WORST_CASE(NON_POLY,?)
proof of input_U4sZjKVJXu.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 1 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 3567 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [FINISHED, 9458 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:

   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(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:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(z0, 0) -> c
   +'(0, z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0) -> c3
   *'(0, z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
S tuples:
   +'(z0, 0) -> c
   +'(0, z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0) -> c3
   *'(0, z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
K tuples:none
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6_2, c7, c8_2, c9, c10_2


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

(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:

   +'(z0, 0) -> c
   +'(0, z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0) -> c3
   *'(0, z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))

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

   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))

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:

   +'(z0, 0') -> c
   +'(0', z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0') -> c3
   *'(0', z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*'(z0, z1), +'(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*'(z0, z1), +'(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))

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

   +'(z0, 0') -> z0
   +'(0', z0) -> z0
   +'(s(z0), s(z1)) -> s(s(+'(z0, z1)))
   *'(z0, 0') -> 0'
   *'(0', z0) -> 0'
   *'(s(z0), s(z1)) -> s(+'(*'(z0, z1), +'(z0, z1)))
   sum(nil) -> 0'
   sum(cons(z0, z1)) -> +'(z0, sum(z1))
   prod(nil) -> s(0')
   prod(cons(z0, z1)) -> *'(z0, prod(z1))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(0', z0) -> c1
+'(s(z0), s(z1)) -> c2(+'(z0, z1))
*'(z0, 0') -> c3
*'(0', z0) -> c4
*'(s(z0), s(z1)) -> c5(+'(*'(z0, z1), +'(z0, z1)), *'(z0, z1))
*'(s(z0), s(z1)) -> c6(+'(*'(z0, z1), +'(z0, z1)), +'(z0, z1))
SUM(nil) -> c7
SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
PROD(nil) -> c9
PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
+'(z0, 0') -> z0
+'(0', z0) -> z0
+'(s(z0), s(z1)) -> s(s(+'(z0, z1)))
*'(z0, 0') -> 0'
*'(0', z0) -> 0'
*'(s(z0), s(z1)) -> s(+'(*'(z0, z1), +'(z0, z1)))
sum(nil) -> 0'
sum(cons(z0, z1)) -> +'(z0, sum(z1))
prod(nil) -> s(0')
prod(cons(z0, z1)) -> *'(z0, prod(z1))

Types:
+' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
0' :: 0':c:c1:s:c2:c3:c4:c5:c6
c :: 0':c:c1:s:c2:c3:c4:c5:c6
c1 :: 0':c:c1:s:c2:c3:c4:c5:c6
s :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c2 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
*' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c3 :: 0':c:c1:s:c2:c3:c4:c5:c6
c4 :: 0':c:c1:s:c2:c3:c4:c5:c6
c5 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c6 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
SUM :: nil:cons -> c7:c8
nil :: nil:cons
c7 :: c7:c8
cons :: 0':c:c1:s:c2:c3:c4:c5:c6 -> nil:cons -> nil:cons
c8 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c7:c8 -> c7:c8
sum :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
PROD :: nil:cons -> c9:c10
c9 :: c9:c10
c10 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c9:c10 -> c9:c10
prod :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
hole_0':c:c1:s:c2:c3:c4:c5:c61_11 :: 0':c:c1:s:c2:c3:c4:c5:c6
hole_c7:c82_11 :: c7:c8
hole_nil:cons3_11 :: nil:cons
hole_c9:c104_11 :: c9:c10
gen_0':c:c1:s:c2:c3:c4:c5:c65_11 :: Nat -> 0':c:c1:s:c2:c3:c4:c5:c6
gen_c7:c86_11 :: Nat -> c7:c8
gen_nil:cons7_11 :: Nat -> nil:cons
gen_c9:c108_11 :: Nat -> c9:c10

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', SUM, sum, PROD, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < SUM
+' < sum
*' < PROD
*' < prod
sum < SUM
prod < PROD

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

(10)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(0', z0) -> c1
+'(s(z0), s(z1)) -> c2(+'(z0, z1))
*'(z0, 0') -> c3
*'(0', z0) -> c4
*'(s(z0), s(z1)) -> c5(+'(*'(z0, z1), +'(z0, z1)), *'(z0, z1))
*'(s(z0), s(z1)) -> c6(+'(*'(z0, z1), +'(z0, z1)), +'(z0, z1))
SUM(nil) -> c7
SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
PROD(nil) -> c9
PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
+'(z0, 0') -> z0
+'(0', z0) -> z0
+'(s(z0), s(z1)) -> s(s(+'(z0, z1)))
*'(z0, 0') -> 0'
*'(0', z0) -> 0'
*'(s(z0), s(z1)) -> s(+'(*'(z0, z1), +'(z0, z1)))
sum(nil) -> 0'
sum(cons(z0, z1)) -> +'(z0, sum(z1))
prod(nil) -> s(0')
prod(cons(z0, z1)) -> *'(z0, prod(z1))

Types:
+' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
0' :: 0':c:c1:s:c2:c3:c4:c5:c6
c :: 0':c:c1:s:c2:c3:c4:c5:c6
c1 :: 0':c:c1:s:c2:c3:c4:c5:c6
s :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c2 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
*' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c3 :: 0':c:c1:s:c2:c3:c4:c5:c6
c4 :: 0':c:c1:s:c2:c3:c4:c5:c6
c5 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c6 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
SUM :: nil:cons -> c7:c8
nil :: nil:cons
c7 :: c7:c8
cons :: 0':c:c1:s:c2:c3:c4:c5:c6 -> nil:cons -> nil:cons
c8 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c7:c8 -> c7:c8
sum :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
PROD :: nil:cons -> c9:c10
c9 :: c9:c10
c10 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c9:c10 -> c9:c10
prod :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
hole_0':c:c1:s:c2:c3:c4:c5:c61_11 :: 0':c:c1:s:c2:c3:c4:c5:c6
hole_c7:c82_11 :: c7:c8
hole_nil:cons3_11 :: nil:cons
hole_c9:c104_11 :: c9:c10
gen_0':c:c1:s:c2:c3:c4:c5:c65_11 :: Nat -> 0':c:c1:s:c2:c3:c4:c5:c6
gen_c7:c86_11 :: Nat -> c7:c8
gen_nil:cons7_11 :: Nat -> nil:cons
gen_c9:c108_11 :: Nat -> c9:c10


Generator Equations:
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(0) <=> 0'
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(x, 1)) <=> s(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(x))
gen_c7:c86_11(0) <=> c7
gen_c7:c86_11(+(x, 1)) <=> c8(0', gen_c7:c86_11(x))
gen_nil:cons7_11(0) <=> nil
gen_nil:cons7_11(+(x, 1)) <=> cons(0', gen_nil:cons7_11(x))
gen_c9:c108_11(0) <=> c9
gen_c9:c108_11(+(x, 1)) <=> c10(0', gen_c9:c108_11(x))


The following defined symbols remain to be analysed:
+', *', SUM, sum, PROD, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < SUM
+' < sum
*' < PROD
*' < prod
sum < SUM
prod < PROD

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11))) -> *9_11, rt in Omega(n10_11)

Induction Base:
+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, 0)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, 0)))

Induction Step:
+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, +(n10_11, 1))), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, +(n10_11, 1)))) ->_R^Omega(1)
c2(+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11)))) ->_IH
c2(*9_11)

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:
+'(z0, 0') -> c
+'(0', z0) -> c1
+'(s(z0), s(z1)) -> c2(+'(z0, z1))
*'(z0, 0') -> c3
*'(0', z0) -> c4
*'(s(z0), s(z1)) -> c5(+'(*'(z0, z1), +'(z0, z1)), *'(z0, z1))
*'(s(z0), s(z1)) -> c6(+'(*'(z0, z1), +'(z0, z1)), +'(z0, z1))
SUM(nil) -> c7
SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
PROD(nil) -> c9
PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
+'(z0, 0') -> z0
+'(0', z0) -> z0
+'(s(z0), s(z1)) -> s(s(+'(z0, z1)))
*'(z0, 0') -> 0'
*'(0', z0) -> 0'
*'(s(z0), s(z1)) -> s(+'(*'(z0, z1), +'(z0, z1)))
sum(nil) -> 0'
sum(cons(z0, z1)) -> +'(z0, sum(z1))
prod(nil) -> s(0')
prod(cons(z0, z1)) -> *'(z0, prod(z1))

Types:
+' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
0' :: 0':c:c1:s:c2:c3:c4:c5:c6
c :: 0':c:c1:s:c2:c3:c4:c5:c6
c1 :: 0':c:c1:s:c2:c3:c4:c5:c6
s :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c2 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
*' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c3 :: 0':c:c1:s:c2:c3:c4:c5:c6
c4 :: 0':c:c1:s:c2:c3:c4:c5:c6
c5 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c6 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
SUM :: nil:cons -> c7:c8
nil :: nil:cons
c7 :: c7:c8
cons :: 0':c:c1:s:c2:c3:c4:c5:c6 -> nil:cons -> nil:cons
c8 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c7:c8 -> c7:c8
sum :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
PROD :: nil:cons -> c9:c10
c9 :: c9:c10
c10 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c9:c10 -> c9:c10
prod :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
hole_0':c:c1:s:c2:c3:c4:c5:c61_11 :: 0':c:c1:s:c2:c3:c4:c5:c6
hole_c7:c82_11 :: c7:c8
hole_nil:cons3_11 :: nil:cons
hole_c9:c104_11 :: c9:c10
gen_0':c:c1:s:c2:c3:c4:c5:c65_11 :: Nat -> 0':c:c1:s:c2:c3:c4:c5:c6
gen_c7:c86_11 :: Nat -> c7:c8
gen_nil:cons7_11 :: Nat -> nil:cons
gen_c9:c108_11 :: Nat -> c9:c10


Generator Equations:
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(0) <=> 0'
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(x, 1)) <=> s(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(x))
gen_c7:c86_11(0) <=> c7
gen_c7:c86_11(+(x, 1)) <=> c8(0', gen_c7:c86_11(x))
gen_nil:cons7_11(0) <=> nil
gen_nil:cons7_11(+(x, 1)) <=> cons(0', gen_nil:cons7_11(x))
gen_c9:c108_11(0) <=> c9
gen_c9:c108_11(+(x, 1)) <=> c10(0', gen_c9:c108_11(x))


The following defined symbols remain to be analysed:
+', *', SUM, sum, PROD, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < SUM
+' < sum
*' < PROD
*' < prod
sum < SUM
prod < PROD

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

(14) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(15)
BOUNDS(n^1, INF)

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(16)
Obligation:
Innermost TRS:
Rules:
+'(z0, 0') -> c
+'(0', z0) -> c1
+'(s(z0), s(z1)) -> c2(+'(z0, z1))
*'(z0, 0') -> c3
*'(0', z0) -> c4
*'(s(z0), s(z1)) -> c5(+'(*'(z0, z1), +'(z0, z1)), *'(z0, z1))
*'(s(z0), s(z1)) -> c6(+'(*'(z0, z1), +'(z0, z1)), +'(z0, z1))
SUM(nil) -> c7
SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
PROD(nil) -> c9
PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
+'(z0, 0') -> z0
+'(0', z0) -> z0
+'(s(z0), s(z1)) -> s(s(+'(z0, z1)))
*'(z0, 0') -> 0'
*'(0', z0) -> 0'
*'(s(z0), s(z1)) -> s(+'(*'(z0, z1), +'(z0, z1)))
sum(nil) -> 0'
sum(cons(z0, z1)) -> +'(z0, sum(z1))
prod(nil) -> s(0')
prod(cons(z0, z1)) -> *'(z0, prod(z1))

Types:
+' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
0' :: 0':c:c1:s:c2:c3:c4:c5:c6
c :: 0':c:c1:s:c2:c3:c4:c5:c6
c1 :: 0':c:c1:s:c2:c3:c4:c5:c6
s :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c2 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
*' :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c3 :: 0':c:c1:s:c2:c3:c4:c5:c6
c4 :: 0':c:c1:s:c2:c3:c4:c5:c6
c5 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
c6 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6 -> 0':c:c1:s:c2:c3:c4:c5:c6
SUM :: nil:cons -> c7:c8
nil :: nil:cons
c7 :: c7:c8
cons :: 0':c:c1:s:c2:c3:c4:c5:c6 -> nil:cons -> nil:cons
c8 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c7:c8 -> c7:c8
sum :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
PROD :: nil:cons -> c9:c10
c9 :: c9:c10
c10 :: 0':c:c1:s:c2:c3:c4:c5:c6 -> c9:c10 -> c9:c10
prod :: nil:cons -> 0':c:c1:s:c2:c3:c4:c5:c6
hole_0':c:c1:s:c2:c3:c4:c5:c61_11 :: 0':c:c1:s:c2:c3:c4:c5:c6
hole_c7:c82_11 :: c7:c8
hole_nil:cons3_11 :: nil:cons
hole_c9:c104_11 :: c9:c10
gen_0':c:c1:s:c2:c3:c4:c5:c65_11 :: Nat -> 0':c:c1:s:c2:c3:c4:c5:c6
gen_c7:c86_11 :: Nat -> c7:c8
gen_nil:cons7_11 :: Nat -> nil:cons
gen_c9:c108_11 :: Nat -> c9:c10


Lemmas:
+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n10_11))) -> *9_11, rt in Omega(n10_11)


Generator Equations:
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(0) <=> 0'
gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(x, 1)) <=> s(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(x))
gen_c7:c86_11(0) <=> c7
gen_c7:c86_11(+(x, 1)) <=> c8(0', gen_c7:c86_11(x))
gen_nil:cons7_11(0) <=> nil
gen_nil:cons7_11(+(x, 1)) <=> cons(0', gen_nil:cons7_11(x))
gen_c9:c108_11(0) <=> c9
gen_c9:c108_11(+(x, 1)) <=> c10(0', gen_c9:c108_11(x))


The following defined symbols remain to be analysed:
*', SUM, sum, PROD, prod

They will be analysed ascendingly in the following order:
*' < PROD
*' < prod
sum < SUM
prod < PROD

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(17) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
*'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11))) -> *9_11, rt in Omega(EXP)

Induction Base:
*'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, 0)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, 0)))

Induction Step:
*'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, +(n2980_11, 1))), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, +(n2980_11, 1)))) ->_R^Omega(1)
c5(+'(*'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11))), +'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)))), *'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)))) ->_IH
c5(+'(*9_11, +'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)))), *'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)))) ->_R^Omega(1)
c5(+'(*9_11, c2(+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(n2980_11), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(n2980_11)))), *'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(+(1, n2980_11)))) ->_IH
c5(+'(*9_11, c2(+'(gen_0':c:c1:s:c2:c3:c4:c5:c65_11(n2980_11), gen_0':c:c1:s:c2:c3:c4:c5:c65_11(n2980_11)))), *9_11)

We have rt in EXP and sz in O(n). Thus, we have irc_R in EXP
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(18)
BOUNDS(EXP, INF)