WORST_CASE(Omega(n^1),O(n^1))
proof of input_gUwcXqva5d.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 454 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (22) 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:

   f(g(X)) -> f(X)

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(g(X)) -> f(X)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(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.

"[2, 3]
{(2,3,[f_1|0, f_1|1]), (3,3,[g_1|0])}"
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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(g(z0)) -> f(z0)
Tuples:
   F(g(z0)) -> c(F(z0))
S tuples:
   F(g(z0)) -> c(F(z0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_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(g(z0)) -> c(F(z0))

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

   f(g(z0)) -> f(z0)

Rewrite Strategy: INNERMOST
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(9) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(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(g(z0)) -> c(F(z0))

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

   f(g(z0)) -> f(z0)

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(12)
Obligation:
Innermost TRS:
Rules:
F(g(z0)) -> c(F(z0))
f(g(z0)) -> f(z0)

Types:
F :: g -> c
g :: g -> g
c :: c -> c
f :: g -> f
hole_c1_1 :: c
hole_g2_1 :: g
hole_f3_1 :: f
gen_c4_1 :: Nat -> c
gen_g5_1 :: Nat -> g

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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(g(z0)) -> c(F(z0))
f(g(z0)) -> f(z0)

Types:
F :: g -> c
g :: g -> g
c :: c -> c
f :: g -> f
hole_c1_1 :: c
hole_g2_1 :: g
hole_f3_1 :: f
gen_c4_1 :: Nat -> c
gen_g5_1 :: Nat -> g


Generator Equations:
gen_c4_1(0) <=> hole_c1_1
gen_c4_1(+(x, 1)) <=> c(gen_c4_1(x))
gen_g5_1(0) <=> hole_g2_1
gen_g5_1(+(x, 1)) <=> g(gen_g5_1(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_g5_1(+(1, n7_1))) -> *6_1, rt in Omega(n7_1)

Induction Base:
F(gen_g5_1(+(1, 0)))

Induction Step:
F(gen_g5_1(+(1, +(n7_1, 1)))) ->_R^Omega(1)
c(F(gen_g5_1(+(1, n7_1)))) ->_IH
c(*6_1)

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(g(z0)) -> c(F(z0))
f(g(z0)) -> f(z0)

Types:
F :: g -> c
g :: g -> g
c :: c -> c
f :: g -> f
hole_c1_1 :: c
hole_g2_1 :: g
hole_f3_1 :: f
gen_c4_1 :: Nat -> c
gen_g5_1 :: Nat -> g


Generator Equations:
gen_c4_1(0) <=> hole_c1_1
gen_c4_1(+(x, 1)) <=> c(gen_c4_1(x))
gen_g5_1(0) <=> hole_g2_1
gen_g5_1(+(x, 1)) <=> g(gen_g5_1(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(g(z0)) -> c(F(z0))
f(g(z0)) -> f(z0)

Types:
F :: g -> c
g :: g -> g
c :: c -> c
f :: g -> f
hole_c1_1 :: c
hole_g2_1 :: g
hole_f3_1 :: f
gen_c4_1 :: Nat -> c
gen_g5_1 :: Nat -> g


Lemmas:
F(gen_g5_1(+(1, n7_1))) -> *6_1, rt in Omega(n7_1)


Generator Equations:
gen_c4_1(0) <=> hole_c1_1
gen_c4_1(+(x, 1)) <=> c(gen_c4_1(x))
gen_g5_1(0) <=> hole_g2_1
gen_g5_1(+(x, 1)) <=> g(gen_g5_1(x))


The following defined symbols remain to be analysed:
f
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_g5_1(+(1, n141_1))) -> *6_1, rt in Omega(0)

Induction Base:
f(gen_g5_1(+(1, 0)))

Induction Step:
f(gen_g5_1(+(1, +(n141_1, 1)))) ->_R^Omega(0)
f(gen_g5_1(+(1, n141_1))) ->_IH
*6_1

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)