KILLED
proof of input_xUlnr3XVbu.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 12 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 159 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 34 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 1793 ms]
            (26) typed CpxTrs
(27) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(28) CpxWeightedTrs
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CpxTypedWeightedTrs
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (34) CpxTypedWeightedCompleteTrs
        (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (38) CpxRNTS
        (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 842 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 186 ms]
        (46) CpxRNTS
    (47) CompletionProof [UPPER BOUND(ID), 0 ms]
    (48) CpxTypedWeightedCompleteTrs
        (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
(51) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(52) CdtProblem
    (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxRelTRS
        (59) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxTRS
        (61) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CpxWeightedTrs
            (63) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (64) CpxTypedWeightedTrs
            (65) CompletionProof [UPPER BOUND(ID), 0 ms]
            (66) CpxTypedWeightedCompleteTrs
                (67) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (68) CpxTypedWeightedCompleteTrs
                (69) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (70) CpxRNTS
                (71) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (72) CpxRNTS
                (73) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (74) CpxRNTS
                (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (76) CpxRNTS
                (77) IntTrsBoundProof [UPPER BOUND(ID), 203 ms]
                (78) CpxRNTS
                (79) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
                (80) CpxRNTS
                (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (82) CpxRNTS
                (83) IntTrsBoundProof [UPPER BOUND(ID), 333 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
                (86) CpxRNTS
                (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (88) CpxRNTS
                (89) IntTrsBoundProof [UPPER BOUND(ID), 518 ms]
                (90) CpxRNTS
                (91) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
                (92) CpxRNTS
            (93) CompletionProof [UPPER BOUND(ID), 0 ms]
            (94) CpxTypedWeightedCompleteTrs
                (95) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (96) CpxRNTS
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
        (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (100) CdtProblem
        (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CdtProblem
        (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(L)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(L)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS
   app(cons(X, XS), YS) -> cons(X, app(XS, YS))
   from(X) -> cons(X, from(s(X)))
   zWadr(nil, YS) -> nil
   zWadr(XS, nil) -> nil
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS))
   prefix(L) -> cons(nil, zWadr(L, prefix(L)))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
Tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   app_2, from_1, zWadr_2, prefix_1

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_1, c6_1, c7_2


----------------------------------------

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))

The (relative) TRS S consists of the following rules:

   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Rewrite Strategy: INNERMOST
----------------------------------------

(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))

The (relative) TRS S consists of the following rules:

   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Rewrite Strategy: INNERMOST
----------------------------------------

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7

----------------------------------------

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)

Induction Base:
APP(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(b)) ->_R^Omega(1)
c

Induction Step:
APP(gen_nil:cons:s7_8(+(n12_8, 1)), gen_nil:cons:s7_8(b)) ->_R^Omega(1)
c1(APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b))) ->_IH
c1(gen_c:c16_8(c13_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Complex Obligation (BEST)

----------------------------------------

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
APP, FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
APP < ZWADR
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

----------------------------------------

(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(19)
BOUNDS(n^1, INF)

----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
FROM, ZWADR, PREFIX, prefix, app, from, zWadr

They will be analysed ascendingly in the following order:
ZWADR < PREFIX
prefix < PREFIX
zWadr < prefix
app < zWadr

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)

Induction Base:
ZWADR(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(0)) ->_R^Omega(1)
c3

Induction Step:
ZWADR(gen_nil:cons:s7_8(+(n632_8, 1)), gen_nil:cons:s7_8(+(n632_8, 1))) ->_R^Omega(1)
c6(ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8))) ->_IH
c6(gen_c3:c4:c5:c69_8(c633_8))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
app, PREFIX, prefix, from, zWadr

They will be analysed ascendingly in the following order:
prefix < PREFIX
zWadr < prefix
app < zWadr

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)

Induction Base:
app(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(b)) ->_R^Omega(0)
gen_nil:cons:s7_8(b)

Induction Step:
app(gen_nil:cons:s7_8(+(n1521_8, 1)), gen_nil:cons:s7_8(b)) ->_R^Omega(0)
cons(nil, app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b))) ->_IH
cons(nil, gen_nil:cons:s7_8(+(b, c1522_8)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
from, PREFIX, prefix, zWadr

They will be analysed ascendingly in the following order:
prefix < PREFIX
zWadr < prefix

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8)) -> *11_8, rt in Omega(0)

Induction Base:
zWadr(gen_nil:cons:s7_8(0), gen_nil:cons:s7_8(0))

Induction Step:
zWadr(gen_nil:cons:s7_8(+(n2878_8, 1)), gen_nil:cons:s7_8(+(n2878_8, 1))) ->_R^Omega(0)
cons(app(nil, cons(nil, nil)), zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8))) ->_L^Omega(0)
cons(gen_nil:cons:s7_8(+(0, +(0, 1))), zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8))) ->_IH
cons(gen_nil:cons:s7_8(1), *11_8)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
APP(nil, z0) -> c
APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
FROM(z0) -> c2(FROM(s(z0)))
ZWADR(nil, z0) -> c3
ZWADR(z0, nil) -> c4
ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
app(nil, z0) -> z0
app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
from(z0) -> cons(z0, from(s(z0)))
zWadr(nil, z0) -> nil
zWadr(z0, nil) -> nil
zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))

Types:
APP :: nil:cons:s -> nil:cons:s -> c:c1
nil :: nil:cons:s
c :: c:c1
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
c1 :: c:c1 -> c:c1
FROM :: nil:cons:s -> c2
c2 :: c2 -> c2
s :: nil:cons:s -> nil:cons:s
ZWADR :: nil:cons:s -> nil:cons:s -> c3:c4:c5:c6
c3 :: c3:c4:c5:c6
c4 :: c3:c4:c5:c6
c5 :: c:c1 -> c3:c4:c5:c6
c6 :: c3:c4:c5:c6 -> c3:c4:c5:c6
PREFIX :: nil:cons:s -> c7
c7 :: c3:c4:c5:c6 -> c7 -> c7
prefix :: nil:cons:s -> nil:cons:s
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
hole_c:c11_8 :: c:c1
hole_nil:cons:s2_8 :: nil:cons:s
hole_c23_8 :: c2
hole_c3:c4:c5:c64_8 :: c3:c4:c5:c6
hole_c75_8 :: c7
gen_c:c16_8 :: Nat -> c:c1
gen_nil:cons:s7_8 :: Nat -> nil:cons:s
gen_c28_8 :: Nat -> c2
gen_c3:c4:c5:c69_8 :: Nat -> c3:c4:c5:c6
gen_c710_8 :: Nat -> c7


Lemmas:
APP(gen_nil:cons:s7_8(n12_8), gen_nil:cons:s7_8(b)) -> gen_c:c16_8(n12_8), rt in Omega(1 + n12_8)
ZWADR(gen_nil:cons:s7_8(n632_8), gen_nil:cons:s7_8(n632_8)) -> gen_c3:c4:c5:c69_8(n632_8), rt in Omega(1 + n632_8)
app(gen_nil:cons:s7_8(n1521_8), gen_nil:cons:s7_8(b)) -> gen_nil:cons:s7_8(+(n1521_8, b)), rt in Omega(0)
zWadr(gen_nil:cons:s7_8(n2878_8), gen_nil:cons:s7_8(n2878_8)) -> *11_8, rt in Omega(0)


Generator Equations:
gen_c:c16_8(0) <=> c
gen_c:c16_8(+(x, 1)) <=> c1(gen_c:c16_8(x))
gen_nil:cons:s7_8(0) <=> nil
gen_nil:cons:s7_8(+(x, 1)) <=> cons(nil, gen_nil:cons:s7_8(x))
gen_c28_8(0) <=> hole_c23_8
gen_c28_8(+(x, 1)) <=> c2(gen_c28_8(x))
gen_c3:c4:c5:c69_8(0) <=> c3
gen_c3:c4:c5:c69_8(+(x, 1)) <=> c6(gen_c3:c4:c5:c69_8(x))
gen_c710_8(0) <=> hole_c75_8
gen_c710_8(+(x, 1)) <=> c7(c3, gen_c710_8(x))


The following defined symbols remain to be analysed:
prefix, PREFIX

They will be analysed ascendingly in the following order:
prefix < PREFIX

----------------------------------------

(27) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(29) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons:s

Rewrite Strategy: INNERMOST
----------------------------------------

(31) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   from_1

(c) The following functions are completely defined:

   prefix_1
   zWadr_2
   app_2

Due to the following rules being added:

   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

And the following fresh constants: none

----------------------------------------

(32)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons:s

Rewrite Strategy: INNERMOST
----------------------------------------

(33) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(34)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, cons(nil, zWadr(L, prefix(L))))) [2]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
app :: nil:cons:s -> nil:cons:s -> nil:cons:s
nil :: nil:cons:s
cons :: nil:cons:s -> nil:cons:s -> nil:cons:s
from :: nil:cons:s -> nil:cons:s
s :: nil:cons:s -> nil:cons:s
zWadr :: nil:cons:s -> nil:cons:s -> nil:cons:s
prefix :: nil:cons:s -> nil:cons:s

Rewrite Strategy: INNERMOST
----------------------------------------

(35) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> YS :|: z' = YS, YS >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + X + app(XS, YS) :|: z' = YS, z = 1 + X + XS, X >= 0, YS >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(L, 1 + 0 + zWadr(L, prefix(L))) :|: z = L, L >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' = YS, YS >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0


----------------------------------------

(37) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0


----------------------------------------

(39) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { from }
   { app }
   { zWadr }
   { prefix }

----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}

----------------------------------------

(41) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}
Previous analysis results are:
  from: runtime: ?, size: O(1) [0]

----------------------------------------

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 1 }-> 1 + X + app(XS, z') :|: z = 1 + X + XS, X >= 0, z' >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0
   prefix(z) -{ 2 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Function symbols to be analyzed: {from}, {app}, {zWadr}, {prefix}
Previous analysis results are:
  from: runtime: INF, size: O(1) [0]

----------------------------------------

(47) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   app(v0, v1) -> null_app [0]
   zWadr(v0, v1) -> null_zWadr [0]

And the following fresh constants:    null_app, null_zWadr

----------------------------------------

(48)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   app(nil, YS) -> YS [1]
   app(cons(X, XS), YS) -> cons(X, app(XS, YS)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   zWadr(nil, YS) -> nil [1]
   zWadr(XS, nil) -> nil [1]
   zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, nil)), zWadr(XS, YS)) [1]
   prefix(L) -> cons(nil, zWadr(L, prefix(L))) [1]
   app(v0, v1) -> null_app [0]
   zWadr(v0, v1) -> null_zWadr [0]

The TRS has the following type information:
app :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
nil :: nil:cons:s:null_app:null_zWadr
cons :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
from :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
s :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
zWadr :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
prefix :: nil:cons:s:null_app:null_zWadr -> nil:cons:s:null_app:null_zWadr
null_app :: nil:cons:s:null_app:null_zWadr
null_zWadr :: nil:cons:s:null_app:null_zWadr

Rewrite Strategy: INNERMOST
----------------------------------------

(49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   null_app => 0
   null_zWadr => 0

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z, z') -{ 1 }-> YS :|: z' = YS, YS >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 1 }-> 1 + X + app(XS, YS) :|: z' = YS, z = 1 + X + XS, X >= 0, YS >= 0, XS >= 0
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X
   prefix(z) -{ 1 }-> 1 + 0 + zWadr(L, prefix(L)) :|: z = L, L >= 0
   zWadr(z, z') -{ 1 }-> 0 :|: z' = YS, YS >= 0, z = 0
   zWadr(z, z') -{ 1 }-> 0 :|: z = XS, XS >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 1 }-> 1 + app(Y, 1 + X + 0) + zWadr(XS, YS) :|: z = 1 + X + XS, Y >= 0, X >= 0, YS >= 0, z' = 1 + Y + YS, XS >= 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(51) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
Tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(nil, z0) -> c
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(nil, z0) -> c3
   ZWADR(z0, nil) -> c4
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   app_2, from_1, zWadr_2, prefix_1

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c, c1_1, c2_1, c3, c4, c5_1, c6_1, c7_2


----------------------------------------

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   APP(nil, z0) -> c
   ZWADR(z0, nil) -> c4
   ZWADR(nil, z0) -> c3

----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
   from(z0) -> cons(z0, from(s(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   app_2, from_1, zWadr_2, prefix_1

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


----------------------------------------

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   from(z0) -> cons(z0, from(s(z0)))

----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


----------------------------------------

(57) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(58)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))

The (relative) TRS S consists of the following rules:

   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))

Rewrite Strategy: INNERMOST
----------------------------------------

(59) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(60)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0))
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(61) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(62)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) [1]
   FROM(z0) -> c2(FROM(s(z0))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) [1]
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0))) [0]
   zWadr(nil, z0) -> nil [0]
   zWadr(z0, nil) -> nil [0]
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3)) [0]
   app(nil, z0) -> z0 [0]
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) [0]

Rewrite Strategy: INNERMOST
----------------------------------------

(63) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(64)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) [1]
   FROM(z0) -> c2(FROM(s(z0))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) [1]
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0))) [0]
   zWadr(nil, z0) -> nil [0]
   zWadr(z0, nil) -> nil [0]
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3)) [0]
   app(nil, z0) -> z0 [0]
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) [0]

The TRS has the following type information:
APP :: cons:nil -> cons:nil -> c1
cons :: cons:nil -> cons:nil -> cons:nil
c1 :: c1 -> c1
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
ZWADR :: cons:nil -> cons:nil -> c5:c6
c5 :: c1 -> c5:c6
nil :: cons:nil
c6 :: c5:c6 -> c5:c6
PREFIX :: cons:nil -> c7
c7 :: c5:c6 -> c7 -> c7
prefix :: cons:nil -> cons:nil
zWadr :: cons:nil -> cons:nil -> cons:nil
app :: cons:nil -> cons:nil -> cons:nil

Rewrite Strategy: INNERMOST
----------------------------------------

(65) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   APP_2
   FROM_1
   ZWADR_2
   PREFIX_1

(c) The following functions are completely defined:

   prefix_1
   zWadr_2
   app_2

Due to the following rules being added:

   prefix(v0) -> nil [0]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

And the following fresh constants:    const, const1, const2, const3, const4

----------------------------------------

(66)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) [1]
   FROM(z0) -> c2(FROM(s(z0))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) [1]
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0))) [0]
   zWadr(nil, z0) -> nil [0]
   zWadr(z0, nil) -> nil [0]
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3)) [0]
   app(nil, z0) -> z0 [0]
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) [0]
   prefix(v0) -> nil [0]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
APP :: cons:nil -> cons:nil -> c1
cons :: cons:nil -> cons:nil -> cons:nil
c1 :: c1 -> c1
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
ZWADR :: cons:nil -> cons:nil -> c5:c6
c5 :: c1 -> c5:c6
nil :: cons:nil
c6 :: c5:c6 -> c5:c6
PREFIX :: cons:nil -> c7
c7 :: c5:c6 -> c7 -> c7
prefix :: cons:nil -> cons:nil
zWadr :: cons:nil -> cons:nil -> cons:nil
app :: cons:nil -> cons:nil -> cons:nil
const :: c1
const1 :: c2
const2 :: s
const3 :: c5:c6
const4 :: c7

Rewrite Strategy: INNERMOST
----------------------------------------

(67) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(68)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) [1]
   FROM(z0) -> c2(FROM(s(z0))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, nil), PREFIX(z0)) [1]
   prefix(z0) -> cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0))))) [0]
   prefix(z0) -> cons(nil, zWadr(z0, nil)) [0]
   zWadr(nil, z0) -> nil [0]
   zWadr(z0, nil) -> nil [0]
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3)) [0]
   app(nil, z0) -> z0 [0]
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) [0]
   prefix(v0) -> nil [0]
   zWadr(v0, v1) -> nil [0]
   app(v0, v1) -> nil [0]

The TRS has the following type information:
APP :: cons:nil -> cons:nil -> c1
cons :: cons:nil -> cons:nil -> cons:nil
c1 :: c1 -> c1
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
ZWADR :: cons:nil -> cons:nil -> c5:c6
c5 :: c1 -> c5:c6
nil :: cons:nil
c6 :: c5:c6 -> c5:c6
PREFIX :: cons:nil -> c7
c7 :: c5:c6 -> c7 -> c7
prefix :: cons:nil -> cons:nil
zWadr :: cons:nil -> cons:nil -> cons:nil
app :: cons:nil -> cons:nil -> cons:nil
const :: c1
const1 :: c2
const2 :: s
const3 :: c5:c6
const4 :: c7

Rewrite Strategy: INNERMOST
----------------------------------------

(69) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const4 => 0

----------------------------------------

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z0) :|: z = z0, z0 >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z0, 0) + PREFIX(z0) :|: z = z0, z0 >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z0, 1 + 0 + zWadr(z0, prefix(z0))) + PREFIX(z0) :|: z = z0, z0 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   prefix(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z0, 0) :|: z = z0, z0 >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z0, 1 + 0 + zWadr(z0, prefix(z0))) :|: z = z0, z0 >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   zWadr(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0


----------------------------------------

(71) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0


----------------------------------------

(73) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { APP }
   { app }
   { FROM }
   { ZWADR }
   { zWadr }
   { prefix }
   { PREFIX }

----------------------------------------

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {APP}, {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}

----------------------------------------

(75) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {APP}, {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}

----------------------------------------

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: APP
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {APP}, {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: ?, size: O(1) [0]

----------------------------------------

(79) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: APP
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]

----------------------------------------

(81) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]

----------------------------------------

(83) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {app}, {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]
  app: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(85) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]
  app: runtime: O(1) [0], size: O(n^1) [z + z']

----------------------------------------

(87) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s1 :|: s1 >= 0, s1 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + s'' + zWadr(z1, z3) :|: s'' >= 0, s'' <= z2 + (1 + z0 + 0), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]
  app: runtime: O(1) [0], size: O(n^1) [z + z']

----------------------------------------

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: FROM
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s1 :|: s1 >= 0, s1 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + s'' + zWadr(z1, z3) :|: s'' >= 0, s'' <= z2 + (1 + z0 + 0), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  FROM: runtime: ?, size: O(1) [0]

----------------------------------------

(91) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: FROM
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 0) + PREFIX(z) :|: z >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z, 1 + 0 + zWadr(z, prefix(z))) + PREFIX(z) :|: z >= 0
   ZWADR(z, z') -{ 1 + z2 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s1 :|: s1 >= 0, s1 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   prefix(z) -{ 0 }-> 0 :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 0) :|: z >= 0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z, 1 + 0 + zWadr(z, prefix(z))) :|: z >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   zWadr(z, z') -{ 0 }-> 1 + s'' + zWadr(z1, z3) :|: s'' >= 0, s'' <= z2 + (1 + z0 + 0), z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {FROM}, {ZWADR}, {zWadr}, {prefix}, {PREFIX}
Previous analysis results are:
  APP: runtime: O(n^1) [z], size: O(1) [0]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  FROM: runtime: INF, size: O(1) [0]

----------------------------------------

(93) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   prefix(v0) -> null_prefix [0]
   zWadr(v0, v1) -> null_zWadr [0]
   app(v0, v1) -> null_app [0]
   APP(v0, v1) -> null_APP [0]
   ZWADR(v0, v1) -> null_ZWADR [0]

And the following fresh constants:    null_prefix, null_zWadr, null_app, null_APP, null_ZWADR, const, const1, const2

----------------------------------------

(94)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) [1]
   FROM(z0) -> c2(FROM(s(z0))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) [1]
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) [1]
   PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) [1]
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0))) [0]
   zWadr(nil, z0) -> nil [0]
   zWadr(z0, nil) -> nil [0]
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3)) [0]
   app(nil, z0) -> z0 [0]
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2)) [0]
   prefix(v0) -> null_prefix [0]
   zWadr(v0, v1) -> null_zWadr [0]
   app(v0, v1) -> null_app [0]
   APP(v0, v1) -> null_APP [0]
   ZWADR(v0, v1) -> null_ZWADR [0]

The TRS has the following type information:
APP :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app -> c1:null_APP
cons :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app
c1 :: c1:null_APP -> c1:null_APP
FROM :: s -> c2
c2 :: c2 -> c2
s :: s -> s
ZWADR :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app -> c5:c6:null_ZWADR
c5 :: c1:null_APP -> c5:c6:null_ZWADR
nil :: cons:nil:null_prefix:null_zWadr:null_app
c6 :: c5:c6:null_ZWADR -> c5:c6:null_ZWADR
PREFIX :: cons:nil:null_prefix:null_zWadr:null_app -> c7
c7 :: c5:c6:null_ZWADR -> c7 -> c7
prefix :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app
zWadr :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app
app :: cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app -> cons:nil:null_prefix:null_zWadr:null_app
null_prefix :: cons:nil:null_prefix:null_zWadr:null_app
null_zWadr :: cons:nil:null_prefix:null_zWadr:null_app
null_app :: cons:nil:null_prefix:null_zWadr:null_app
null_APP :: c1:null_APP
null_ZWADR :: c5:c6:null_ZWADR
const :: c2
const1 :: s
const2 :: c7

Rewrite Strategy: INNERMOST
----------------------------------------

(95) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   nil => 0
   null_prefix => 0
   null_zWadr => 0
   null_app => 0
   null_APP => 0
   null_ZWADR => 0
   const => 0
   const1 => 0
   const2 => 0

----------------------------------------

(96)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   APP(z, z') -{ 1 }-> 1 + APP(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   FROM(z) -{ 1 }-> 1 + FROM(1 + z0) :|: z = z0, z0 >= 0
   PREFIX(z) -{ 1 }-> 1 + ZWADR(z0, prefix(z0)) + PREFIX(z0) :|: z = z0, z0 >= 0
   ZWADR(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   ZWADR(z, z') -{ 1 }-> 1 + ZWADR(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   ZWADR(z, z') -{ 1 }-> 1 + APP(z2, 1 + z0 + 0) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   app(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   prefix(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   prefix(z) -{ 0 }-> 1 + 0 + zWadr(z0, prefix(z0)) :|: z = z0, z0 >= 0
   zWadr(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   zWadr(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   zWadr(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   zWadr(z, z') -{ 0 }-> 1 + app(z2, 1 + z0 + 0) + zWadr(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PREFIX(z0) -> c7(ZWADR(z0, prefix(z0)), PREFIX(z0)) by 
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


----------------------------------------

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, prefix(z0)))), PREFIX(z0)) by 
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(nil) -> c7(ZWADR(nil, cons(nil, nil)), PREFIX(nil))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2


----------------------------------------

(101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   FROM(z0) -> c2(FROM(s(z0)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, FROM_1, ZWADR_2, PREFIX_1

Compound Symbols:   c1_1, c2_1, c5_1, c6_1, c7_2, c7_1


----------------------------------------

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by 
   FROM(s(x0)) -> c2(FROM(s(s(x0))))

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, ZWADR_2, PREFIX_1, FROM_1

Compound Symbols:   c1_1, c5_1, c6_1, c7_2, c7_1, c2_1


----------------------------------------

(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))

----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
S tuples:
   APP(cons(z0, z1), z2) -> c1(APP(z1, z2))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   APP_2, ZWADR_2, PREFIX_1, FROM_1

Compound Symbols:   c1_1, c5_1, c6_1, c7_2, c7_1, c2_1


----------------------------------------

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(cons(z0, z1), z2) -> c1(APP(z1, z2)) by 
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))

----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
S tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil)))
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   ZWADR_2, PREFIX_1, FROM_1, APP_2

Compound Symbols:   c5_1, c6_1, c7_2, c7_1, c2_1, c1_1


----------------------------------------

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(z2, z3)) -> c5(APP(z2, cons(z0, nil))) by 
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))

----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
S tuples:
   ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3))
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   ZWADR_2, PREFIX_1, FROM_1, APP_2

Compound Symbols:   c6_1, c7_2, c7_1, c2_1, c1_1, c5_1


----------------------------------------

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(z2, z3)) -> c6(ZWADR(z1, z3)) by 
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))

----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, APP_2, ZWADR_2

Compound Symbols:   c7_2, c7_1, c2_1, c1_1, c5_1, c6_1


----------------------------------------

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace APP(cons(z0, cons(y0, y1)), z2) -> c1(APP(cons(y0, y1), z2)) by 
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))

----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c5_1, c6_1, c1_1


----------------------------------------

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, z1), cons(cons(y0, cons(y1, y2)), z3)) -> c5(APP(cons(y0, cons(y1, y2)), cons(z0, nil))) by 
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))

----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3)))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c6_1, c1_1, c5_1


----------------------------------------

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(y2, y3))) -> c6(ZWADR(cons(y0, y1), cons(y2, y3))) by 
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))

----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5)))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, ZWADR_2, APP_2

Compound Symbols:   c7_2, c7_1, c2_1, c6_1, c1_1, c5_1


----------------------------------------

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ZWADR(cons(z0, cons(y0, y1)), cons(z2, cons(cons(y2, cons(y3, y4)), y5))) -> c6(ZWADR(cons(y0, y1), cons(cons(y2, cons(y3, y4)), y5))) by 
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(z4, cons(z5, cons(y4, y5))), z7))) -> c6(ZWADR(cons(z1, z2), cons(cons(z4, cons(z5, cons(y4, y5))), z7)))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))

----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   prefix(z0) -> cons(nil, zWadr(z0, prefix(z0)))
   zWadr(nil, z0) -> nil
   zWadr(z0, nil) -> nil
   zWadr(cons(z0, z1), cons(z2, z3)) -> cons(app(z2, cons(z0, nil)), zWadr(z1, z3))
   app(nil, z0) -> z0
   app(cons(z0, z1), z2) -> cons(z0, app(z1, z2))
Tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))
S tuples:
   PREFIX(z0) -> c7(ZWADR(z0, cons(nil, zWadr(z0, cons(nil, zWadr(z0, prefix(z0)))))), PREFIX(z0))
   PREFIX(nil) -> c7(PREFIX(nil))
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
   APP(cons(z0, cons(z1, cons(y1, y2))), z3) -> c1(APP(cons(z1, cons(y1, y2)), z3))
   ZWADR(cons(z0, z1), cons(cons(z2, cons(z3, cons(y2, y3))), z5)) -> c5(APP(cons(z2, cons(z3, cons(y2, y3))), cons(z0, nil)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(y4, y5)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(y4, y5))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(z4, cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(z4, cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, z2)), cons(z3, cons(cons(y2, cons(y3, cons(y4, y5))), z5))) -> c6(ZWADR(cons(z1, z2), cons(cons(y2, cons(y3, cons(y4, y5))), z5)))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, y6)), y7))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(y6, y7)))))
   ZWADR(cons(z0, cons(z1, cons(y1, cons(y2, y3)))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9))))) -> c6(ZWADR(cons(z1, cons(y1, cons(y2, y3))), cons(cons(z4, cons(z5, z6)), cons(y5, cons(cons(y6, cons(y7, y8)), y9)))))
   ZWADR(cons(z0, cons(z1, cons(y1, y2))), cons(z3, cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8)))) -> c6(ZWADR(cons(z1, cons(y1, y2)), cons(cons(z4, cons(z5, z6)), cons(cons(y4, cons(y5, cons(y6, y7))), y8))))
K tuples:none
Defined Rule Symbols:   prefix_1, zWadr_2, app_2

Defined Pair Symbols:   PREFIX_1, FROM_1, APP_2, ZWADR_2

Compound Symbols:   c7_2, c7_1, c2_1, c1_1, c5_1, c6_1