the following rules:

goalA_in_g(0) → U1_g(listB_in_)
listB_in_listB_out_
U1_g(listB_out_) → goalA_out_g(0)
goalA_in_g(s(T9)) → U2_g(T9, pC_in_gaa(T9, X36, X35))
pC_in_gaa(T9, T10, X35) → U7_gaa(T9, T10, X35, s2lD_in_ga(T9, T10))
s2lD_in_ga(0, []) → s2lD_out_ga(0, [])
s2lD_in_ga(s(T16), .(X75, X76)) → U3_ga(T16, X75, X76, s2lD_in_ga(T16, X76))
U3_ga(T16, X75, X76, s2lD_out_ga(T16, X76)) → s2lD_out_ga(s(T16), .(X75, X76))
U7_gaa(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_gaa(T9, T10, X35, listF_in_ag(X35, T10))
listF_in_ag(X100, T26) → U6_ag(X100, T26, listE_in_g(T26))
listE_in_g([]) → listE_out_g([])
listE_in_g([]) → U4_g(listB_in_)
U4_g(listB_out_) → listE_out_g([])
listE_in_g(.(T36, T38)) → U5_g(T36, T38, listE_in_g(T38))
U5_g(T36, T38, listE_out_g(T38)) → listE_out_g(.(T36, T38))
U6_ag(X100, T26, listE_out_g(T26)) → listF_out_ag(X100, T26)
U8_gaa(T9, T10, X35, listF_out_ag(X35, T10)) → pC_out_gaa(T9, T10, X35)
U2_g(T9, pC_out_gaa(T9, X36, X35)) → goalA_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalA_in_g(x1)  =  goalA_in_g(x1)
0  =  0
U1_g(x1)  =  U1_g(x1)
listB_in_  =  listB_in_
listB_out_  =  listB_out_
goalA_out_g(x1)  =  goalA_out_g(x1)
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
s2lD_in_ga(x1, x2)  =  s2lD_in_ga(x1)
s2lD_out_ga(x1, x2)  =  s2lD_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
.(x1, x2)  =  .(x2)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x2, x4)
listF_in_ag(x1, x2)  =  listF_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
listE_in_g(x1)  =  listE_in_g(x1)
[]  =  []
listE_out_g(x1)  =  listE_out_g(x1)
U4_g(x1)  =  U4_g(x1)
U5_g(x1, x2, x3)  =  U5_g(x2, x3)
listF_out_ag(x1, x2)  =  listF_out_ag(x2)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2)
GOALA_IN_G(x1)  =  GOALA_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
LISTB_IN_  =  LISTB_IN_
U2_G(x1, x2)  =  U2_G(x1, x2)
PC_IN_GAA(x1, x2, x3)  =  PC_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
S2LD_IN_GA(x1, x2)  =  S2LD_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x2, x4)
LISTF_IN_AG(x1, x2)  =  LISTF_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
LISTE_IN_G(x1)  =  LISTE_IN_G(x1)
U4_G(x1)  =  U4_G(x1)
U5_G(x1, x2, x3)  =  U5_G(x2, x3)

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

(4) Obligation:

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

GOALA_IN_G(0) → U1_G(listB_in_)
GOALA_IN_G(0) → LISTB_IN_
GOALA_IN_G(s(T9)) → U2_G(T9, pC_in_gaa(T9, X36, X35))
GOALA_IN_G(s(T9)) → PC_IN_GAA(T9, X36, X35)
PC_IN_GAA(T9, T10, X35) → U7_GAA(T9, T10, X35, s2lD_in_ga(T9, T10))
PC_IN_GAA(T9, T10, X35) → S2LD_IN_GA(T9, T10)
S2LD_IN_GA(s(T16), .(X75, X76)) → U3_GA(T16, X75, X76, s2lD_in_ga(T16, X76))
S2LD_IN_GA(s(T16), .(X75, X76)) → S2LD_IN_GA(T16, X76)
U7_GAA(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_GAA(T9, T10, X35, listF_in_ag(X35, T10))
U7_GAA(T9, T10, X35, s2lD_out_ga(T9, T10)) → LISTF_IN_AG(X35, T10)
LISTF_IN_AG(X100, T26) → U6_AG(X100, T26, listE_in_g(T26))
LISTF_IN_AG(X100, T26) → LISTE_IN_G(T26)
LISTE_IN_G([]) → U4_G(listB_in_)
LISTE_IN_G([]) → LISTB_IN_
LISTE_IN_G(.(T36, T38)) → U5_G(T36, T38, listE_in_g(T38))
LISTE_IN_G(.(T36, T38)) → LISTE_IN_G(T38)

The TRS R consists of the following rules:

goalA_in_g(0) → U1_g(listB_in_)
listB_in_listB_out_
U1_g(listB_out_) → goalA_out_g(0)
goalA_in_g(s(T9)) → U2_g(T9, pC_in_gaa(T9, X36, X35))
pC_in_gaa(T9, T10, X35) → U7_gaa(T9, T10, X35, s2lD_in_ga(T9, T10))
s2lD_in_ga(0, []) → s2lD_out_ga(0, [])
s2lD_in_ga(s(T16), .(X75, X76)) → U3_ga(T16, X75, X76, s2lD_in_ga(T16, X76))
U3_ga(T16, X75, X76, s2lD_out_ga(T16, X76)) → s2lD_out_ga(s(T16), .(X75, X76))
U7_gaa(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_gaa(T9, T10, X35, listF_in_ag(X35, T10))
listF_in_ag(X100, T26) → U6_ag(X100, T26, listE_in_g(T26))
listE_in_g([]) → listE_out_g([])
listE_in_g([]) → U4_g(listB_in_)
U4_g(listB_out_) → listE_out_g([])
listE_in_g(.(T36, T38)) → U5_g(T36, T38, listE_in_g(T38))
U5_g(T36, T38, listE_out_g(T38)) → listE_out_g(.(T36, T38))
U6_ag(X100, T26, listE_out_g(T26)) → listF_out_ag(X100, T26)
U8_gaa(T9, T10, X35, listF_out_ag(X35, T10)) → pC_out_gaa(T9, T10, X35)
U2_g(T9, pC_out_gaa(T9, X36, X35)) → goalA_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalA_in_g(x1)  =  goalA_in_g(x1)
0  =  0
U1_g(x1)  =  U1_g(x1)
listB_in_  =  listB_in_
listB_out_  =  listB_out_
goalA_out_g(x1)  =  goalA_out_g(x1)
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
s2lD_in_ga(x1, x2)  =  s2lD_in_ga(x1)
s2lD_out_ga(x1, x2)  =  s2lD_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
.(x1, x2)  =  .(x2)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x2, x4)
listF_in_ag(x1, x2)  =  listF_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
listE_in_g(x1)  =  listE_in_g(x1)
[]  =  []
listE_out_g(x1)  =  listE_out_g(x1)
U4_g(x1)  =  U4_g(x1)
U5_g(x1, x2, x3)  =  U5_g(x2, x3)
listF_out_ag(x1, x2)  =  listF_out_ag(x2)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2)
GOALA_IN_G(x1)  =  GOALA_IN_G(x1)
U1_G(x1)  =  U1_G(x1)
LISTB_IN_  =  LISTB_IN_
U2_G(x1, x2)  =  U2_G(x1, x2)
PC_IN_GAA(x1, x2, x3)  =  PC_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
S2LD_IN_GA(x1, x2)  =  S2LD_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U8_GAA(x1, x2, x3, x4)  =  U8_GAA(x1, x2, x4)
LISTF_IN_AG(x1, x2)  =  LISTF_IN_AG(x2)
U6_AG(x1, x2, x3)  =  U6_AG(x2, x3)
LISTE_IN_G(x1)  =  LISTE_IN_G(x1)
U4_G(x1)  =  U4_G(x1)
U5_G(x1, x2, x3)  =  U5_G(x2, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LISTE_IN_G(.(T36, T38)) → LISTE_IN_G(T38)

The TRS R consists of the following rules:

goalA_in_g(0) → U1_g(listB_in_)
listB_in_listB_out_
U1_g(listB_out_) → goalA_out_g(0)
goalA_in_g(s(T9)) → U2_g(T9, pC_in_gaa(T9, X36, X35))
pC_in_gaa(T9, T10, X35) → U7_gaa(T9, T10, X35, s2lD_in_ga(T9, T10))
s2lD_in_ga(0, []) → s2lD_out_ga(0, [])
s2lD_in_ga(s(T16), .(X75, X76)) → U3_ga(T16, X75, X76, s2lD_in_ga(T16, X76))
U3_ga(T16, X75, X76, s2lD_out_ga(T16, X76)) → s2lD_out_ga(s(T16), .(X75, X76))
U7_gaa(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_gaa(T9, T10, X35, listF_in_ag(X35, T10))
listF_in_ag(X100, T26) → U6_ag(X100, T26, listE_in_g(T26))
listE_in_g([]) → listE_out_g([])
listE_in_g([]) → U4_g(listB_in_)
U4_g(listB_out_) → listE_out_g([])
listE_in_g(.(T36, T38)) → U5_g(T36, T38, listE_in_g(T38))
U5_g(T36, T38, listE_out_g(T38)) → listE_out_g(.(T36, T38))
U6_ag(X100, T26, listE_out_g(T26)) → listF_out_ag(X100, T26)
U8_gaa(T9, T10, X35, listF_out_ag(X35, T10)) → pC_out_gaa(T9, T10, X35)
U2_g(T9, pC_out_gaa(T9, X36, X35)) → goalA_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalA_in_g(x1)  =  goalA_in_g(x1)
0  =  0
U1_g(x1)  =  U1_g(x1)
listB_in_  =  listB_in_
listB_out_  =  listB_out_
goalA_out_g(x1)  =  goalA_out_g(x1)
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
s2lD_in_ga(x1, x2)  =  s2lD_in_ga(x1)
s2lD_out_ga(x1, x2)  =  s2lD_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
.(x1, x2)  =  .(x2)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x2, x4)
listF_in_ag(x1, x2)  =  listF_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
listE_in_g(x1)  =  listE_in_g(x1)
[]  =  []
listE_out_g(x1)  =  listE_out_g(x1)
U4_g(x1)  =  U4_g(x1)
U5_g(x1, x2, x3)  =  U5_g(x2, x3)
listF_out_ag(x1, x2)  =  listF_out_ag(x2)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2)
LISTE_IN_G(x1)  =  LISTE_IN_G(x1)

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:

LISTE_IN_G(.(T36, T38)) → LISTE_IN_G(T38)

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

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:

LISTE_IN_G(.(T38)) → LISTE_IN_G(T38)

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

(12) 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:

  • LISTE_IN_G(.(T38)) → LISTE_IN_G(T38)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

S2LD_IN_GA(s(T16), .(X75, X76)) → S2LD_IN_GA(T16, X76)

The TRS R consists of the following rules:

goalA_in_g(0) → U1_g(listB_in_)
listB_in_listB_out_
U1_g(listB_out_) → goalA_out_g(0)
goalA_in_g(s(T9)) → U2_g(T9, pC_in_gaa(T9, X36, X35))
pC_in_gaa(T9, T10, X35) → U7_gaa(T9, T10, X35, s2lD_in_ga(T9, T10))
s2lD_in_ga(0, []) → s2lD_out_ga(0, [])
s2lD_in_ga(s(T16), .(X75, X76)) → U3_ga(T16, X75, X76, s2lD_in_ga(T16, X76))
U3_ga(T16, X75, X76, s2lD_out_ga(T16, X76)) → s2lD_out_ga(s(T16), .(X75, X76))
U7_gaa(T9, T10, X35, s2lD_out_ga(T9, T10)) → U8_gaa(T9, T10, X35, listF_in_ag(X35, T10))
listF_in_ag(X100, T26) → U6_ag(X100, T26, listE_in_g(T26))
listE_in_g([]) → listE_out_g([])
listE_in_g([]) → U4_g(listB_in_)
U4_g(listB_out_) → listE_out_g([])
listE_in_g(.(T36, T38)) → U5_g(T36, T38, listE_in_g(T38))
U5_g(T36, T38, listE_out_g(T38)) → listE_out_g(.(T36, T38))
U6_ag(X100, T26, listE_out_g(T26)) → listF_out_ag(X100, T26)
U8_gaa(T9, T10, X35, listF_out_ag(X35, T10)) → pC_out_gaa(T9, T10, X35)
U2_g(T9, pC_out_gaa(T9, X36, X35)) → goalA_out_g(s(T9))

The argument filtering Pi contains the following mapping:
goalA_in_g(x1)  =  goalA_in_g(x1)
0  =  0
U1_g(x1)  =  U1_g(x1)
listB_in_  =  listB_in_
listB_out_  =  listB_out_
goalA_out_g(x1)  =  goalA_out_g(x1)
s(x1)  =  s(x1)
U2_g(x1, x2)  =  U2_g(x1, x2)
pC_in_gaa(x1, x2, x3)  =  pC_in_gaa(x1)
U7_gaa(x1, x2, x3, x4)  =  U7_gaa(x1, x4)
s2lD_in_ga(x1, x2)  =  s2lD_in_ga(x1)
s2lD_out_ga(x1, x2)  =  s2lD_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
.(x1, x2)  =  .(x2)
U8_gaa(x1, x2, x3, x4)  =  U8_gaa(x1, x2, x4)
listF_in_ag(x1, x2)  =  listF_in_ag(x2)
U6_ag(x1, x2, x3)  =  U6_ag(x2, x3)
listE_in_g(x1)  =  listE_in_g(x1)
[]  =  []
listE_out_g(x1)  =  listE_out_g(x1)
U4_g(x1)  =  U4_g(x1)
U5_g(x1, x2, x3)  =  U5_g(x2, x3)
listF_out_ag(x1, x2)  =  listF_out_ag(x2)
pC_out_gaa(x1, x2, x3)  =  pC_out_gaa(x1, x2)
S2LD_IN_GA(x1, x2)  =  S2LD_IN_GA(x1)

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:

S2LD_IN_GA(s(T16), .(X75, X76)) → S2LD_IN_GA(T16, X76)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2LD_IN_GA(x1, x2)  =  S2LD_IN_GA(x1)

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:

S2LD_IN_GA(s(T16)) → S2LD_IN_GA(T16)

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:

  • S2LD_IN_GA(s(T16)) → S2LD_IN_GA(T16)
    The graph contains the following edges 1 > 1

(20) YES