WORST_CASE(Omega(n^1),O(n^1))
proof of input_97yrun43yu.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsMatchBoundsProof [FINISHED, 0 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), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 219 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 1362 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 1378 ms]
            (24) 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:

   g(s(x)) -> f(x)
   f(0) -> s(0)
   f(s(x)) -> s(s(g(x)))
   g(0) -> 0

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:

   g(s(x)) -> f(x)
   f(0) -> s(0)
   f(s(x)) -> s(s(g(x)))
   g(0) -> 0

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

(3) 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.

"[5, 6, 7, 8, 9]
{(5,6,[g_1|0, f_1|0, f_1|1, 0|1]), (5,7,[s_1|1]), (5,8,[s_1|1]), (6,6,[s_1|0, 0|0]), (7,6,[0|1]), (8,9,[s_1|1]), (9,6,[g_1|1, f_1|1, 0|1]), (9,7,[s_1|1]), (9,8,[s_1|1])}"
----------------------------------------

(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
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(s(z0)) -> f(z0)
   g(0) -> 0
   f(0) -> s(0)
   f(s(z0)) -> s(s(g(z0)))
Tuples:
   G(s(z0)) -> c(F(z0))
   G(0) -> c1
   F(0) -> c2
   F(s(z0)) -> c3(G(z0))
S tuples:
   G(s(z0)) -> c(F(z0))
   G(0) -> c1
   F(0) -> c2
   F(s(z0)) -> c3(G(z0))
K tuples:none
Defined Rule Symbols:   g_1, f_1

Defined Pair Symbols:   G_1, F_1

Compound Symbols:   c_1, c1, c2, 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 CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   G(s(z0)) -> c(F(z0))
   G(0) -> c1
   F(0) -> c2
   F(s(z0)) -> c3(G(z0))

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

   g(s(z0)) -> f(z0)
   g(0) -> 0
   f(0) -> s(0)
   f(s(z0)) -> s(s(g(z0)))

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:

   G(s(z0)) -> c(F(z0))
   G(0') -> c1
   F(0') -> c2
   F(s(z0)) -> c3(G(z0))

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

   g(s(z0)) -> f(z0)
   g(0') -> 0'
   f(0') -> s(0')
   f(s(z0)) -> s(s(g(z0)))

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

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

(12)
Obligation:
Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'

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(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
G, F, g, f

They will be analysed ascendingly in the following order:
G = F
g = f

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(14)
Obligation:
Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_4(0) <=> 0'
gen_s:0'4_4(+(x, 1)) <=> s(gen_s:0'4_4(x))


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

They will be analysed ascendingly in the following order:
G = F
g = f

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_s:0'4_4(*(2, n6_4))) -> gen_s:0'4_4(+(1, *(2, n6_4))), rt in Omega(0)

Induction Base:
f(gen_s:0'4_4(*(2, 0))) ->_R^Omega(0)
s(0')

Induction Step:
f(gen_s:0'4_4(*(2, +(n6_4, 1)))) ->_R^Omega(0)
s(s(g(gen_s:0'4_4(+(1, *(2, n6_4)))))) ->_R^Omega(0)
s(s(f(gen_s:0'4_4(*(2, n6_4))))) ->_IH
s(s(gen_s:0'4_4(+(1, *(2, c7_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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(16)
Obligation:
Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'


Lemmas:
f(gen_s:0'4_4(*(2, n6_4))) -> gen_s:0'4_4(+(1, *(2, n6_4))), rt in Omega(0)


Generator Equations:
gen_s:0'4_4(0) <=> 0'
gen_s:0'4_4(+(x, 1)) <=> s(gen_s:0'4_4(x))


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

They will be analysed ascendingly in the following order:
G = F
g = f

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(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(gen_s:0'4_4(*(2, n239_4))) -> *5_4, rt in Omega(n239_4)

Induction Base:
F(gen_s:0'4_4(*(2, 0)))

Induction Step:
F(gen_s:0'4_4(*(2, +(n239_4, 1)))) ->_R^Omega(1)
c3(G(gen_s:0'4_4(+(1, *(2, n239_4))))) ->_R^Omega(1)
c3(c(F(gen_s:0'4_4(*(2, n239_4))))) ->_IH
c3(c(*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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(18)
Complex Obligation (BEST)

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(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'


Lemmas:
f(gen_s:0'4_4(*(2, n6_4))) -> gen_s:0'4_4(+(1, *(2, n6_4))), rt in Omega(0)


Generator Equations:
gen_s:0'4_4(0) <=> 0'
gen_s:0'4_4(+(x, 1)) <=> s(gen_s:0'4_4(x))


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

They will be analysed ascendingly in the following order:
G = F

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(20) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(21)
BOUNDS(n^1, INF)

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(22)
Obligation:
Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'


Lemmas:
f(gen_s:0'4_4(*(2, n6_4))) -> gen_s:0'4_4(+(1, *(2, n6_4))), rt in Omega(0)
F(gen_s:0'4_4(*(2, n239_4))) -> *5_4, rt in Omega(n239_4)


Generator Equations:
gen_s:0'4_4(0) <=> 0'
gen_s:0'4_4(+(x, 1)) <=> s(gen_s:0'4_4(x))


The following defined symbols remain to be analysed:
G

They will be analysed ascendingly in the following order:
G = F

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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(gen_s:0'4_4(+(1, *(2, n485_4)))) -> *5_4, rt in Omega(n485_4)

Induction Base:
G(gen_s:0'4_4(+(1, *(2, 0))))

Induction Step:
G(gen_s:0'4_4(+(1, *(2, +(n485_4, 1))))) ->_R^Omega(1)
c(F(gen_s:0'4_4(+(2, *(2, n485_4))))) ->_R^Omega(1)
c(c3(G(gen_s:0'4_4(+(1, *(2, n485_4)))))) ->_IH
c(c3(*5_4))

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

(24)
Obligation:
Innermost TRS:
Rules:
G(s(z0)) -> c(F(z0))
G(0') -> c1
F(0') -> c2
F(s(z0)) -> c3(G(z0))
g(s(z0)) -> f(z0)
g(0') -> 0'
f(0') -> s(0')
f(s(z0)) -> s(s(g(z0)))

Types:
G :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c2:c3 -> c:c1
F :: s:0' -> c2:c3
0' :: s:0'
c1 :: c:c1
c2 :: c2:c3
c3 :: c:c1 -> c2:c3
g :: s:0' -> s:0'
f :: s:0' -> s:0'
hole_c:c11_4 :: c:c1
hole_s:0'2_4 :: s:0'
hole_c2:c33_4 :: c2:c3
gen_s:0'4_4 :: Nat -> s:0'


Lemmas:
f(gen_s:0'4_4(*(2, n6_4))) -> gen_s:0'4_4(+(1, *(2, n6_4))), rt in Omega(0)
F(gen_s:0'4_4(*(2, n239_4))) -> *5_4, rt in Omega(n239_4)
G(gen_s:0'4_4(+(1, *(2, n485_4)))) -> *5_4, rt in Omega(n485_4)


Generator Equations:
gen_s:0'4_4(0) <=> 0'
gen_s:0'4_4(+(x, 1)) <=> s(gen_s:0'4_4(x))


The following defined symbols remain to be analysed:
F

They will be analysed ascendingly in the following order:
G = F