WORST_CASE(Omega(n^1),O(n^1))
proof of input_e4TBzSgODr.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 86 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 14 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 2230 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) 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, n^1).


The TRS R consists of the following rules:

   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(X), c) -> f(X, c)
   f(c, c) -> f(a, a)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 4. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
a0() -> 0
b0() -> 0
s0(0) -> 0
c0() -> 0
f0(0, 0) -> 1
a1() -> 2
b1() -> 3
f1(2, 3) -> 1
a1() -> 5
s1(5) -> 4
c1() -> 6
f1(4, 6) -> 1
f1(0, 6) -> 1
a1() -> 7
f1(2, 7) -> 1
a2() -> 8
b2() -> 9
f2(8, 9) -> 1
a2() -> 11
s2(11) -> 10
c2() -> 12
f2(10, 12) -> 1
f2(5, 12) -> 1
a3() -> 14
s3(14) -> 13
c3() -> 15
f3(13, 15) -> 1
f3(11, 15) -> 1
c4() -> 16
f4(14, 16) -> 1

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(4)
BOUNDS(1, n^1)

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(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(z0), c) -> f(z0, c)
   f(c, c) -> f(a, a)
Tuples:
   F(a, a) -> c1(F(a, b))
   F(a, b) -> c2(F(s(a), c))
   F(s(z0), c) -> c3(F(z0, c))
   F(c, c) -> c4(F(a, a))
S tuples:
   F(a, a) -> c1(F(a, b))
   F(a, b) -> c2(F(s(a), c))
   F(s(z0), c) -> c3(F(z0, c))
   F(c, c) -> c4(F(a, a))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1, c2_1, c3_1, c4_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.
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(8)
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:

   F(a, a) -> c1(F(a, b))
   F(a, b) -> c2(F(s(a), c))
   F(s(z0), c) -> c3(F(z0, c))
   F(c, c) -> c4(F(a, a))

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

   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(z0), c) -> f(z0, c)
   f(c, c) -> f(a, a)

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

(10)
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:

   F(a, a) -> c1(F(a, b))
   F(a, b) -> c2(F(s(a), c))
   F(s(z0), c) -> c3(F(z0, c))
   F(c, c) -> c4(F(a, a))

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

   f(a, a) -> f(a, b)
   f(a, b) -> f(s(a), c)
   f(s(z0), c) -> f(z0, c)
   f(c, c) -> f(a, a)

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
F(a, a) -> c1(F(a, b))
F(a, b) -> c2(F(s(a), c))
F(s(z0), c) -> c3(F(z0, c))
F(c, c) -> c4(F(a, a))
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(z0), c) -> f(z0, c)
f(c, c) -> f(a, a)

Types:
F :: a:b:s:c -> a:b:s:c -> c1:c2:c3:c4
a :: a:b:s:c
c1 :: c1:c2:c3:c4 -> c1:c2:c3:c4
b :: a:b:s:c
c2 :: c1:c2:c3:c4 -> c1:c2:c3:c4
s :: a:b:s:c -> a:b:s:c
c :: a:b:s:c
c3 :: c1:c2:c3:c4 -> c1:c2:c3:c4
c4 :: c1:c2:c3:c4 -> c1:c2:c3:c4
f :: a:b:s:c -> a:b:s:c -> f
hole_c1:c2:c3:c41_5 :: c1:c2:c3:c4
hole_a:b:s:c2_5 :: a:b:s:c
hole_f3_5 :: f
gen_c1:c2:c3:c44_5 :: Nat -> c1:c2:c3:c4
gen_a:b:s:c5_5 :: Nat -> a:b:s:c

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(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, f
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(14)
Obligation:
Innermost TRS:
Rules:
F(a, a) -> c1(F(a, b))
F(a, b) -> c2(F(s(a), c))
F(s(z0), c) -> c3(F(z0, c))
F(c, c) -> c4(F(a, a))
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(z0), c) -> f(z0, c)
f(c, c) -> f(a, a)

Types:
F :: a:b:s:c -> a:b:s:c -> c1:c2:c3:c4
a :: a:b:s:c
c1 :: c1:c2:c3:c4 -> c1:c2:c3:c4
b :: a:b:s:c
c2 :: c1:c2:c3:c4 -> c1:c2:c3:c4
s :: a:b:s:c -> a:b:s:c
c :: a:b:s:c
c3 :: c1:c2:c3:c4 -> c1:c2:c3:c4
c4 :: c1:c2:c3:c4 -> c1:c2:c3:c4
f :: a:b:s:c -> a:b:s:c -> f
hole_c1:c2:c3:c41_5 :: c1:c2:c3:c4
hole_a:b:s:c2_5 :: a:b:s:c
hole_f3_5 :: f
gen_c1:c2:c3:c44_5 :: Nat -> c1:c2:c3:c4
gen_a:b:s:c5_5 :: Nat -> a:b:s:c


Generator Equations:
gen_c1:c2:c3:c44_5(0) <=> hole_c1:c2:c3:c41_5
gen_c1:c2:c3:c44_5(+(x, 1)) <=> c1(gen_c1:c2:c3:c44_5(x))
gen_a:b:s:c5_5(0) <=> c
gen_a:b:s:c5_5(+(x, 1)) <=> s(gen_a:b:s:c5_5(x))


The following defined symbols remain to be analysed:
F, f
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(gen_a:b:s:c5_5(+(1, n7_5)), gen_a:b:s:c5_5(0)) -> *6_5, rt in Omega(n7_5)

Induction Base:
F(gen_a:b:s:c5_5(+(1, 0)), gen_a:b:s:c5_5(0))

Induction Step:
F(gen_a:b:s:c5_5(+(1, +(n7_5, 1))), gen_a:b:s:c5_5(0)) ->_R^Omega(1)
c3(F(gen_a:b:s:c5_5(+(1, n7_5)), c)) ->_IH
c3(*6_5)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
Complex Obligation (BEST)

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(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(a, a) -> c1(F(a, b))
F(a, b) -> c2(F(s(a), c))
F(s(z0), c) -> c3(F(z0, c))
F(c, c) -> c4(F(a, a))
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(z0), c) -> f(z0, c)
f(c, c) -> f(a, a)

Types:
F :: a:b:s:c -> a:b:s:c -> c1:c2:c3:c4
a :: a:b:s:c
c1 :: c1:c2:c3:c4 -> c1:c2:c3:c4
b :: a:b:s:c
c2 :: c1:c2:c3:c4 -> c1:c2:c3:c4
s :: a:b:s:c -> a:b:s:c
c :: a:b:s:c
c3 :: c1:c2:c3:c4 -> c1:c2:c3:c4
c4 :: c1:c2:c3:c4 -> c1:c2:c3:c4
f :: a:b:s:c -> a:b:s:c -> f
hole_c1:c2:c3:c41_5 :: c1:c2:c3:c4
hole_a:b:s:c2_5 :: a:b:s:c
hole_f3_5 :: f
gen_c1:c2:c3:c44_5 :: Nat -> c1:c2:c3:c4
gen_a:b:s:c5_5 :: Nat -> a:b:s:c


Generator Equations:
gen_c1:c2:c3:c44_5(0) <=> hole_c1:c2:c3:c41_5
gen_c1:c2:c3:c44_5(+(x, 1)) <=> c1(gen_c1:c2:c3:c44_5(x))
gen_a:b:s:c5_5(0) <=> c
gen_a:b:s:c5_5(+(x, 1)) <=> s(gen_a:b:s:c5_5(x))


The following defined symbols remain to be analysed:
F, f
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(18) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(19)
BOUNDS(n^1, INF)

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(20)
Obligation:
Innermost TRS:
Rules:
F(a, a) -> c1(F(a, b))
F(a, b) -> c2(F(s(a), c))
F(s(z0), c) -> c3(F(z0, c))
F(c, c) -> c4(F(a, a))
f(a, a) -> f(a, b)
f(a, b) -> f(s(a), c)
f(s(z0), c) -> f(z0, c)
f(c, c) -> f(a, a)

Types:
F :: a:b:s:c -> a:b:s:c -> c1:c2:c3:c4
a :: a:b:s:c
c1 :: c1:c2:c3:c4 -> c1:c2:c3:c4
b :: a:b:s:c
c2 :: c1:c2:c3:c4 -> c1:c2:c3:c4
s :: a:b:s:c -> a:b:s:c
c :: a:b:s:c
c3 :: c1:c2:c3:c4 -> c1:c2:c3:c4
c4 :: c1:c2:c3:c4 -> c1:c2:c3:c4
f :: a:b:s:c -> a:b:s:c -> f
hole_c1:c2:c3:c41_5 :: c1:c2:c3:c4
hole_a:b:s:c2_5 :: a:b:s:c
hole_f3_5 :: f
gen_c1:c2:c3:c44_5 :: Nat -> c1:c2:c3:c4
gen_a:b:s:c5_5 :: Nat -> a:b:s:c


Lemmas:
F(gen_a:b:s:c5_5(+(1, n7_5)), gen_a:b:s:c5_5(0)) -> *6_5, rt in Omega(n7_5)


Generator Equations:
gen_c1:c2:c3:c44_5(0) <=> hole_c1:c2:c3:c41_5
gen_c1:c2:c3:c44_5(+(x, 1)) <=> c1(gen_c1:c2:c3:c44_5(x))
gen_a:b:s:c5_5(0) <=> c
gen_a:b:s:c5_5(+(x, 1)) <=> s(gen_a:b:s:c5_5(x))


The following defined symbols remain to be analysed:
f