WORST_CASE(NON_POLY,?)
proof of input_Lrk7uKblpj.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(EXP, 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), 10 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 2134 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [FINISHED, 348 ms]
        (18) BOUNDS(EXP, INF)


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

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


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(x, *(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0)))))

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:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))))
Tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(0) -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(0) -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, twice_1, -_2, f_1

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_1, c6, c7_1, c8_3, c9_4


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

(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(EXP, INF).


The TRS R consists of the following rules:

   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(0) -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))

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

   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))))

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(EXP, INF).


The TRS R consists of the following rules:

   +'(0', z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0') -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *'(z0, z1)), *'(z0, z1))
   TWICE(0') -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0') -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), +'(*'(s(z0), s(s(z0))), s(s(0'))), *'(s(z0), s(s(z0))))

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

   +'(0', z0) -> z0
   +'(s(z0), z1) -> s(+'(z0, z1))
   *'(z0, 0') -> 0'
   *'(z0, s(z1)) -> +'(z0, *'(z0, z1))
   twice(0') -> 0'
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0') -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   f(s(z0)) -> f(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
+'(0', z0) -> c
+'(s(z0), z1) -> c1(+'(z0, z1))
*'(z0, 0') -> c2
*'(z0, s(z1)) -> c3(+'(z0, *'(z0, z1)), *'(z0, z1))
TWICE(0') -> c4
TWICE(s(z0)) -> c5(TWICE(z0))
-'(z0, 0') -> c6
-'(s(z0), s(z1)) -> c7(-'(z0, z1))
F(s(z0)) -> c8(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), *'(s(s(z0)), s(s(z0))))
F(s(z0)) -> c9(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), +'(*'(s(z0), s(s(z0))), s(s(0'))), *'(s(z0), s(s(z0))))
+'(0', z0) -> z0
+'(s(z0), z1) -> s(+'(z0, z1))
*'(z0, 0') -> 0'
*'(z0, s(z1)) -> +'(z0, *'(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
f(s(z0)) -> f(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))))

Types:
+' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
0' :: 0':c:s:c1:c2:c3
c :: 0':c:s:c1:c2:c3
s :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c1 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
*' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c2 :: 0':c:s:c1:c2:c3
c3 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
TWICE :: 0':c:s:c1:c2:c3 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
-' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
F :: 0':c:s:c1:c2:c3 -> c8:c9
c8 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> c8:c9
- :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c9 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c8:c9
twice :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
f :: 0':c:s:c1:c2:c3 -> f
hole_0':c:s:c1:c2:c31_10 :: 0':c:s:c1:c2:c3
hole_c4:c52_10 :: c4:c5
hole_c6:c73_10 :: c6:c7
hole_c8:c94_10 :: c8:c9
hole_f5_10 :: f
gen_0':c:s:c1:c2:c36_10 :: Nat -> 0':c:s:c1:c2:c3
gen_c4:c57_10 :: Nat -> c4:c5
gen_c6:c78_10 :: Nat -> c6:c7
gen_c8:c99_10 :: Nat -> c8:c9

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', TWICE, -', F, -, twice, f

They will be analysed ascendingly in the following order:
+' < *'
+' < F
+' < f
*' < F
*' < f
-' < F
- < F
- < f

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

(10)
Obligation:
Innermost TRS:
Rules:
+'(0', z0) -> c
+'(s(z0), z1) -> c1(+'(z0, z1))
*'(z0, 0') -> c2
*'(z0, s(z1)) -> c3(+'(z0, *'(z0, z1)), *'(z0, z1))
TWICE(0') -> c4
TWICE(s(z0)) -> c5(TWICE(z0))
-'(z0, 0') -> c6
-'(s(z0), s(z1)) -> c7(-'(z0, z1))
F(s(z0)) -> c8(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), *'(s(s(z0)), s(s(z0))))
F(s(z0)) -> c9(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), +'(*'(s(z0), s(s(z0))), s(s(0'))), *'(s(z0), s(s(z0))))
+'(0', z0) -> z0
+'(s(z0), z1) -> s(+'(z0, z1))
*'(z0, 0') -> 0'
*'(z0, s(z1)) -> +'(z0, *'(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
f(s(z0)) -> f(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))))

Types:
+' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
0' :: 0':c:s:c1:c2:c3
c :: 0':c:s:c1:c2:c3
s :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c1 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
*' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c2 :: 0':c:s:c1:c2:c3
c3 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
TWICE :: 0':c:s:c1:c2:c3 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
-' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
F :: 0':c:s:c1:c2:c3 -> c8:c9
c8 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> c8:c9
- :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c9 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c8:c9
twice :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
f :: 0':c:s:c1:c2:c3 -> f
hole_0':c:s:c1:c2:c31_10 :: 0':c:s:c1:c2:c3
hole_c4:c52_10 :: c4:c5
hole_c6:c73_10 :: c6:c7
hole_c8:c94_10 :: c8:c9
hole_f5_10 :: f
gen_0':c:s:c1:c2:c36_10 :: Nat -> 0':c:s:c1:c2:c3
gen_c4:c57_10 :: Nat -> c4:c5
gen_c6:c78_10 :: Nat -> c6:c7
gen_c8:c99_10 :: Nat -> c8:c9


Generator Equations:
gen_0':c:s:c1:c2:c36_10(0) <=> 0'
gen_0':c:s:c1:c2:c36_10(+(x, 1)) <=> s(gen_0':c:s:c1:c2:c36_10(x))
gen_c4:c57_10(0) <=> c4
gen_c4:c57_10(+(x, 1)) <=> c5(gen_c4:c57_10(x))
gen_c6:c78_10(0) <=> c6
gen_c6:c78_10(+(x, 1)) <=> c7(gen_c6:c78_10(x))
gen_c8:c99_10(0) <=> hole_c8:c94_10
gen_c8:c99_10(+(x, 1)) <=> c8(gen_c8:c99_10(x), c6, 0')


The following defined symbols remain to be analysed:
+', *', TWICE, -', F, -, twice, f

They will be analysed ascendingly in the following order:
+' < *'
+' < F
+' < f
*' < F
*' < f
-' < F
- < F
- < f

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':c:s:c1:c2:c36_10(+(1, n11_10)), gen_0':c:s:c1:c2:c36_10(b)) -> *10_10, rt in Omega(n11_10)

Induction Base:
+'(gen_0':c:s:c1:c2:c36_10(+(1, 0)), gen_0':c:s:c1:c2:c36_10(b))

Induction Step:
+'(gen_0':c:s:c1:c2:c36_10(+(1, +(n11_10, 1))), gen_0':c:s:c1:c2:c36_10(b)) ->_R^Omega(1)
c1(+'(gen_0':c:s:c1:c2:c36_10(+(1, n11_10)), gen_0':c:s:c1:c2:c36_10(b))) ->_IH
c1(*10_10)

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:
+'(0', z0) -> c
+'(s(z0), z1) -> c1(+'(z0, z1))
*'(z0, 0') -> c2
*'(z0, s(z1)) -> c3(+'(z0, *'(z0, z1)), *'(z0, z1))
TWICE(0') -> c4
TWICE(s(z0)) -> c5(TWICE(z0))
-'(z0, 0') -> c6
-'(s(z0), s(z1)) -> c7(-'(z0, z1))
F(s(z0)) -> c8(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), *'(s(s(z0)), s(s(z0))))
F(s(z0)) -> c9(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), +'(*'(s(z0), s(s(z0))), s(s(0'))), *'(s(z0), s(s(z0))))
+'(0', z0) -> z0
+'(s(z0), z1) -> s(+'(z0, z1))
*'(z0, 0') -> 0'
*'(z0, s(z1)) -> +'(z0, *'(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
f(s(z0)) -> f(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))))

Types:
+' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
0' :: 0':c:s:c1:c2:c3
c :: 0':c:s:c1:c2:c3
s :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c1 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
*' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c2 :: 0':c:s:c1:c2:c3
c3 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
TWICE :: 0':c:s:c1:c2:c3 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
-' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
F :: 0':c:s:c1:c2:c3 -> c8:c9
c8 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> c8:c9
- :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c9 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c8:c9
twice :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
f :: 0':c:s:c1:c2:c3 -> f
hole_0':c:s:c1:c2:c31_10 :: 0':c:s:c1:c2:c3
hole_c4:c52_10 :: c4:c5
hole_c6:c73_10 :: c6:c7
hole_c8:c94_10 :: c8:c9
hole_f5_10 :: f
gen_0':c:s:c1:c2:c36_10 :: Nat -> 0':c:s:c1:c2:c3
gen_c4:c57_10 :: Nat -> c4:c5
gen_c6:c78_10 :: Nat -> c6:c7
gen_c8:c99_10 :: Nat -> c8:c9


Generator Equations:
gen_0':c:s:c1:c2:c36_10(0) <=> 0'
gen_0':c:s:c1:c2:c36_10(+(x, 1)) <=> s(gen_0':c:s:c1:c2:c36_10(x))
gen_c4:c57_10(0) <=> c4
gen_c4:c57_10(+(x, 1)) <=> c5(gen_c4:c57_10(x))
gen_c6:c78_10(0) <=> c6
gen_c6:c78_10(+(x, 1)) <=> c7(gen_c6:c78_10(x))
gen_c8:c99_10(0) <=> hole_c8:c94_10
gen_c8:c99_10(+(x, 1)) <=> c8(gen_c8:c99_10(x), c6, 0')


The following defined symbols remain to be analysed:
+', *', TWICE, -', F, -, twice, f

They will be analysed ascendingly in the following order:
+' < *'
+' < F
+' < f
*' < F
*' < f
-' < F
- < F
- < f

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
+'(0', z0) -> c
+'(s(z0), z1) -> c1(+'(z0, z1))
*'(z0, 0') -> c2
*'(z0, s(z1)) -> c3(+'(z0, *'(z0, z1)), *'(z0, z1))
TWICE(0') -> c4
TWICE(s(z0)) -> c5(TWICE(z0))
-'(z0, 0') -> c6
-'(s(z0), s(z1)) -> c7(-'(z0, z1))
F(s(z0)) -> c8(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), *'(s(s(z0)), s(s(z0))))
F(s(z0)) -> c9(F(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0'))))), -'(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))), +'(*'(s(z0), s(s(z0))), s(s(0'))), *'(s(z0), s(s(z0))))
+'(0', z0) -> z0
+'(s(z0), z1) -> s(+'(z0, z1))
*'(z0, 0') -> 0'
*'(z0, s(z1)) -> +'(z0, *'(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
f(s(z0)) -> f(-(*'(s(s(z0)), s(s(z0))), +'(*'(s(z0), s(s(z0))), s(s(0')))))

Types:
+' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
0' :: 0':c:s:c1:c2:c3
c :: 0':c:s:c1:c2:c3
s :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c1 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
*' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c2 :: 0':c:s:c1:c2:c3
c3 :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
TWICE :: 0':c:s:c1:c2:c3 -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
-' :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c6:c7
c6 :: c6:c7
c7 :: c6:c7 -> c6:c7
F :: 0':c:s:c1:c2:c3 -> c8:c9
c8 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> c8:c9
- :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
c9 :: c8:c9 -> c6:c7 -> 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3 -> c8:c9
twice :: 0':c:s:c1:c2:c3 -> 0':c:s:c1:c2:c3
f :: 0':c:s:c1:c2:c3 -> f
hole_0':c:s:c1:c2:c31_10 :: 0':c:s:c1:c2:c3
hole_c4:c52_10 :: c4:c5
hole_c6:c73_10 :: c6:c7
hole_c8:c94_10 :: c8:c9
hole_f5_10 :: f
gen_0':c:s:c1:c2:c36_10 :: Nat -> 0':c:s:c1:c2:c3
gen_c4:c57_10 :: Nat -> c4:c5
gen_c6:c78_10 :: Nat -> c6:c7
gen_c8:c99_10 :: Nat -> c8:c9


Lemmas:
+'(gen_0':c:s:c1:c2:c36_10(+(1, n11_10)), gen_0':c:s:c1:c2:c36_10(b)) -> *10_10, rt in Omega(n11_10)


Generator Equations:
gen_0':c:s:c1:c2:c36_10(0) <=> 0'
gen_0':c:s:c1:c2:c36_10(+(x, 1)) <=> s(gen_0':c:s:c1:c2:c36_10(x))
gen_c4:c57_10(0) <=> c4
gen_c4:c57_10(+(x, 1)) <=> c5(gen_c4:c57_10(x))
gen_c6:c78_10(0) <=> c6
gen_c6:c78_10(+(x, 1)) <=> c7(gen_c6:c78_10(x))
gen_c8:c99_10(0) <=> hole_c8:c94_10
gen_c8:c99_10(+(x, 1)) <=> c8(gen_c8:c99_10(x), c6, 0')


The following defined symbols remain to be analysed:
*', TWICE, -', F, -, twice, f

They will be analysed ascendingly in the following order:
*' < F
*' < f
-' < F
- < F
- < f

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

(17) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
*'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, n2103_10))) -> *10_10, rt in Omega(EXP)

Induction Base:
*'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, 0)))

Induction Step:
*'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, +(n2103_10, 1)))) ->_R^Omega(1)
c3(+'(gen_0':c:s:c1:c2:c36_10(a), *'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, n2103_10)))), *'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, n2103_10)))) ->_IH
c3(+'(gen_0':c:s:c1:c2:c36_10(a), *10_10), *'(gen_0':c:s:c1:c2:c36_10(a), gen_0':c:s:c1:c2:c36_10(+(1, n2103_10)))) ->_IH
c3(+'(gen_0':c:s:c1:c2:c36_10(a), *10_10), *10_10)

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

(18)
BOUNDS(EXP, INF)