WORST_CASE(NON_POLY,?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 317 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 4 ms]
(4) CpxRelTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 1906 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [FINISHED, 2589 ms]
        (16) BOUNDS(EXP, INF)


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

(0)
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:

   F(0) -> c
   F(s(0)) -> c1
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(0) -> c5
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   H(z0) -> c12(P(z0))
   P(pair(z0, z1)) -> c13
   Q(pair(z0, z1)) -> c14
   +'(z0, 0) -> c15
   +'(z0, s(z1)) -> c16(+'(z0, z1))

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

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
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:

   F(0) -> c
   F(s(0)) -> c1
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(0) -> c5
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   H(z0) -> c12(P(z0))
   P(pair(z0, z1)) -> c13
   Q(pair(z0, z1)) -> c14
   +'(z0, 0) -> c15
   +'(z0, s(z1)) -> c16(+'(z0, z1))

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

   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

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

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

(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:

   F(0') -> c
   F(s(0')) -> c1
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(0') -> c5
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   H(z0) -> c12(P(z0))
   P(pair(z0, z1)) -> c13
   Q(pair(z0, z1)) -> c14
   +'(z0, 0') -> c15
   +'(z0, s(z1)) -> c16(+'(z0, z1))

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

   f(0') -> 0'
   f(s(0')) -> s(0')
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +'(p(g(z0)), q(g(z0)))
   g(0') -> pair(s(0'), s(0'))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+'(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+'(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +'(z0, 0') -> z0
   +'(z0, s(z1)) -> s(+'(z0, z1))

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

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

(6)
Obligation:
Innermost TRS:
Rules:
F(0') -> c
F(s(0')) -> c1
F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(0') -> c5
G(s(z0)) -> c6(H(g(z0)), G(z0))
G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(s(z0)) -> c9(P(g(z0)), G(z0))
H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
H(z0) -> c12(P(z0))
P(pair(z0, z1)) -> c13
Q(pair(z0, z1)) -> c14
+'(z0, 0') -> c15
+'(z0, s(z1)) -> c16(+'(z0, z1))
f(0') -> 0'
f(s(0')) -> s(0')
f(s(s(z0))) -> p(h(g(z0)))
f(s(s(z0))) -> +'(p(g(z0)), q(g(z0)))
g(0') -> pair(s(0'), s(0'))
g(s(z0)) -> h(g(z0))
g(s(z0)) -> pair(+'(p(g(z0)), q(g(z0))), p(g(z0)))
h(z0) -> pair(+'(p(z0), q(z0)), p(z0))
p(pair(z0, z1)) -> z0
q(pair(z0, z1)) -> z1
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
F :: 0':s:c15:c16 -> c:c1:c2:c3:c4
0' :: 0':s:c15:c16
c :: c:c1:c2:c3:c4
s :: 0':s:c15:c16 -> 0':s:c15:c16
c1 :: c:c1:c2:c3:c4
c2 :: c13 -> c10:c11:c12 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
P :: pair -> c13
h :: pair -> pair
g :: 0':s:c15:c16 -> pair
H :: pair -> c10:c11:c12
G :: 0':s:c15:c16 -> c5:c6:c7:c8:c9
c3 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
+' :: 0':s:c15:c16 -> 0':s:c15:c16 -> 0':s:c15:c16
p :: pair -> 0':s:c15:c16
q :: pair -> 0':s:c15:c16
c4 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
Q :: pair -> c14
c5 :: c5:c6:c7:c8:c9
c6 :: c10:c11:c12 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c7 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c8 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c9 :: c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c10 :: 0':s:c15:c16 -> c13 -> c10:c11:c12
c11 :: 0':s:c15:c16 -> c14 -> c10:c11:c12
c12 :: c13 -> c10:c11:c12
pair :: 0':s:c15:c16 -> 0':s:c15:c16 -> pair
c13 :: c13
c14 :: c14
c15 :: 0':s:c15:c16
c16 :: 0':s:c15:c16 -> 0':s:c15:c16
f :: 0':s:c15:c16 -> 0':s:c15:c16
hole_c:c1:c2:c3:c41_17 :: c:c1:c2:c3:c4
hole_0':s:c15:c162_17 :: 0':s:c15:c16
hole_c133_17 :: c13
hole_c10:c11:c124_17 :: c10:c11:c12
hole_c5:c6:c7:c8:c95_17 :: c5:c6:c7:c8:c9
hole_pair6_17 :: pair
hole_c147_17 :: c14
gen_0':s:c15:c168_17 :: Nat -> 0':s:c15:c16
gen_c5:c6:c7:c8:c99_17 :: Nat -> c5:c6:c7:c8:c9

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, G, +'

They will be analysed ascendingly in the following order:
g < G
+' < g
+' < G

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

(8)
Obligation:
Innermost TRS:
Rules:
F(0') -> c
F(s(0')) -> c1
F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(0') -> c5
G(s(z0)) -> c6(H(g(z0)), G(z0))
G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(s(z0)) -> c9(P(g(z0)), G(z0))
H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
H(z0) -> c12(P(z0))
P(pair(z0, z1)) -> c13
Q(pair(z0, z1)) -> c14
+'(z0, 0') -> c15
+'(z0, s(z1)) -> c16(+'(z0, z1))
f(0') -> 0'
f(s(0')) -> s(0')
f(s(s(z0))) -> p(h(g(z0)))
f(s(s(z0))) -> +'(p(g(z0)), q(g(z0)))
g(0') -> pair(s(0'), s(0'))
g(s(z0)) -> h(g(z0))
g(s(z0)) -> pair(+'(p(g(z0)), q(g(z0))), p(g(z0)))
h(z0) -> pair(+'(p(z0), q(z0)), p(z0))
p(pair(z0, z1)) -> z0
q(pair(z0, z1)) -> z1
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
F :: 0':s:c15:c16 -> c:c1:c2:c3:c4
0' :: 0':s:c15:c16
c :: c:c1:c2:c3:c4
s :: 0':s:c15:c16 -> 0':s:c15:c16
c1 :: c:c1:c2:c3:c4
c2 :: c13 -> c10:c11:c12 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
P :: pair -> c13
h :: pair -> pair
g :: 0':s:c15:c16 -> pair
H :: pair -> c10:c11:c12
G :: 0':s:c15:c16 -> c5:c6:c7:c8:c9
c3 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
+' :: 0':s:c15:c16 -> 0':s:c15:c16 -> 0':s:c15:c16
p :: pair -> 0':s:c15:c16
q :: pair -> 0':s:c15:c16
c4 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
Q :: pair -> c14
c5 :: c5:c6:c7:c8:c9
c6 :: c10:c11:c12 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c7 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c8 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c9 :: c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c10 :: 0':s:c15:c16 -> c13 -> c10:c11:c12
c11 :: 0':s:c15:c16 -> c14 -> c10:c11:c12
c12 :: c13 -> c10:c11:c12
pair :: 0':s:c15:c16 -> 0':s:c15:c16 -> pair
c13 :: c13
c14 :: c14
c15 :: 0':s:c15:c16
c16 :: 0':s:c15:c16 -> 0':s:c15:c16
f :: 0':s:c15:c16 -> 0':s:c15:c16
hole_c:c1:c2:c3:c41_17 :: c:c1:c2:c3:c4
hole_0':s:c15:c162_17 :: 0':s:c15:c16
hole_c133_17 :: c13
hole_c10:c11:c124_17 :: c10:c11:c12
hole_c5:c6:c7:c8:c95_17 :: c5:c6:c7:c8:c9
hole_pair6_17 :: pair
hole_c147_17 :: c14
gen_0':s:c15:c168_17 :: Nat -> 0':s:c15:c16
gen_c5:c6:c7:c8:c99_17 :: Nat -> c5:c6:c7:c8:c9


Generator Equations:
gen_0':s:c15:c168_17(0) <=> 0'
gen_0':s:c15:c168_17(+(x, 1)) <=> s(gen_0':s:c15:c168_17(x))
gen_c5:c6:c7:c8:c99_17(0) <=> c5
gen_c5:c6:c7:c8:c99_17(+(x, 1)) <=> c6(c10(0', c13), gen_c5:c6:c7:c8:c99_17(x))


The following defined symbols remain to be analysed:
+', g, G

They will be analysed ascendingly in the following order:
g < G
+' < g
+' < G

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:c15:c168_17(a), gen_0':s:c15:c168_17(+(1, n11_17))) -> *10_17, rt in Omega(n11_17)

Induction Base:
+'(gen_0':s:c15:c168_17(a), gen_0':s:c15:c168_17(+(1, 0)))

Induction Step:
+'(gen_0':s:c15:c168_17(a), gen_0':s:c15:c168_17(+(1, +(n11_17, 1)))) ->_R^Omega(1)
c16(+'(gen_0':s:c15:c168_17(a), gen_0':s:c15:c168_17(+(1, n11_17)))) ->_IH
c16(*10_17)

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(0') -> c
F(s(0')) -> c1
F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(0') -> c5
G(s(z0)) -> c6(H(g(z0)), G(z0))
G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(s(z0)) -> c9(P(g(z0)), G(z0))
H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
H(z0) -> c12(P(z0))
P(pair(z0, z1)) -> c13
Q(pair(z0, z1)) -> c14
+'(z0, 0') -> c15
+'(z0, s(z1)) -> c16(+'(z0, z1))
f(0') -> 0'
f(s(0')) -> s(0')
f(s(s(z0))) -> p(h(g(z0)))
f(s(s(z0))) -> +'(p(g(z0)), q(g(z0)))
g(0') -> pair(s(0'), s(0'))
g(s(z0)) -> h(g(z0))
g(s(z0)) -> pair(+'(p(g(z0)), q(g(z0))), p(g(z0)))
h(z0) -> pair(+'(p(z0), q(z0)), p(z0))
p(pair(z0, z1)) -> z0
q(pair(z0, z1)) -> z1
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
F :: 0':s:c15:c16 -> c:c1:c2:c3:c4
0' :: 0':s:c15:c16
c :: c:c1:c2:c3:c4
s :: 0':s:c15:c16 -> 0':s:c15:c16
c1 :: c:c1:c2:c3:c4
c2 :: c13 -> c10:c11:c12 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
P :: pair -> c13
h :: pair -> pair
g :: 0':s:c15:c16 -> pair
H :: pair -> c10:c11:c12
G :: 0':s:c15:c16 -> c5:c6:c7:c8:c9
c3 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
+' :: 0':s:c15:c16 -> 0':s:c15:c16 -> 0':s:c15:c16
p :: pair -> 0':s:c15:c16
q :: pair -> 0':s:c15:c16
c4 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
Q :: pair -> c14
c5 :: c5:c6:c7:c8:c9
c6 :: c10:c11:c12 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c7 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c8 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c9 :: c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c10 :: 0':s:c15:c16 -> c13 -> c10:c11:c12
c11 :: 0':s:c15:c16 -> c14 -> c10:c11:c12
c12 :: c13 -> c10:c11:c12
pair :: 0':s:c15:c16 -> 0':s:c15:c16 -> pair
c13 :: c13
c14 :: c14
c15 :: 0':s:c15:c16
c16 :: 0':s:c15:c16 -> 0':s:c15:c16
f :: 0':s:c15:c16 -> 0':s:c15:c16
hole_c:c1:c2:c3:c41_17 :: c:c1:c2:c3:c4
hole_0':s:c15:c162_17 :: 0':s:c15:c16
hole_c133_17 :: c13
hole_c10:c11:c124_17 :: c10:c11:c12
hole_c5:c6:c7:c8:c95_17 :: c5:c6:c7:c8:c9
hole_pair6_17 :: pair
hole_c147_17 :: c14
gen_0':s:c15:c168_17 :: Nat -> 0':s:c15:c16
gen_c5:c6:c7:c8:c99_17 :: Nat -> c5:c6:c7:c8:c9


Generator Equations:
gen_0':s:c15:c168_17(0) <=> 0'
gen_0':s:c15:c168_17(+(x, 1)) <=> s(gen_0':s:c15:c168_17(x))
gen_c5:c6:c7:c8:c99_17(0) <=> c5
gen_c5:c6:c7:c8:c99_17(+(x, 1)) <=> c6(c10(0', c13), gen_c5:c6:c7:c8:c99_17(x))


The following defined symbols remain to be analysed:
+', g, G

They will be analysed ascendingly in the following order:
g < G
+' < g
+' < G

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
Innermost TRS:
Rules:
F(0') -> c
F(s(0')) -> c1
F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(0') -> c5
G(s(z0)) -> c6(H(g(z0)), G(z0))
G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
G(s(z0)) -> c9(P(g(z0)), G(z0))
H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
H(z0) -> c12(P(z0))
P(pair(z0, z1)) -> c13
Q(pair(z0, z1)) -> c14
+'(z0, 0') -> c15
+'(z0, s(z1)) -> c16(+'(z0, z1))
f(0') -> 0'
f(s(0')) -> s(0')
f(s(s(z0))) -> p(h(g(z0)))
f(s(s(z0))) -> +'(p(g(z0)), q(g(z0)))
g(0') -> pair(s(0'), s(0'))
g(s(z0)) -> h(g(z0))
g(s(z0)) -> pair(+'(p(g(z0)), q(g(z0))), p(g(z0)))
h(z0) -> pair(+'(p(z0), q(z0)), p(z0))
p(pair(z0, z1)) -> z0
q(pair(z0, z1)) -> z1
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
F :: 0':s:c15:c16 -> c:c1:c2:c3:c4
0' :: 0':s:c15:c16
c :: c:c1:c2:c3:c4
s :: 0':s:c15:c16 -> 0':s:c15:c16
c1 :: c:c1:c2:c3:c4
c2 :: c13 -> c10:c11:c12 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
P :: pair -> c13
h :: pair -> pair
g :: 0':s:c15:c16 -> pair
H :: pair -> c10:c11:c12
G :: 0':s:c15:c16 -> c5:c6:c7:c8:c9
c3 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
+' :: 0':s:c15:c16 -> 0':s:c15:c16 -> 0':s:c15:c16
p :: pair -> 0':s:c15:c16
q :: pair -> 0':s:c15:c16
c4 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c:c1:c2:c3:c4
Q :: pair -> c14
c5 :: c5:c6:c7:c8:c9
c6 :: c10:c11:c12 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c7 :: 0':s:c15:c16 -> c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c8 :: 0':s:c15:c16 -> c14 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c9 :: c13 -> c5:c6:c7:c8:c9 -> c5:c6:c7:c8:c9
c10 :: 0':s:c15:c16 -> c13 -> c10:c11:c12
c11 :: 0':s:c15:c16 -> c14 -> c10:c11:c12
c12 :: c13 -> c10:c11:c12
pair :: 0':s:c15:c16 -> 0':s:c15:c16 -> pair
c13 :: c13
c14 :: c14
c15 :: 0':s:c15:c16
c16 :: 0':s:c15:c16 -> 0':s:c15:c16
f :: 0':s:c15:c16 -> 0':s:c15:c16
hole_c:c1:c2:c3:c41_17 :: c:c1:c2:c3:c4
hole_0':s:c15:c162_17 :: 0':s:c15:c16
hole_c133_17 :: c13
hole_c10:c11:c124_17 :: c10:c11:c12
hole_c5:c6:c7:c8:c95_17 :: c5:c6:c7:c8:c9
hole_pair6_17 :: pair
hole_c147_17 :: c14
gen_0':s:c15:c168_17 :: Nat -> 0':s:c15:c16
gen_c5:c6:c7:c8:c99_17 :: Nat -> c5:c6:c7:c8:c9


Lemmas:
+'(gen_0':s:c15:c168_17(a), gen_0':s:c15:c168_17(+(1, n11_17))) -> *10_17, rt in Omega(n11_17)


Generator Equations:
gen_0':s:c15:c168_17(0) <=> 0'
gen_0':s:c15:c168_17(+(x, 1)) <=> s(gen_0':s:c15:c168_17(x))
gen_c5:c6:c7:c8:c99_17(0) <=> c5
gen_c5:c6:c7:c8:c99_17(+(x, 1)) <=> c6(c10(0', c13), gen_c5:c6:c7:c8:c99_17(x))


The following defined symbols remain to be analysed:
g, G

They will be analysed ascendingly in the following order:
g < G

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

(15) RewriteLemmaProof (FINISHED)
Proved the following rewrite lemma:
g(gen_0':s:c15:c168_17(+(1, n2395_17))) -> *10_17, rt in Omega(EXP)

Induction Base:
g(gen_0':s:c15:c168_17(+(1, 0)))

Induction Step:
g(gen_0':s:c15:c168_17(+(1, +(n2395_17, 1)))) ->_R^Omega(0)
pair(+'(p(g(gen_0':s:c15:c168_17(+(1, n2395_17)))), q(g(gen_0':s:c15:c168_17(+(1, n2395_17))))), p(g(gen_0':s:c15:c168_17(+(1, n2395_17))))) ->_IH
pair(+'(p(*10_17), q(g(gen_0':s:c15:c168_17(+(1, n2395_17))))), p(g(gen_0':s:c15:c168_17(+(1, n2395_17))))) ->_IH
pair(+'(p(*10_17), q(*10_17)), p(g(gen_0':s:c15:c168_17(+(1, n2395_17))))) ->_IH
pair(+'(p(*10_17), q(*10_17)), p(*10_17))

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

(16)
BOUNDS(EXP, INF)