WORST_CASE(Omega(n^1),?)
proof of input_GU0mc4Kkc5.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 9 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 271 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), 118 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 30 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
        (22) 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:

   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fib(x) -> fibiter(x, 0, 0, s(0))
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
   if(false, b, c, x, y) -> x
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))

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:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fib(z0) -> fibiter(z0, 0, 0, s(0))
   fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
   if(false, z0, z1, z2, z3) -> z2
   if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))
Tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
S tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
K tuples:none
Defined Rule Symbols:   plus_2, lt_2, fib_1, fibiter_4, if_5

Defined Pair Symbols:   PLUS_2, LT_2, FIB_1, FIBITER_4, IF_5

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5_1, c6_2, c7, c8_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(n^1, INF).


The TRS R consists of the following rules:

   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))

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

   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fib(z0) -> fibiter(z0, 0, 0, s(0))
   fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
   if(false, z0, z1, z2, z3) -> z2
   if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

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:

   PLUS(0', z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0', s(z0)) -> c2
   LT(z0, 0') -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))

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

   plus(0', z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   lt(0', s(z0)) -> true
   lt(z0, 0') -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fib(z0) -> fibiter(z0, 0', 0', s(0'))
   fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
   if(false, z0, z1, z2, z3) -> z2
   if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

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

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

(8)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
PLUS, LT, FIBITER, lt, plus, fibiter

They will be analysed ascendingly in the following order:
PLUS < FIBITER
LT < FIBITER
lt < FIBITER
plus < FIBITER
lt < fibiter
plus < fibiter

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

(10)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
PLUS, LT, FIBITER, lt, plus, fibiter

They will be analysed ascendingly in the following order:
PLUS < FIBITER
LT < FIBITER
lt < FIBITER
plus < FIBITER
lt < fibiter
plus < fibiter

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b)) -> gen_c:c18_9(n12_9), rt in Omega(1 + n12_9)

Induction Base:
PLUS(gen_0':s9_9(0), gen_0':s9_9(b)) ->_R^Omega(1)
c

Induction Step:
PLUS(gen_0':s9_9(+(n12_9, 1)), gen_0':s9_9(b)) ->_R^Omega(1)
c1(PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b))) ->_IH
c1(gen_c:c18_9(c13_9))

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:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
PLUS, LT, FIBITER, lt, plus, fibiter

They will be analysed ascendingly in the following order:
PLUS < FIBITER
LT < FIBITER
lt < FIBITER
plus < FIBITER
lt < fibiter
plus < fibiter

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Lemmas:
PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b)) -> gen_c:c18_9(n12_9), rt in Omega(1 + n12_9)


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
LT, FIBITER, lt, plus, fibiter

They will be analysed ascendingly in the following order:
LT < FIBITER
lt < FIBITER
plus < FIBITER
lt < fibiter
plus < fibiter

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s9_9(n503_9), gen_0':s9_9(+(1, n503_9))) -> gen_c2:c3:c410_9(n503_9), rt in Omega(1 + n503_9)

Induction Base:
LT(gen_0':s9_9(0), gen_0':s9_9(+(1, 0))) ->_R^Omega(1)
c2

Induction Step:
LT(gen_0':s9_9(+(n503_9, 1)), gen_0':s9_9(+(1, +(n503_9, 1)))) ->_R^Omega(1)
c4(LT(gen_0':s9_9(n503_9), gen_0':s9_9(+(1, n503_9)))) ->_IH
c4(gen_c2:c3:c410_9(c504_9))

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

(18)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Lemmas:
PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b)) -> gen_c:c18_9(n12_9), rt in Omega(1 + n12_9)
LT(gen_0':s9_9(n503_9), gen_0':s9_9(+(1, n503_9))) -> gen_c2:c3:c410_9(n503_9), rt in Omega(1 + n503_9)


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
lt, FIBITER, plus, fibiter

They will be analysed ascendingly in the following order:
lt < FIBITER
plus < FIBITER
lt < fibiter
plus < fibiter

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s9_9(n1133_9), gen_0':s9_9(+(1, n1133_9))) -> true, rt in Omega(0)

Induction Base:
lt(gen_0':s9_9(0), gen_0':s9_9(+(1, 0))) ->_R^Omega(0)
true

Induction Step:
lt(gen_0':s9_9(+(n1133_9, 1)), gen_0':s9_9(+(1, +(n1133_9, 1)))) ->_R^Omega(0)
lt(gen_0':s9_9(n1133_9), gen_0':s9_9(+(1, n1133_9))) ->_IH
true

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:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Lemmas:
PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b)) -> gen_c:c18_9(n12_9), rt in Omega(1 + n12_9)
LT(gen_0':s9_9(n503_9), gen_0':s9_9(+(1, n503_9))) -> gen_c2:c3:c410_9(n503_9), rt in Omega(1 + n503_9)
lt(gen_0':s9_9(n1133_9), gen_0':s9_9(+(1, n1133_9))) -> true, rt in Omega(0)


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
plus, FIBITER, fibiter

They will be analysed ascendingly in the following order:
plus < FIBITER
plus < fibiter

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s9_9(n1481_9), gen_0':s9_9(b)) -> gen_0':s9_9(+(n1481_9, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s9_9(0), gen_0':s9_9(b)) ->_R^Omega(0)
gen_0':s9_9(b)

Induction Step:
plus(gen_0':s9_9(+(n1481_9, 1)), gen_0':s9_9(b)) ->_R^Omega(0)
s(plus(gen_0':s9_9(n1481_9), gen_0':s9_9(b))) ->_IH
s(gen_0':s9_9(+(b, c1482_9)))

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

(22)
Obligation:
Innermost TRS:
Rules:
PLUS(0', z0) -> c
PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
LT(0', s(z0)) -> c2
LT(z0, 0') -> c3
LT(s(z0), s(z1)) -> c4(LT(z0, z1))
FIB(z0) -> c5(FIBITER(z0, 0', 0', s(0')))
FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
IF(false, z0, z1, z2, z3) -> c7
IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
plus(0', z0) -> z0
plus(s(z0), z1) -> s(plus(z0, z1))
lt(0', s(z0)) -> true
lt(z0, 0') -> false
lt(s(z0), s(z1)) -> lt(z0, z1)
fib(z0) -> fibiter(z0, 0', 0', s(0'))
fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
if(false, z0, z1, z2, z3) -> z2
if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))

Types:
PLUS :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
LT :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
FIB :: 0':s -> c5
c5 :: c6 -> c5
FIBITER :: 0':s -> 0':s -> 0':s -> 0':s -> c6
c6 :: c7:c8 -> c2:c3:c4 -> c6
IF :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> c7:c8
lt :: 0':s -> 0':s -> false:true
false :: false:true
c7 :: c7:c8
true :: false:true
c8 :: c6 -> c:c1 -> c7:c8
plus :: 0':s -> 0':s -> 0':s
fib :: 0':s -> 0':s
fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c11_9 :: c:c1
hole_0':s2_9 :: 0':s
hole_c2:c3:c43_9 :: c2:c3:c4
hole_c54_9 :: c5
hole_c65_9 :: c6
hole_c7:c86_9 :: c7:c8
hole_false:true7_9 :: false:true
gen_c:c18_9 :: Nat -> c:c1
gen_0':s9_9 :: Nat -> 0':s
gen_c2:c3:c410_9 :: Nat -> c2:c3:c4


Lemmas:
PLUS(gen_0':s9_9(n12_9), gen_0':s9_9(b)) -> gen_c:c18_9(n12_9), rt in Omega(1 + n12_9)
LT(gen_0':s9_9(n503_9), gen_0':s9_9(+(1, n503_9))) -> gen_c2:c3:c410_9(n503_9), rt in Omega(1 + n503_9)
lt(gen_0':s9_9(n1133_9), gen_0':s9_9(+(1, n1133_9))) -> true, rt in Omega(0)
plus(gen_0':s9_9(n1481_9), gen_0':s9_9(b)) -> gen_0':s9_9(+(n1481_9, b)), rt in Omega(0)


Generator Equations:
gen_c:c18_9(0) <=> c
gen_c:c18_9(+(x, 1)) <=> c1(gen_c:c18_9(x))
gen_0':s9_9(0) <=> 0'
gen_0':s9_9(+(x, 1)) <=> s(gen_0':s9_9(x))
gen_c2:c3:c410_9(0) <=> c2
gen_c2:c3:c410_9(+(x, 1)) <=> c4(gen_c2:c3:c410_9(x))


The following defined symbols remain to be analysed:
FIBITER, fibiter