(0) Obligation:
Clauses:
goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(app(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])) :- !.
last(X, Y) :- ','(tail(Y, T), last(X, T)).
app([], Y, Z) :- ','(!, eq(Y, Z)).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).
Queries:
goal(g,a,a).
(1) CutEliminatorProof (SOUND transformation)
Eliminated all cuts by simply ignoring them[PROLOG].
(2) Obligation:
Clauses:
goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(app(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])) :- true.
last(X, Y) :- ','(tail(Y, T), last(X, T)).
app([], Y, Z) :- eq(Y, Z).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
s2l(0, L) :- eq(L, []).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).
Queries:
goal(g,a,a).
(3) UndefinedPredicateHandlerProof (SOUND transformation)
Added facts for all undefined predicates [PROLOG].
(4) Obligation:
Clauses:
goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)).
applast(L, X, Last) :- ','(app(L, .(X, []), LX), last(Last, LX)).
last(X, .(X, [])) :- true.
last(X, Y) :- ','(tail(Y, T), last(X, T)).
app([], Y, Z) :- eq(Y, Z).
app(X, Y, .(H, Z)) :- ','(head(X, H), ','(tail(X, T), app(T, Y, Z))).
s2l(0, L) :- eq(L, []).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).
true.
Queries:
goal(g,a,a).
(5) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b,f,f)
s2l_in: (b,f)
applast_in: (b,f,f)
app_in: (b,b,f)
last_in: (f,b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(6) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
(7) 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:
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2l_in_ga(A, D))
GOAL_IN_GAA(A, B, C) → S2L_IN_GA(A, D)
S2L_IN_GA(0, L) → U12_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U13_GA(X, X1, Xs, p_out_ga(X, P)) → U14_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_GAA(A, B, C, s2l_out_ga(A, D)) → U2_GAA(A, B, C, applast_in_gaa(D, B, C))
U1_GAA(A, B, C, s2l_out_ga(A, D)) → APPLAST_IN_GAA(D, B, C)
APPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, app_in_gga(L, .(X, []), LX))
APPLAST_IN_GAA(L, X, Last) → APP_IN_GGA(L, .(X, []), LX)
APP_IN_GGA([], Y, Z) → U8_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GGA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → U11_GGA(X, Y, H, Z, app_in_gga(T, Y, Z))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_GAA(L, X, Last, last_in_ag(Last, LX))
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → LAST_IN_AG(Last, LX)
LAST_IN_AG(X, .(X, [])) → U5_AG(X, true_in_)
LAST_IN_AG(X, .(X, [])) → TRUE_IN_
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
LAST_IN_AG(X, Y) → TAIL_IN_GA(Y, T)
U6_AG(X, Y, tail_out_ga(Y, T)) → U7_AG(X, Y, last_in_ag(X, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
GOAL_IN_GAA(
x1,
x2,
x3) =
GOAL_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4) =
U1_GAA(
x1,
x4)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U12_GA(
x1,
x2) =
U12_GA(
x2)
EQ_IN_AG(
x1,
x2) =
EQ_IN_AG(
x2)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x1,
x4)
P_IN_GA(
x1,
x2) =
P_IN_GA(
x1)
U14_GA(
x1,
x2,
x3,
x4) =
U14_GA(
x1,
x4)
U2_GAA(
x1,
x2,
x3,
x4) =
U2_GAA(
x1,
x4)
APPLAST_IN_GAA(
x1,
x2,
x3) =
APPLAST_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3,
x4) =
U3_GAA(
x1,
x4)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x1,
x3)
EQ_IN_GA(
x1,
x2) =
EQ_IN_GA(
x1)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
HEAD_IN_GA(
x1,
x2) =
HEAD_IN_GA(
x1)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x1,
x2,
x5)
TAIL_IN_GA(
x1,
x2) =
TAIL_IN_GA(
x1)
U11_GGA(
x1,
x2,
x3,
x4,
x5) =
U11_GGA(
x1,
x2,
x5)
U4_GAA(
x1,
x2,
x3,
x4) =
U4_GAA(
x1,
x4)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U5_AG(
x1,
x2) =
U5_AG(
x2)
TRUE_IN_ =
TRUE_IN_
U6_AG(
x1,
x2,
x3) =
U6_AG(
x2,
x3)
U7_AG(
x1,
x2,
x3) =
U7_AG(
x2,
x3)
We have to consider all (P,R,Pi)-chains
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2l_in_ga(A, D))
GOAL_IN_GAA(A, B, C) → S2L_IN_GA(A, D)
S2L_IN_GA(0, L) → U12_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U13_GA(X, X1, Xs, p_out_ga(X, P)) → U14_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_GAA(A, B, C, s2l_out_ga(A, D)) → U2_GAA(A, B, C, applast_in_gaa(D, B, C))
U1_GAA(A, B, C, s2l_out_ga(A, D)) → APPLAST_IN_GAA(D, B, C)
APPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, app_in_gga(L, .(X, []), LX))
APPLAST_IN_GAA(L, X, Last) → APP_IN_GGA(L, .(X, []), LX)
APP_IN_GGA([], Y, Z) → U8_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GGA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → U11_GGA(X, Y, H, Z, app_in_gga(T, Y, Z))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_GAA(L, X, Last, last_in_ag(Last, LX))
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → LAST_IN_AG(Last, LX)
LAST_IN_AG(X, .(X, [])) → U5_AG(X, true_in_)
LAST_IN_AG(X, .(X, [])) → TRUE_IN_
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
LAST_IN_AG(X, Y) → TAIL_IN_GA(Y, T)
U6_AG(X, Y, tail_out_ga(Y, T)) → U7_AG(X, Y, last_in_ag(X, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
GOAL_IN_GAA(
x1,
x2,
x3) =
GOAL_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4) =
U1_GAA(
x1,
x4)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U12_GA(
x1,
x2) =
U12_GA(
x2)
EQ_IN_AG(
x1,
x2) =
EQ_IN_AG(
x2)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x1,
x4)
P_IN_GA(
x1,
x2) =
P_IN_GA(
x1)
U14_GA(
x1,
x2,
x3,
x4) =
U14_GA(
x1,
x4)
U2_GAA(
x1,
x2,
x3,
x4) =
U2_GAA(
x1,
x4)
APPLAST_IN_GAA(
x1,
x2,
x3) =
APPLAST_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3,
x4) =
U3_GAA(
x1,
x4)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x1,
x3)
EQ_IN_GA(
x1,
x2) =
EQ_IN_GA(
x1)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
HEAD_IN_GA(
x1,
x2) =
HEAD_IN_GA(
x1)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x1,
x2,
x5)
TAIL_IN_GA(
x1,
x2) =
TAIL_IN_GA(
x1)
U11_GGA(
x1,
x2,
x3,
x4,
x5) =
U11_GGA(
x1,
x2,
x5)
U4_GAA(
x1,
x2,
x3,
x4) =
U4_GAA(
x1,
x4)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U5_AG(
x1,
x2) =
U5_AG(
x2)
TRUE_IN_ =
TRUE_IN_
U6_AG(
x1,
x2,
x3) =
U6_AG(
x2,
x3)
U7_AG(
x1,
x2,
x3) =
U7_AG(
x2,
x3)
We have to consider all (P,R,Pi)-chains
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 21 less nodes.
(10) Complex Obligation (AND)
(11) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U6_AG(
x1,
x2,
x3) =
U6_AG(
x2,
x3)
We have to consider all (P,R,Pi)-chains
(12) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(13) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x2)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U6_AG(
x1,
x2,
x3) =
U6_AG(
x2,
x3)
We have to consider all (P,R,Pi)-chains
(14) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(15) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(Y) → U6_AG(Y, tail_in_ga(Y))
U6_AG(Y, tail_out_ga(Y, T)) → LAST_IN_AG(T)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(16) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
LAST_IN_AG(
Y) →
U6_AG(
Y,
tail_in_ga(
Y)) at position [1] we obtained the following new rules [LPAR04]:
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
(17) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(Y, tail_out_ga(Y, T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(18) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(19) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(Y, tail_out_ga(Y, T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
R is empty.
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(20) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
tail_in_ga(x0)
(21) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(Y, tail_out_ga(Y, T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(22) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U6_AG(
Y,
tail_out_ga(
Y,
T)) →
LAST_IN_AG(
T) we obtained the following new rules [LPAR04]:
U6_AG([], tail_out_ga([], [])) → LAST_IN_AG([])
U6_AG(.(z0), tail_out_ga(.(z0), z0)) → LAST_IN_AG(z0)
(23) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
U6_AG([], tail_out_ga([], [])) → LAST_IN_AG([])
U6_AG(.(z0), tail_out_ga(.(z0), z0)) → LAST_IN_AG(z0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(24) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(25) Complex Obligation (AND)
(26) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG([], tail_out_ga([], [])) → LAST_IN_AG([])
LAST_IN_AG([]) → U6_AG([], tail_out_ga([], []))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(27) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
LAST_IN_AG(
[]) evaluates to t =
LAST_IN_AG(
[])
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequenceLAST_IN_AG([]) →
U6_AG(
[],
tail_out_ga(
[],
[]))
with rule
LAST_IN_AG(
[]) →
U6_AG(
[],
tail_out_ga(
[],
[])) at position [] and matcher [ ]
U6_AG([], tail_out_ga([], [])) →
LAST_IN_AG(
[])
with rule
U6_AG(
[],
tail_out_ga(
[],
[])) →
LAST_IN_AG(
[])
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(28) FALSE
(29) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
U6_AG(.(z0), tail_out_ga(.(z0), z0)) → LAST_IN_AG(z0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(30) 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:
- U6_AG(.(z0), tail_out_ga(.(z0), z0)) → LAST_IN_AG(z0)
The graph contains the following edges 1 > 1, 2 > 1
- LAST_IN_AG(.(x0)) → U6_AG(.(x0), tail_out_ga(.(x0), x0))
The graph contains the following edges 1 >= 1
(31) TRUE
(32) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x1,
x2,
x5)
We have to consider all (P,R,Pi)-chains
(33) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(34) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
The TRS R consists of the following rules:
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x2)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x1,
x2,
x5)
We have to consider all (P,R,Pi)-chains
(35) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(36) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y) → U9_GGA(X, Y, head_in_ga(X))
U9_GGA(X, Y, head_out_ga(X)) → U10_GGA(X, Y, tail_in_ga(X))
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
The TRS R consists of the following rules:
head_in_ga([]) → head_out_ga([])
head_in_ga(.(X3)) → head_out_ga(.(X3))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(37) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
APP_IN_GGA(
X,
Y) →
U9_GGA(
X,
Y,
head_in_ga(
X)) at position [2] we obtained the following new rules [LPAR04]:
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
(38) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga(X)) → U10_GGA(X, Y, tail_in_ga(X))
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
The TRS R consists of the following rules:
head_in_ga([]) → head_out_ga([])
head_in_ga(.(X3)) → head_out_ga(.(X3))
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(39) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(40) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga(X)) → U10_GGA(X, Y, tail_in_ga(X))
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(41) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
head_in_ga(x0)
(42) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga(X)) → U10_GGA(X, Y, tail_in_ga(X))
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(43) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
U9_GGA(
X,
Y,
head_out_ga(
X)) →
U10_GGA(
X,
Y,
tail_in_ga(
X)) at position [2] we obtained the following new rules [LPAR04]:
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
(44) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([], [])
tail_in_ga(.(Xs)) → tail_out_ga(.(Xs), Xs)
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(45) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(46) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
R is empty.
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(47) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
tail_in_ga(x0)
(48) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(X, Y, tail_out_ga(X, T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(49) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U10_GGA(
X,
Y,
tail_out_ga(
X,
T)) →
APP_IN_GGA(
T,
Y) we obtained the following new rules [LPAR04]:
U10_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)
U10_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → APP_IN_GGA(z0, z1)
(50) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
U10_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)
U10_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → APP_IN_GGA(z0, z1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(51) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(52) Complex Obligation (AND)
(53) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA([], y1, head_out_ga([])) → U10_GGA([], y1, tail_out_ga([], []))
U10_GGA([], z0, tail_out_ga([], [])) → APP_IN_GGA([], z0)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga([]))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(54) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
U10_GGA(
[],
z0,
tail_out_ga(
[],
[])) evaluates to t =
U10_GGA(
[],
z0,
tail_out_ga(
[],
[]))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequenceU10_GGA([], z0, tail_out_ga([], [])) →
APP_IN_GGA(
[],
z0)
with rule
U10_GGA(
[],
z0',
tail_out_ga(
[],
[])) →
APP_IN_GGA(
[],
z0') at position [] and matcher [
z0' /
z0]
APP_IN_GGA([], z0) →
U9_GGA(
[],
z0,
head_out_ga(
[]))
with rule
APP_IN_GGA(
[],
y1') →
U9_GGA(
[],
y1',
head_out_ga(
[])) at position [] and matcher [
y1' /
z0]
U9_GGA([], z0, head_out_ga([])) →
U10_GGA(
[],
z0,
tail_out_ga(
[],
[]))
with rule
U9_GGA(
[],
y1,
head_out_ga(
[])) →
U10_GGA(
[],
y1,
tail_out_ga(
[],
[]))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(55) FALSE
(56) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
U10_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → APP_IN_GGA(z0, z1)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(57) 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:
- U9_GGA(.(x0), y1, head_out_ga(.(x0))) → U10_GGA(.(x0), y1, tail_out_ga(.(x0), x0))
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
- U10_GGA(.(z0), z1, tail_out_ga(.(z0), z0)) → APP_IN_GGA(z0, z1)
The graph contains the following edges 1 > 1, 3 > 1, 2 >= 2
- APP_IN_GGA(.(x0), y1) → U9_GGA(.(x0), y1, head_out_ga(.(x0)))
The graph contains the following edges 1 >= 1, 2 >= 2
(58) TRUE
(59) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x1,
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1,
x2)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x1,
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x1,
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x1,
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x1,
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x1,
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x1,
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x1,
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga(
x1)
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x1,
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x1,
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x1,
x2,
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x1,
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag(
x2)
U6_ag(
x1,
x2,
x3) =
U6_ag(
x2,
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x2,
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa(
x1)
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa(
x1)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x1,
x4)
We have to consider all (P,R,Pi)-chains
(60) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(61) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
The TRS R consists of the following rules:
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
The argument filtering Pi contains the following mapping:
0 =
0
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x1,
x2)
s(
x1) =
s(
x1)
.(
x1,
x2) =
.(
x2)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x1,
x4)
We have to consider all (P,R,Pi)-chains
(62) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(63) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X) → U13_GA(X, p_in_ga(X))
U13_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
The TRS R consists of the following rules:
p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(64) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
S2L_IN_GA(
X) →
U13_GA(
X,
p_in_ga(
X)) at position [1] we obtained the following new rules [LPAR04]:
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
(65) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
The TRS R consists of the following rules:
p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(66) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(67) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
R is empty.
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(68) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
p_in_ga(x0)
(69) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(X, p_out_ga(X, P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(70) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U13_GA(
X,
p_out_ga(
X,
P)) →
S2L_IN_GA(
P) we obtained the following new rules [LPAR04]:
U13_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)
(71) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U13_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(72) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
(73) Complex Obligation (AND)
(74) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(0, p_out_ga(0, 0)) → S2L_IN_GA(0)
S2L_IN_GA(0) → U13_GA(0, p_out_ga(0, 0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(75) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
S2L_IN_GA(
0) evaluates to t =
S2L_IN_GA(
0)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequenceS2L_IN_GA(0) →
U13_GA(
0,
p_out_ga(
0,
0))
with rule
S2L_IN_GA(
0) →
U13_GA(
0,
p_out_ga(
0,
0)) at position [] and matcher [ ]
U13_GA(0, p_out_ga(0, 0)) →
S2L_IN_GA(
0)
with rule
U13_GA(
0,
p_out_ga(
0,
0)) →
S2L_IN_GA(
0)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(76) FALSE
(77) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(78) 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:
- U13_GA(s(z0), p_out_ga(s(z0), z0)) → S2L_IN_GA(z0)
The graph contains the following edges 1 > 1, 2 > 1
- S2L_IN_GA(s(x0)) → U13_GA(s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1
(79) TRUE
(80) PrologToPiTRSProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b,f,f)
s2l_in: (b,f)
applast_in: (b,f,f)
app_in: (b,b,f)
last_in: (f,b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(81) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
(82) 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:
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2l_in_ga(A, D))
GOAL_IN_GAA(A, B, C) → S2L_IN_GA(A, D)
S2L_IN_GA(0, L) → U12_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U13_GA(X, X1, Xs, p_out_ga(X, P)) → U14_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_GAA(A, B, C, s2l_out_ga(A, D)) → U2_GAA(A, B, C, applast_in_gaa(D, B, C))
U1_GAA(A, B, C, s2l_out_ga(A, D)) → APPLAST_IN_GAA(D, B, C)
APPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, app_in_gga(L, .(X, []), LX))
APPLAST_IN_GAA(L, X, Last) → APP_IN_GGA(L, .(X, []), LX)
APP_IN_GGA([], Y, Z) → U8_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GGA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → U11_GGA(X, Y, H, Z, app_in_gga(T, Y, Z))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_GAA(L, X, Last, last_in_ag(Last, LX))
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → LAST_IN_AG(Last, LX)
LAST_IN_AG(X, .(X, [])) → U5_AG(X, true_in_)
LAST_IN_AG(X, .(X, [])) → TRUE_IN_
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
LAST_IN_AG(X, Y) → TAIL_IN_GA(Y, T)
U6_AG(X, Y, tail_out_ga(Y, T)) → U7_AG(X, Y, last_in_ag(X, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
GOAL_IN_GAA(
x1,
x2,
x3) =
GOAL_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4) =
U1_GAA(
x4)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U12_GA(
x1,
x2) =
U12_GA(
x2)
EQ_IN_AG(
x1,
x2) =
EQ_IN_AG(
x2)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x4)
P_IN_GA(
x1,
x2) =
P_IN_GA(
x1)
U14_GA(
x1,
x2,
x3,
x4) =
U14_GA(
x4)
U2_GAA(
x1,
x2,
x3,
x4) =
U2_GAA(
x4)
APPLAST_IN_GAA(
x1,
x2,
x3) =
APPLAST_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3,
x4) =
U3_GAA(
x4)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x3)
EQ_IN_GA(
x1,
x2) =
EQ_IN_GA(
x1)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
HEAD_IN_GA(
x1,
x2) =
HEAD_IN_GA(
x1)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x2,
x5)
TAIL_IN_GA(
x1,
x2) =
TAIL_IN_GA(
x1)
U11_GGA(
x1,
x2,
x3,
x4,
x5) =
U11_GGA(
x5)
U4_GAA(
x1,
x2,
x3,
x4) =
U4_GAA(
x4)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U5_AG(
x1,
x2) =
U5_AG(
x2)
TRUE_IN_ =
TRUE_IN_
U6_AG(
x1,
x2,
x3) =
U6_AG(
x3)
U7_AG(
x1,
x2,
x3) =
U7_AG(
x3)
We have to consider all (P,R,Pi)-chains
(83) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
GOAL_IN_GAA(A, B, C) → U1_GAA(A, B, C, s2l_in_ga(A, D))
GOAL_IN_GAA(A, B, C) → S2L_IN_GA(A, D)
S2L_IN_GA(0, L) → U12_GA(L, eq_in_ag(L, []))
S2L_IN_GA(0, L) → EQ_IN_AG(L, [])
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
S2L_IN_GA(X, .(X1, Xs)) → P_IN_GA(X, P)
U13_GA(X, X1, Xs, p_out_ga(X, P)) → U14_GA(X, X1, Xs, s2l_in_ga(P, Xs))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
U1_GAA(A, B, C, s2l_out_ga(A, D)) → U2_GAA(A, B, C, applast_in_gaa(D, B, C))
U1_GAA(A, B, C, s2l_out_ga(A, D)) → APPLAST_IN_GAA(D, B, C)
APPLAST_IN_GAA(L, X, Last) → U3_GAA(L, X, Last, app_in_gga(L, .(X, []), LX))
APPLAST_IN_GAA(L, X, Last) → APP_IN_GGA(L, .(X, []), LX)
APP_IN_GGA([], Y, Z) → U8_GGA(Y, Z, eq_in_ga(Y, Z))
APP_IN_GGA([], Y, Z) → EQ_IN_GA(Y, Z)
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
APP_IN_GGA(X, Y, .(H, Z)) → HEAD_IN_GA(X, H)
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → TAIL_IN_GA(X, T)
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → U11_GGA(X, Y, H, Z, app_in_gga(T, Y, Z))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_GAA(L, X, Last, last_in_ag(Last, LX))
U3_GAA(L, X, Last, app_out_gga(L, .(X, []), LX)) → LAST_IN_AG(Last, LX)
LAST_IN_AG(X, .(X, [])) → U5_AG(X, true_in_)
LAST_IN_AG(X, .(X, [])) → TRUE_IN_
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
LAST_IN_AG(X, Y) → TAIL_IN_GA(Y, T)
U6_AG(X, Y, tail_out_ga(Y, T)) → U7_AG(X, Y, last_in_ag(X, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
GOAL_IN_GAA(
x1,
x2,
x3) =
GOAL_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4) =
U1_GAA(
x4)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U12_GA(
x1,
x2) =
U12_GA(
x2)
EQ_IN_AG(
x1,
x2) =
EQ_IN_AG(
x2)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x4)
P_IN_GA(
x1,
x2) =
P_IN_GA(
x1)
U14_GA(
x1,
x2,
x3,
x4) =
U14_GA(
x4)
U2_GAA(
x1,
x2,
x3,
x4) =
U2_GAA(
x4)
APPLAST_IN_GAA(
x1,
x2,
x3) =
APPLAST_IN_GAA(
x1)
U3_GAA(
x1,
x2,
x3,
x4) =
U3_GAA(
x4)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U8_GGA(
x1,
x2,
x3) =
U8_GGA(
x3)
EQ_IN_GA(
x1,
x2) =
EQ_IN_GA(
x1)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
HEAD_IN_GA(
x1,
x2) =
HEAD_IN_GA(
x1)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x2,
x5)
TAIL_IN_GA(
x1,
x2) =
TAIL_IN_GA(
x1)
U11_GGA(
x1,
x2,
x3,
x4,
x5) =
U11_GGA(
x5)
U4_GAA(
x1,
x2,
x3,
x4) =
U4_GAA(
x4)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U5_AG(
x1,
x2) =
U5_AG(
x2)
TRUE_IN_ =
TRUE_IN_
U6_AG(
x1,
x2,
x3) =
U6_AG(
x3)
U7_AG(
x1,
x2,
x3) =
U7_AG(
x3)
We have to consider all (P,R,Pi)-chains
(84) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 21 less nodes.
(85) Complex Obligation (AND)
(86) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U6_AG(
x1,
x2,
x3) =
U6_AG(
x3)
We have to consider all (P,R,Pi)-chains
(87) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(88) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(X, Y) → U6_AG(X, Y, tail_in_ga(Y, T))
U6_AG(X, Y, tail_out_ga(Y, T)) → LAST_IN_AG(X, T)
The TRS R consists of the following rules:
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x2)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
LAST_IN_AG(
x1,
x2) =
LAST_IN_AG(
x2)
U6_AG(
x1,
x2,
x3) =
U6_AG(
x3)
We have to consider all (P,R,Pi)-chains
(89) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(90) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(Y) → U6_AG(tail_in_ga(Y))
U6_AG(tail_out_ga(T)) → LAST_IN_AG(T)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(91) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
tail_in_ga(.(Xs)) → tail_out_ga(Xs)
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(.(x1)) = 2·x1
POL(LAST_IN_AG(x1)) = x1
POL(U6_AG(x1)) = x1
POL([]) = 0
POL(tail_in_ga(x1)) = x1
POL(tail_out_ga(x1)) = 2·x1
(92) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG(Y) → U6_AG(tail_in_ga(Y))
U6_AG(tail_out_ga(T)) → LAST_IN_AG(T)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(93) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
LAST_IN_AG(
Y) →
U6_AG(
tail_in_ga(
Y)) at position [0] we obtained the following new rules [LPAR04]:
LAST_IN_AG([]) → U6_AG(tail_out_ga([]))
(94) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(tail_out_ga(T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG(tail_out_ga([]))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(95) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(96) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(tail_out_ga(T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG(tail_out_ga([]))
R is empty.
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(97) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
tail_in_ga(x0)
(98) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U6_AG(tail_out_ga(T)) → LAST_IN_AG(T)
LAST_IN_AG([]) → U6_AG(tail_out_ga([]))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(99) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U6_AG(
tail_out_ga(
T)) →
LAST_IN_AG(
T) we obtained the following new rules [LPAR04]:
U6_AG(tail_out_ga([])) → LAST_IN_AG([])
(100) Obligation:
Q DP problem:
The TRS P consists of the following rules:
LAST_IN_AG([]) → U6_AG(tail_out_ga([]))
U6_AG(tail_out_ga([])) → LAST_IN_AG([])
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(101) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
U6_AG(
tail_out_ga(
[])) evaluates to t =
U6_AG(
tail_out_ga(
[]))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequenceU6_AG(tail_out_ga([])) →
LAST_IN_AG(
[])
with rule
U6_AG(
tail_out_ga(
[])) →
LAST_IN_AG(
[]) at position [] and matcher [ ]
LAST_IN_AG([]) →
U6_AG(
tail_out_ga(
[]))
with rule
LAST_IN_AG(
[]) →
U6_AG(
tail_out_ga(
[]))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(102) FALSE
(103) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x2,
x5)
We have to consider all (P,R,Pi)-chains
(104) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(105) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y, .(H, Z)) → U9_GGA(X, Y, H, Z, head_in_ga(X, H))
U9_GGA(X, Y, H, Z, head_out_ga(X, H)) → U10_GGA(X, Y, H, Z, tail_in_ga(X, T))
U10_GGA(X, Y, H, Z, tail_out_ga(X, T)) → APP_IN_GGA(T, Y, Z)
The TRS R consists of the following rules:
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x2)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
APP_IN_GGA(
x1,
x2,
x3) =
APP_IN_GGA(
x1,
x2)
U9_GGA(
x1,
x2,
x3,
x4,
x5) =
U9_GGA(
x1,
x2,
x5)
U10_GGA(
x1,
x2,
x3,
x4,
x5) =
U10_GGA(
x2,
x5)
We have to consider all (P,R,Pi)-chains
(106) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(107) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y) → U9_GGA(X, Y, head_in_ga(X))
U9_GGA(X, Y, head_out_ga) → U10_GGA(Y, tail_in_ga(X))
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
The TRS R consists of the following rules:
head_in_ga([]) → head_out_ga
head_in_ga(.(X3)) → head_out_ga
tail_in_ga([]) → tail_out_ga([])
tail_in_ga(.(Xs)) → tail_out_ga(Xs)
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(108) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
tail_in_ga(.(Xs)) → tail_out_ga(Xs)
head_in_ga(.(X3)) → head_out_ga
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(.(x1)) = 2·x1
POL(APP_IN_GGA(x1, x2)) = 2·x1 + x2
POL(U10_GGA(x1, x2)) = x1 + x2
POL(U9_GGA(x1, x2, x3)) = x1 + x2 + x3
POL([]) = 0
POL(head_in_ga(x1)) = x1
POL(head_out_ga) = 0
POL(tail_in_ga(x1)) = x1
POL(tail_out_ga(x1)) = 2·x1
(109) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA(X, Y) → U9_GGA(X, Y, head_in_ga(X))
U9_GGA(X, Y, head_out_ga) → U10_GGA(Y, tail_in_ga(X))
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
head_in_ga([]) → head_out_ga
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(110) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
APP_IN_GGA(
X,
Y) →
U9_GGA(
X,
Y,
head_in_ga(
X)) at position [2] we obtained the following new rules [LPAR04]:
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
(111) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga) → U10_GGA(Y, tail_in_ga(X))
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
head_in_ga([]) → head_out_ga
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(112) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(113) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga) → U10_GGA(Y, tail_in_ga(X))
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
The set Q consists of the following terms:
head_in_ga(x0)
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(114) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
head_in_ga(x0)
(115) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U9_GGA(X, Y, head_out_ga) → U10_GGA(Y, tail_in_ga(X))
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(116) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
U9_GGA(
X,
Y,
head_out_ga) →
U10_GGA(
Y,
tail_in_ga(
X)) at position [1] we obtained the following new rules [LPAR04]:
U9_GGA([], y1, head_out_ga) → U10_GGA(y1, tail_out_ga([]))
(117) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
U9_GGA([], y1, head_out_ga) → U10_GGA(y1, tail_out_ga([]))
The TRS R consists of the following rules:
tail_in_ga([]) → tail_out_ga([])
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(118) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(119) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
U9_GGA([], y1, head_out_ga) → U10_GGA(y1, tail_out_ga([]))
R is empty.
The set Q consists of the following terms:
tail_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(120) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
tail_in_ga(x0)
(121) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U10_GGA(Y, tail_out_ga(T)) → APP_IN_GGA(T, Y)
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
U9_GGA([], y1, head_out_ga) → U10_GGA(y1, tail_out_ga([]))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(122) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U10_GGA(
Y,
tail_out_ga(
T)) →
APP_IN_GGA(
T,
Y) we obtained the following new rules [LPAR04]:
U10_GGA(z0, tail_out_ga([])) → APP_IN_GGA([], z0)
(123) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APP_IN_GGA([], y1) → U9_GGA([], y1, head_out_ga)
U9_GGA([], y1, head_out_ga) → U10_GGA(y1, tail_out_ga([]))
U10_GGA(z0, tail_out_ga([])) → APP_IN_GGA([], z0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(124) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
U9_GGA(
[],
y1,
head_out_ga) evaluates to t =
U9_GGA(
[],
y1,
head_out_ga)
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Semiunifier: [ ]
- Matcher: [ ]
Rewriting sequenceU9_GGA([], y1, head_out_ga) →
U10_GGA(
y1,
tail_out_ga(
[]))
with rule
U9_GGA(
[],
y1',
head_out_ga) →
U10_GGA(
y1',
tail_out_ga(
[])) at position [] and matcher [
y1' /
y1]
U10_GGA(y1, tail_out_ga([])) →
APP_IN_GGA(
[],
y1)
with rule
U10_GGA(
z0,
tail_out_ga(
[])) →
APP_IN_GGA(
[],
z0) at position [] and matcher [
z0 /
y1]
APP_IN_GGA([], y1) →
U9_GGA(
[],
y1,
head_out_ga)
with rule
APP_IN_GGA(
[],
y1) →
U9_GGA(
[],
y1,
head_out_ga)
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(125) FALSE
(126) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
The TRS R consists of the following rules:
goal_in_gaa(A, B, C) → U1_gaa(A, B, C, s2l_in_ga(A, D))
s2l_in_ga(0, L) → U12_ga(L, eq_in_ag(L, []))
eq_in_ag(X, X) → eq_out_ag(X, X)
U12_ga(L, eq_out_ag(L, [])) → s2l_out_ga(0, L)
s2l_in_ga(X, .(X1, Xs)) → U13_ga(X, X1, Xs, p_in_ga(X, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U13_ga(X, X1, Xs, p_out_ga(X, P)) → U14_ga(X, X1, Xs, s2l_in_ga(P, Xs))
U14_ga(X, X1, Xs, s2l_out_ga(P, Xs)) → s2l_out_ga(X, .(X1, Xs))
U1_gaa(A, B, C, s2l_out_ga(A, D)) → U2_gaa(A, B, C, applast_in_gaa(D, B, C))
applast_in_gaa(L, X, Last) → U3_gaa(L, X, Last, app_in_gga(L, .(X, []), LX))
app_in_gga([], Y, Z) → U8_gga(Y, Z, eq_in_ga(Y, Z))
eq_in_ga(X, X) → eq_out_ga(X, X)
U8_gga(Y, Z, eq_out_ga(Y, Z)) → app_out_gga([], Y, Z)
app_in_gga(X, Y, .(H, Z)) → U9_gga(X, Y, H, Z, head_in_ga(X, H))
head_in_ga([], X2) → head_out_ga([], X2)
head_in_ga(.(X, X3), X) → head_out_ga(.(X, X3), X)
U9_gga(X, Y, H, Z, head_out_ga(X, H)) → U10_gga(X, Y, H, Z, tail_in_ga(X, T))
tail_in_ga([], []) → tail_out_ga([], [])
tail_in_ga(.(X4, Xs), Xs) → tail_out_ga(.(X4, Xs), Xs)
U10_gga(X, Y, H, Z, tail_out_ga(X, T)) → U11_gga(X, Y, H, Z, app_in_gga(T, Y, Z))
U11_gga(X, Y, H, Z, app_out_gga(T, Y, Z)) → app_out_gga(X, Y, .(H, Z))
U3_gaa(L, X, Last, app_out_gga(L, .(X, []), LX)) → U4_gaa(L, X, Last, last_in_ag(Last, LX))
last_in_ag(X, .(X, [])) → U5_ag(X, true_in_)
true_in_ → true_out_
U5_ag(X, true_out_) → last_out_ag(X, .(X, []))
last_in_ag(X, Y) → U6_ag(X, Y, tail_in_ga(Y, T))
U6_ag(X, Y, tail_out_ga(Y, T)) → U7_ag(X, Y, last_in_ag(X, T))
U7_ag(X, Y, last_out_ag(X, T)) → last_out_ag(X, Y)
U4_gaa(L, X, Last, last_out_ag(Last, LX)) → applast_out_gaa(L, X, Last)
U2_gaa(A, B, C, applast_out_gaa(D, B, C)) → goal_out_gaa(A, B, C)
The argument filtering Pi contains the following mapping:
goal_in_gaa(
x1,
x2,
x3) =
goal_in_gaa(
x1)
U1_gaa(
x1,
x2,
x3,
x4) =
U1_gaa(
x4)
s2l_in_ga(
x1,
x2) =
s2l_in_ga(
x1)
0 =
0
U12_ga(
x1,
x2) =
U12_ga(
x2)
eq_in_ag(
x1,
x2) =
eq_in_ag(
x2)
eq_out_ag(
x1,
x2) =
eq_out_ag(
x1)
[] =
[]
s2l_out_ga(
x1,
x2) =
s2l_out_ga(
x2)
U13_ga(
x1,
x2,
x3,
x4) =
U13_ga(
x4)
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
U14_ga(
x1,
x2,
x3,
x4) =
U14_ga(
x4)
.(
x1,
x2) =
.(
x2)
U2_gaa(
x1,
x2,
x3,
x4) =
U2_gaa(
x4)
applast_in_gaa(
x1,
x2,
x3) =
applast_in_gaa(
x1)
U3_gaa(
x1,
x2,
x3,
x4) =
U3_gaa(
x4)
app_in_gga(
x1,
x2,
x3) =
app_in_gga(
x1,
x2)
U8_gga(
x1,
x2,
x3) =
U8_gga(
x3)
eq_in_ga(
x1,
x2) =
eq_in_ga(
x1)
eq_out_ga(
x1,
x2) =
eq_out_ga(
x2)
app_out_gga(
x1,
x2,
x3) =
app_out_gga(
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x5)
head_in_ga(
x1,
x2) =
head_in_ga(
x1)
head_out_ga(
x1,
x2) =
head_out_ga
U10_gga(
x1,
x2,
x3,
x4,
x5) =
U10_gga(
x2,
x5)
tail_in_ga(
x1,
x2) =
tail_in_ga(
x1)
tail_out_ga(
x1,
x2) =
tail_out_ga(
x2)
U11_gga(
x1,
x2,
x3,
x4,
x5) =
U11_gga(
x5)
U4_gaa(
x1,
x2,
x3,
x4) =
U4_gaa(
x4)
last_in_ag(
x1,
x2) =
last_in_ag(
x2)
U5_ag(
x1,
x2) =
U5_ag(
x2)
true_in_ =
true_in_
true_out_ =
true_out_
last_out_ag(
x1,
x2) =
last_out_ag
U6_ag(
x1,
x2,
x3) =
U6_ag(
x3)
U7_ag(
x1,
x2,
x3) =
U7_ag(
x3)
applast_out_gaa(
x1,
x2,
x3) =
applast_out_gaa
goal_out_gaa(
x1,
x2,
x3) =
goal_out_gaa
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x4)
We have to consider all (P,R,Pi)-chains
(127) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(128) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X, .(X1, Xs)) → U13_GA(X, X1, Xs, p_in_ga(X, P))
U13_GA(X, X1, Xs, p_out_ga(X, P)) → S2L_IN_GA(P, Xs)
The TRS R consists of the following rules:
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
The argument filtering Pi contains the following mapping:
0 =
0
p_in_ga(
x1,
x2) =
p_in_ga(
x1)
p_out_ga(
x1,
x2) =
p_out_ga(
x2)
s(
x1) =
s(
x1)
.(
x1,
x2) =
.(
x2)
S2L_IN_GA(
x1,
x2) =
S2L_IN_GA(
x1)
U13_GA(
x1,
x2,
x3,
x4) =
U13_GA(
x4)
We have to consider all (P,R,Pi)-chains
(129) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(130) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2L_IN_GA(P)
The TRS R consists of the following rules:
p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(131) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
No dependency pairs are removed.
The following rules are removed from R:
p_in_ga(s(X)) → p_out_ga(X)
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(0) = 0
POL(S2L_IN_GA(x1)) = x1
POL(U13_GA(x1)) = x1
POL(p_in_ga(x1)) = x1
POL(p_out_ga(x1)) = 2·x1
POL(s(x1)) = 2·x1
(132) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(X) → U13_GA(p_in_ga(X))
U13_GA(p_out_ga(P)) → S2L_IN_GA(P)
The TRS R consists of the following rules:
p_in_ga(0) → p_out_ga(0)
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(133) Narrowing (SOUND transformation)
By narrowing [LPAR04] the rule
S2L_IN_GA(
X) →
U13_GA(
p_in_ga(
X)) at position [0] we obtained the following new rules [LPAR04]:
S2L_IN_GA(0) → U13_GA(p_out_ga(0))
(134) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(p_out_ga(0))
The TRS R consists of the following rules:
p_in_ga(0) → p_out_ga(0)
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(135) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(136) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(p_out_ga(0))
R is empty.
The set Q consists of the following terms:
p_in_ga(x0)
We have to consider all (P,Q,R)-chains.
(137) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
p_in_ga(x0)
(138) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U13_GA(p_out_ga(P)) → S2L_IN_GA(P)
S2L_IN_GA(0) → U13_GA(p_out_ga(0))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(139) Instantiation (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
U13_GA(
p_out_ga(
P)) →
S2L_IN_GA(
P) we obtained the following new rules [LPAR04]:
U13_GA(p_out_ga(0)) → S2L_IN_GA(0)
(140) Obligation:
Q DP problem:
The TRS P consists of the following rules:
S2L_IN_GA(0) → U13_GA(p_out_ga(0))
U13_GA(p_out_ga(0)) → S2L_IN_GA(0)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(141) NonTerminationProof (EQUIVALENT transformation)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:
s =
U13_GA(
p_out_ga(
0)) evaluates to t =
U13_GA(
p_out_ga(
0))
Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
- Matcher: [ ]
- Semiunifier: [ ]
Rewriting sequenceU13_GA(p_out_ga(0)) →
S2L_IN_GA(
0)
with rule
U13_GA(
p_out_ga(
0)) →
S2L_IN_GA(
0) at position [] and matcher [ ]
S2L_IN_GA(0) →
U13_GA(
p_out_ga(
0))
with rule
S2L_IN_GA(
0) →
U13_GA(
p_out_ga(
0))
Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence
All these steps are and every following step will be a correct step w.r.t to Q.
(142) FALSE