WORST_CASE(Omega(n^1),?)
proof of input_kOSPO8WC2P.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 4 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 1866 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), 445 ms]
        (18) typed CpxTrs


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(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:

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> +(fib(s(x)), fib(x))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
   FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
   +'(z0, 0) -> c4
   +'(z0, s(z1)) -> c5(+'(z0, z1))
S tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
   FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
   +'(z0, 0) -> c4
   +'(z0, s(z1)) -> c5(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   fib_1, +_2

Defined Pair Symbols:   FIB_1, +'_2

Compound Symbols:   c, c1, c2_2, c3_2, c4, c5_1


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

   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
   FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
   +'(z0, 0) -> c4
   +'(z0, s(z1)) -> c5(+'(z0, z1))

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

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(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:

   FIB(0') -> c
   FIB(s(0')) -> c1
   FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
   FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
   +'(z0, 0') -> c4
   +'(z0, s(z1)) -> c5(+'(z0, z1))

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

   fib(0') -> 0'
   fib(s(0')) -> s(0')
   fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
   +'(z0, 0') -> z0
   +'(z0, s(z1)) -> s(+'(z0, z1))

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
+'(z0, 0') -> c4
+'(z0, s(z1)) -> c5(+'(z0, z1))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
FIB :: 0':s:c4:c5 -> c:c1:c2:c3
0' :: 0':s:c4:c5
c :: c:c1:c2:c3
s :: 0':s:c4:c5 -> 0':s:c4:c5
c1 :: c:c1:c2:c3
c2 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
+' :: 0':s:c4:c5 -> 0':s:c4:c5 -> 0':s:c4:c5
fib :: 0':s:c4:c5 -> 0':s:c4:c5
c3 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
c4 :: 0':s:c4:c5
c5 :: 0':s:c4:c5 -> 0':s:c4:c5
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s:c4:c52_6 :: 0':s:c4:c5
gen_c:c1:c2:c33_6 :: Nat -> c:c1:c2:c3
gen_0':s:c4:c54_6 :: Nat -> 0':s:c4:c5

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FIB, +', fib

They will be analysed ascendingly in the following order:
+' < FIB
fib < FIB
+' < fib

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(10)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
+'(z0, 0') -> c4
+'(z0, s(z1)) -> c5(+'(z0, z1))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
FIB :: 0':s:c4:c5 -> c:c1:c2:c3
0' :: 0':s:c4:c5
c :: c:c1:c2:c3
s :: 0':s:c4:c5 -> 0':s:c4:c5
c1 :: c:c1:c2:c3
c2 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
+' :: 0':s:c4:c5 -> 0':s:c4:c5 -> 0':s:c4:c5
fib :: 0':s:c4:c5 -> 0':s:c4:c5
c3 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
c4 :: 0':s:c4:c5
c5 :: 0':s:c4:c5 -> 0':s:c4:c5
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s:c4:c52_6 :: 0':s:c4:c5
gen_c:c1:c2:c33_6 :: Nat -> c:c1:c2:c3
gen_0':s:c4:c54_6 :: Nat -> 0':s:c4:c5


Generator Equations:
gen_c:c1:c2:c33_6(0) <=> c
gen_c:c1:c2:c33_6(+(x, 1)) <=> c2(0', gen_c:c1:c2:c33_6(x))
gen_0':s:c4:c54_6(0) <=> 0'
gen_0':s:c4:c54_6(+(x, 1)) <=> s(gen_0':s:c4:c54_6(x))


The following defined symbols remain to be analysed:
+', FIB, fib

They will be analysed ascendingly in the following order:
+' < FIB
fib < FIB
+' < fib

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, n6_6))) -> *5_6, rt in Omega(n6_6)

Induction Base:
+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, 0)))

Induction Step:
+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, +(n6_6, 1)))) ->_R^Omega(1)
c5(+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, n6_6)))) ->_IH
c5(*5_6)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(12)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
+'(z0, 0') -> c4
+'(z0, s(z1)) -> c5(+'(z0, z1))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
FIB :: 0':s:c4:c5 -> c:c1:c2:c3
0' :: 0':s:c4:c5
c :: c:c1:c2:c3
s :: 0':s:c4:c5 -> 0':s:c4:c5
c1 :: c:c1:c2:c3
c2 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
+' :: 0':s:c4:c5 -> 0':s:c4:c5 -> 0':s:c4:c5
fib :: 0':s:c4:c5 -> 0':s:c4:c5
c3 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
c4 :: 0':s:c4:c5
c5 :: 0':s:c4:c5 -> 0':s:c4:c5
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s:c4:c52_6 :: 0':s:c4:c5
gen_c:c1:c2:c33_6 :: Nat -> c:c1:c2:c3
gen_0':s:c4:c54_6 :: Nat -> 0':s:c4:c5


Generator Equations:
gen_c:c1:c2:c33_6(0) <=> c
gen_c:c1:c2:c33_6(+(x, 1)) <=> c2(0', gen_c:c1:c2:c33_6(x))
gen_0':s:c4:c54_6(0) <=> 0'
gen_0':s:c4:c54_6(+(x, 1)) <=> s(gen_0':s:c4:c54_6(x))


The following defined symbols remain to be analysed:
+', FIB, fib

They will be analysed ascendingly in the following order:
+' < FIB
fib < FIB
+' < fib

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

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(16)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
+'(z0, 0') -> c4
+'(z0, s(z1)) -> c5(+'(z0, z1))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
FIB :: 0':s:c4:c5 -> c:c1:c2:c3
0' :: 0':s:c4:c5
c :: c:c1:c2:c3
s :: 0':s:c4:c5 -> 0':s:c4:c5
c1 :: c:c1:c2:c3
c2 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
+' :: 0':s:c4:c5 -> 0':s:c4:c5 -> 0':s:c4:c5
fib :: 0':s:c4:c5 -> 0':s:c4:c5
c3 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
c4 :: 0':s:c4:c5
c5 :: 0':s:c4:c5 -> 0':s:c4:c5
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s:c4:c52_6 :: 0':s:c4:c5
gen_c:c1:c2:c33_6 :: Nat -> c:c1:c2:c3
gen_0':s:c4:c54_6 :: Nat -> 0':s:c4:c5


Lemmas:
+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, n6_6))) -> *5_6, rt in Omega(n6_6)


Generator Equations:
gen_c:c1:c2:c33_6(0) <=> c
gen_c:c1:c2:c33_6(+(x, 1)) <=> c2(0', gen_c:c1:c2:c33_6(x))
gen_0':s:c4:c54_6(0) <=> 0'
gen_0':s:c4:c54_6(+(x, 1)) <=> s(gen_0':s:c4:c54_6(x))


The following defined symbols remain to be analysed:
fib, FIB

They will be analysed ascendingly in the following order:
fib < FIB

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fib(gen_0':s:c4:c54_6(+(2, n1700_6))) -> *5_6, rt in Omega(0)

Induction Base:
fib(gen_0':s:c4:c54_6(+(2, 0)))

Induction Step:
fib(gen_0':s:c4:c54_6(+(2, +(n1700_6, 1)))) ->_R^Omega(0)
+'(fib(s(gen_0':s:c4:c54_6(+(1, n1700_6)))), fib(gen_0':s:c4:c54_6(+(1, n1700_6)))) ->_IH
+'(*5_6, fib(gen_0':s:c4:c54_6(+(1, n1700_6))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(18)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(+'(fib(s(z0)), fib(z0)), FIB(s(z0)))
FIB(s(s(z0))) -> c3(+'(fib(s(z0)), fib(z0)), FIB(z0))
+'(z0, 0') -> c4
+'(z0, s(z1)) -> c5(+'(z0, z1))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))
+'(z0, 0') -> z0
+'(z0, s(z1)) -> s(+'(z0, z1))

Types:
FIB :: 0':s:c4:c5 -> c:c1:c2:c3
0' :: 0':s:c4:c5
c :: c:c1:c2:c3
s :: 0':s:c4:c5 -> 0':s:c4:c5
c1 :: c:c1:c2:c3
c2 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
+' :: 0':s:c4:c5 -> 0':s:c4:c5 -> 0':s:c4:c5
fib :: 0':s:c4:c5 -> 0':s:c4:c5
c3 :: 0':s:c4:c5 -> c:c1:c2:c3 -> c:c1:c2:c3
c4 :: 0':s:c4:c5
c5 :: 0':s:c4:c5 -> 0':s:c4:c5
hole_c:c1:c2:c31_6 :: c:c1:c2:c3
hole_0':s:c4:c52_6 :: 0':s:c4:c5
gen_c:c1:c2:c33_6 :: Nat -> c:c1:c2:c3
gen_0':s:c4:c54_6 :: Nat -> 0':s:c4:c5


Lemmas:
+'(gen_0':s:c4:c54_6(a), gen_0':s:c4:c54_6(+(1, n6_6))) -> *5_6, rt in Omega(n6_6)
fib(gen_0':s:c4:c54_6(+(2, n1700_6))) -> *5_6, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c33_6(0) <=> c
gen_c:c1:c2:c33_6(+(x, 1)) <=> c2(0', gen_c:c1:c2:c33_6(x))
gen_0':s:c4:c54_6(0) <=> 0'
gen_0':s:c4:c54_6(+(x, 1)) <=> s(gen_0':s:c4:c54_6(x))


The following defined symbols remain to be analysed:
FIB