WORST_CASE(Omega(n^1),O(n^1))
proof of input_TQWre31C8c.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTRS
    (9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTRS
    (11) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (12) BOUNDS(1, n^1)
(13) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CdtProblem
    (15) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRelTRS
    (17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRelTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 13 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 99.7 s]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 543 ms]
            (30) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


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))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(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))
Tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
S tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
K tuples:none
Defined Rule Symbols:   fib_1

Defined Pair Symbols:   FIB_1

Compound Symbols:   c, c1, c2_1, c3_1


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   FIB(0) -> c
   FIB(s(0)) -> c1

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))
Tuples:
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
S tuples:
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
K tuples:none
Defined Rule Symbols:   fib_1

Defined Pair Symbols:   FIB_1

Compound Symbols:   c2_1, c3_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> +(fib(s(z0)), fib(z0))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
S tuples:
   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FIB_1

Compound Symbols:   c2_1, c3_1


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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   FIB(s(s(z0))) -> c2(FIB(s(z0)))
   FIB(s(s(z0))) -> c3(FIB(z0))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(11) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[37, 38, 39, 40, 41]
{(37,38,[FIB_1|0]), (37,39,[c2_1|1]), (37,41,[c3_1|1]), (38,38,[s_1|0, c2_1|0, c3_1|0]), (39,40,[FIB_1|1]), (39,39,[c2_1|1]), (39,41,[c3_1|1]), (40,38,[s_1|1]), (41,38,[FIB_1|1]), (41,39,[c2_1|1]), (41,41,[c3_1|1])}"
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(12)
BOUNDS(1, n^1)

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(13) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   FIB_1

Compound Symbols:   c, c1, c2_1, c3_1


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

(16)
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(s(s(z0))) -> c3(FIB(z0))

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))

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

(18)
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(s(s(z0))) -> c3(FIB(z0))

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))

Rewrite Strategy: INNERMOST
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(19) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(20)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(FIB(s(z0)))
FIB(s(s(z0))) -> c3(FIB(z0))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))

Types:
FIB :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
fib :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'

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(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
FIB, fib
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(22)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(FIB(s(z0)))
FIB(s(s(z0))) -> c3(FIB(z0))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))

Types:
FIB :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
fib :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
FIB, fib
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
FIB(gen_0':s:+'4_4(+(2, n6_4))) -> *5_4, rt in Omega(n6_4)

Induction Base:
FIB(gen_0':s:+'4_4(+(2, 0)))

Induction Step:
FIB(gen_0':s:+'4_4(+(2, +(n6_4, 1)))) ->_R^Omega(1)
c2(FIB(s(gen_0':s:+'4_4(+(1, n6_4))))) ->_IH
c2(*5_4)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Complex Obligation (BEST)

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(25)
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(s(s(z0))) -> c3(FIB(z0))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))

Types:
FIB :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
fib :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
FIB, fib
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(26) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(27)
BOUNDS(n^1, INF)

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(28)
Obligation:
Innermost TRS:
Rules:
FIB(0') -> c
FIB(s(0')) -> c1
FIB(s(s(z0))) -> c2(FIB(s(z0)))
FIB(s(s(z0))) -> c3(FIB(z0))
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(z0))) -> +'(fib(s(z0)), fib(z0))

Types:
FIB :: 0':s:+' -> c:c1:c2:c3
0' :: 0':s:+'
c :: c:c1:c2:c3
s :: 0':s:+' -> 0':s:+'
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
fib :: 0':s:+' -> 0':s:+'
+' :: 0':s:+' -> 0':s:+' -> 0':s:+'
hole_c:c1:c2:c31_4 :: c:c1:c2:c3
hole_0':s:+'2_4 :: 0':s:+'
gen_c:c1:c2:c33_4 :: Nat -> c:c1:c2:c3
gen_0':s:+'4_4 :: Nat -> 0':s:+'


Lemmas:
FIB(gen_0':s:+'4_4(+(2, n6_4))) -> *5_4, rt in Omega(n6_4)


Generator Equations:
gen_c:c1:c2:c33_4(0) <=> c
gen_c:c1:c2:c33_4(+(x, 1)) <=> c2(gen_c:c1:c2:c33_4(x))
gen_0':s:+'4_4(0) <=> 0'
gen_0':s:+'4_4(+(x, 1)) <=> s(gen_0':s:+'4_4(x))


The following defined symbols remain to be analysed:
fib
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(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fib(gen_0':s:+'4_4(+(2, n73396_4))) -> *5_4, rt in Omega(0)

Induction Base:
fib(gen_0':s:+'4_4(+(2, 0)))

Induction Step:
fib(gen_0':s:+'4_4(+(2, +(n73396_4, 1)))) ->_R^Omega(0)
+'(fib(s(gen_0':s:+'4_4(+(1, n73396_4)))), fib(gen_0':s:+'4_4(+(1, n73396_4)))) ->_IH
+'(*5_4, fib(gen_0':s:+'4_4(+(1, n73396_4))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(30)
BOUNDS(1, INF)