WORST_CASE(Omega(n^1),O(n^1))
proof of input_zRdJwv6qXO.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 8 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), 6 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 500 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), 91 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)) -> g(g(f(x)))
   f(g(x)) -> g(g(g(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)) -> g(g(f(x)))
   f(g(x)) -> g(g(g(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.

"[3, 4, 5, 6, 7, 8]
{(3,4,[f_1|0]), (3,5,[g_1|1]), (3,7,[g_1|1]), (4,4,[g_1|0]), (5,6,[g_1|1]), (6,4,[f_1|1]), (6,5,[g_1|1]), (6,7,[g_1|1]), (7,8,[g_1|1]), (8,4,[g_1|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(g(z0)) -> g(g(f(z0)))
   f(g(z0)) -> g(g(g(z0)))
Tuples:
   F(g(z0)) -> c(F(z0))
   F(g(z0)) -> c1
S tuples:
   F(g(z0)) -> c(F(z0))
   F(g(z0)) -> c1
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1


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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))
   F(g(z0)) -> c1

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

   f(g(z0)) -> g(g(f(z0)))
   f(g(z0)) -> g(g(g(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))
   F(g(z0)) -> c1

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

   f(g(z0)) -> g(g(f(z0)))
   f(g(z0)) -> g(g(g(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)) -> c1
f(g(z0)) -> g(g(f(z0)))
f(g(z0)) -> g(g(g(z0)))

Types:
F :: g -> c:c1
g :: g -> g
c :: c:c1 -> c:c1
c1 :: c:c1
f :: g -> g
hole_c:c11_2 :: c:c1
hole_g2_2 :: g
gen_c:c13_2 :: Nat -> c:c1
gen_g4_2 :: 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)) -> c1
f(g(z0)) -> g(g(f(z0)))
f(g(z0)) -> g(g(g(z0)))

Types:
F :: g -> c:c1
g :: g -> g
c :: c:c1 -> c:c1
c1 :: c:c1
f :: g -> g
hole_c:c11_2 :: c:c1
hole_g2_2 :: g
gen_c:c13_2 :: Nat -> c:c1
gen_g4_2 :: Nat -> g


Generator Equations:
gen_c:c13_2(0) <=> c1
gen_c:c13_2(+(x, 1)) <=> c(gen_c:c13_2(x))
gen_g4_2(0) <=> hole_g2_2
gen_g4_2(+(x, 1)) <=> g(gen_g4_2(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_g4_2(+(1, n6_2))) -> *5_2, rt in Omega(n6_2)

Induction Base:
F(gen_g4_2(+(1, 0)))

Induction Step:
F(gen_g4_2(+(1, +(n6_2, 1)))) ->_R^Omega(1)
c(F(gen_g4_2(+(1, n6_2)))) ->_IH
c(*5_2)

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)) -> c1
f(g(z0)) -> g(g(f(z0)))
f(g(z0)) -> g(g(g(z0)))

Types:
F :: g -> c:c1
g :: g -> g
c :: c:c1 -> c:c1
c1 :: c:c1
f :: g -> g
hole_c:c11_2 :: c:c1
hole_g2_2 :: g
gen_c:c13_2 :: Nat -> c:c1
gen_g4_2 :: Nat -> g


Generator Equations:
gen_c:c13_2(0) <=> c1
gen_c:c13_2(+(x, 1)) <=> c(gen_c:c13_2(x))
gen_g4_2(0) <=> hole_g2_2
gen_g4_2(+(x, 1)) <=> g(gen_g4_2(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)) -> c1
f(g(z0)) -> g(g(f(z0)))
f(g(z0)) -> g(g(g(z0)))

Types:
F :: g -> c:c1
g :: g -> g
c :: c:c1 -> c:c1
c1 :: c:c1
f :: g -> g
hole_c:c11_2 :: c:c1
hole_g2_2 :: g
gen_c:c13_2 :: Nat -> c:c1
gen_g4_2 :: Nat -> g


Lemmas:
F(gen_g4_2(+(1, n6_2))) -> *5_2, rt in Omega(n6_2)


Generator Equations:
gen_c:c13_2(0) <=> c1
gen_c:c13_2(+(x, 1)) <=> c(gen_c:c13_2(x))
gen_g4_2(0) <=> hole_g2_2
gen_g4_2(+(x, 1)) <=> g(gen_g4_2(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_g4_2(+(1, n205_2))) -> *5_2, rt in Omega(0)

Induction Base:
f(gen_g4_2(+(1, 0)))

Induction Step:
f(gen_g4_2(+(1, +(n205_2, 1)))) ->_R^Omega(0)
g(g(f(gen_g4_2(+(1, n205_2))))) ->_IH
g(g(*5_2))

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)