WORST_CASE(Omega(n^1),?)
proof of input_XvRI5d1cPs.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 299 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 22 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 586 ms]
        (20) typed CpxTrs


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

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


The TRS R consists of the following rules:

   rev1(0, nil) -> 0
   rev1(s(X), nil) -> s(X)
   rev1(X, cons(Y, L)) -> rev1(Y, L)
   rev(nil) -> nil
   rev(cons(X, L)) -> cons(rev1(X, L), rev2(X, L))
   rev2(X, nil) -> nil
   rev2(X, cons(Y, L)) -> rev(cons(X, rev(rev2(Y, L))))

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

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))
Tuples:
   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
S tuples:
   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
K tuples:none
Defined Rule Symbols:   rev1_2, rev_1, rev2_2

Defined Pair Symbols:   REV1_2, REV_1, REV2_2

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5_1, c6, c7_3


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

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

(4)
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:

   REV1(0, nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0, nil) -> 0
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

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

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

(6)
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:

   REV1(0', nil) -> c
   REV1(s(z0), nil) -> c1
   REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
   REV(nil) -> c3
   REV(cons(z0, z1)) -> c4(REV1(z0, z1))
   REV(cons(z0, z1)) -> c5(REV2(z0, z1))
   REV2(z0, nil) -> c6
   REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))

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

   rev1(0', nil) -> 0'
   rev1(s(z0), nil) -> s(z0)
   rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
   rev(nil) -> nil
   rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
   rev2(z0, nil) -> nil
   rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
REV1, REV, REV2, rev, rev2, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2
rev1 < rev

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

(10)
Obligation:
Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c27_8(0) <=> c
gen_c:c1:c27_8(+(x, 1)) <=> c2(gen_c:c1:c27_8(x))
gen_nil:cons8_8(0) <=> nil
gen_nil:cons8_8(+(x, 1)) <=> cons(0', gen_nil:cons8_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c3, c3, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
REV1, REV, REV2, rev, rev2, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2
rev1 < rev

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REV1(0', gen_nil:cons8_8(n11_8)) -> gen_c:c1:c27_8(n11_8), rt in Omega(1 + n11_8)

Induction Base:
REV1(0', gen_nil:cons8_8(0)) ->_R^Omega(1)
c

Induction Step:
REV1(0', gen_nil:cons8_8(+(n11_8, 1))) ->_R^Omega(1)
c2(REV1(0', gen_nil:cons8_8(n11_8))) ->_IH
c2(gen_c:c1:c27_8(c12_8))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7


Generator Equations:
gen_c:c1:c27_8(0) <=> c
gen_c:c1:c27_8(+(x, 1)) <=> c2(gen_c:c1:c27_8(x))
gen_nil:cons8_8(0) <=> nil
gen_nil:cons8_8(+(x, 1)) <=> cons(0', gen_nil:cons8_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c3, c3, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
REV1, REV, REV2, rev, rev2, rev1

They will be analysed ascendingly in the following order:
REV1 < REV
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2
rev1 < rev

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7


Lemmas:
REV1(0', gen_nil:cons8_8(n11_8)) -> gen_c:c1:c27_8(n11_8), rt in Omega(1 + n11_8)


Generator Equations:
gen_c:c1:c27_8(0) <=> c
gen_c:c1:c27_8(+(x, 1)) <=> c2(gen_c:c1:c27_8(x))
gen_nil:cons8_8(0) <=> nil
gen_nil:cons8_8(+(x, 1)) <=> cons(0', gen_nil:cons8_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c3, c3, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
rev1, REV, REV2, rev, rev2

They will be analysed ascendingly in the following order:
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2
rev1 < rev

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev1(0', gen_nil:cons8_8(n383_8)) -> 0', rt in Omega(0)

Induction Base:
rev1(0', gen_nil:cons8_8(0)) ->_R^Omega(0)
0'

Induction Step:
rev1(0', gen_nil:cons8_8(+(n383_8, 1))) ->_R^Omega(0)
rev1(0', gen_nil:cons8_8(n383_8)) ->_IH
0'

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

(18)
Obligation:
Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7


Lemmas:
REV1(0', gen_nil:cons8_8(n11_8)) -> gen_c:c1:c27_8(n11_8), rt in Omega(1 + n11_8)
rev1(0', gen_nil:cons8_8(n383_8)) -> 0', rt in Omega(0)


Generator Equations:
gen_c:c1:c27_8(0) <=> c
gen_c:c1:c27_8(+(x, 1)) <=> c2(gen_c:c1:c27_8(x))
gen_nil:cons8_8(0) <=> nil
gen_nil:cons8_8(+(x, 1)) <=> cons(0', gen_nil:cons8_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c3, c3, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
rev2, REV, REV2, rev

They will be analysed ascendingly in the following order:
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev2(0', gen_nil:cons8_8(+(1, n630_8))) -> *10_8, rt in Omega(0)

Induction Base:
rev2(0', gen_nil:cons8_8(+(1, 0)))

Induction Step:
rev2(0', gen_nil:cons8_8(+(1, +(n630_8, 1)))) ->_R^Omega(0)
rev(cons(0', rev(rev2(0', gen_nil:cons8_8(+(1, n630_8)))))) ->_IH
rev(cons(0', rev(*10_8)))

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

(20)
Obligation:
Innermost TRS:
Rules:
REV1(0', nil) -> c
REV1(s(z0), nil) -> c1
REV1(z0, cons(z1, z2)) -> c2(REV1(z1, z2))
REV(nil) -> c3
REV(cons(z0, z1)) -> c4(REV1(z0, z1))
REV(cons(z0, z1)) -> c5(REV2(z0, z1))
REV2(z0, nil) -> c6
REV2(z0, cons(z1, z2)) -> c7(REV(cons(z0, rev(rev2(z1, z2)))), REV(rev2(z1, z2)), REV2(z1, z2))
rev1(0', nil) -> 0'
rev1(s(z0), nil) -> s(z0)
rev1(z0, cons(z1, z2)) -> rev1(z1, z2)
rev(nil) -> nil
rev(cons(z0, z1)) -> cons(rev1(z0, z1), rev2(z0, z1))
rev2(z0, nil) -> nil
rev2(z0, cons(z1, z2)) -> rev(cons(z0, rev(rev2(z1, z2))))

Types:
REV1 :: 0':s -> nil:cons -> c:c1:c2
0' :: 0':s
nil :: nil:cons
c :: c:c1:c2
s :: a -> 0':s
c1 :: c:c1:c2
cons :: 0':s -> nil:cons -> nil:cons
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4:c5
c3 :: c3:c4:c5
c4 :: c:c1:c2 -> c3:c4:c5
c5 :: c6:c7 -> c3:c4:c5
REV2 :: 0':s -> nil:cons -> c6:c7
c6 :: c6:c7
c7 :: c3:c4:c5 -> c3:c4:c5 -> c6:c7 -> c6:c7
rev :: nil:cons -> nil:cons
rev2 :: 0':s -> nil:cons -> nil:cons
rev1 :: 0':s -> nil:cons -> 0':s
hole_c:c1:c21_8 :: c:c1:c2
hole_0':s2_8 :: 0':s
hole_nil:cons3_8 :: nil:cons
hole_a4_8 :: a
hole_c3:c4:c55_8 :: c3:c4:c5
hole_c6:c76_8 :: c6:c7
gen_c:c1:c27_8 :: Nat -> c:c1:c2
gen_nil:cons8_8 :: Nat -> nil:cons
gen_c6:c79_8 :: Nat -> c6:c7


Lemmas:
REV1(0', gen_nil:cons8_8(n11_8)) -> gen_c:c1:c27_8(n11_8), rt in Omega(1 + n11_8)
rev1(0', gen_nil:cons8_8(n383_8)) -> 0', rt in Omega(0)
rev2(0', gen_nil:cons8_8(+(1, n630_8))) -> *10_8, rt in Omega(0)


Generator Equations:
gen_c:c1:c27_8(0) <=> c
gen_c:c1:c27_8(+(x, 1)) <=> c2(gen_c:c1:c27_8(x))
gen_nil:cons8_8(0) <=> nil
gen_nil:cons8_8(+(x, 1)) <=> cons(0', gen_nil:cons8_8(x))
gen_c6:c79_8(0) <=> c6
gen_c6:c79_8(+(x, 1)) <=> c7(c3, c3, gen_c6:c79_8(x))


The following defined symbols remain to be analysed:
rev, REV, REV2

They will be analysed ascendingly in the following order:
REV = REV2
rev < REV2
rev2 < REV2
rev = rev2