WORST_CASE(Omega(n^1),O(n^1))
proof of input_CZJ1Se3uGZ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 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), 5 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 241 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), 76 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(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

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(x, a) -> x
   f(x, g(y)) -> f(g(x), y)

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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
a0() -> 0
g0(0) -> 0
f0(0, 0) -> 1
g1(0) -> 2
f1(2, 0) -> 1
g1(2) -> 2
0 -> 1
2 -> 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(z0, a) -> z0
   f(z0, g(z1)) -> f(g(z0), z1)
Tuples:
   F(z0, a) -> c
   F(z0, g(z1)) -> c1(F(g(z0), z1))
S tuples:
   F(z0, a) -> c
   F(z0, g(z1)) -> c1(F(g(z0), z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c, c1_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(z0, a) -> c
   F(z0, g(z1)) -> c1(F(g(z0), z1))

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

   f(z0, a) -> z0
   f(z0, g(z1)) -> f(g(z0), z1)

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(z0, a) -> c
   F(z0, g(z1)) -> c1(F(g(z0), z1))

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

   f(z0, a) -> z0
   f(z0, g(z1)) -> f(g(z0), z1)

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(z0, a) -> c
F(z0, g(z1)) -> c1(F(g(z0), z1))
f(z0, a) -> z0
f(z0, g(z1)) -> f(g(z0), z1)

Types:
F :: a:g -> a:g -> c:c1
a :: a:g
c :: c:c1
g :: a:g -> a:g
c1 :: c:c1 -> c:c1
f :: a:g -> a:g -> a:g
hole_c:c11_2 :: c:c1
hole_a:g2_2 :: a:g
gen_c:c13_2 :: Nat -> c:c1
gen_a:g4_2 :: Nat -> a: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(z0, a) -> c
F(z0, g(z1)) -> c1(F(g(z0), z1))
f(z0, a) -> z0
f(z0, g(z1)) -> f(g(z0), z1)

Types:
F :: a:g -> a:g -> c:c1
a :: a:g
c :: c:c1
g :: a:g -> a:g
c1 :: c:c1 -> c:c1
f :: a:g -> a:g -> a:g
hole_c:c11_2 :: c:c1
hole_a:g2_2 :: a:g
gen_c:c13_2 :: Nat -> c:c1
gen_a:g4_2 :: Nat -> a:g


Generator Equations:
gen_c:c13_2(0) <=> c
gen_c:c13_2(+(x, 1)) <=> c1(gen_c:c13_2(x))
gen_a:g4_2(0) <=> a
gen_a:g4_2(+(x, 1)) <=> g(gen_a: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_a:g4_2(a), gen_a:g4_2(n6_2)) -> gen_c:c13_2(n6_2), rt in Omega(1 + n6_2)

Induction Base:
F(gen_a:g4_2(a), gen_a:g4_2(0)) ->_R^Omega(1)
c

Induction Step:
F(gen_a:g4_2(a), gen_a:g4_2(+(n6_2, 1))) ->_R^Omega(1)
c1(F(g(gen_a:g4_2(a)), gen_a:g4_2(n6_2))) ->_IH
c1(gen_c:c13_2(c7_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(z0, a) -> c
F(z0, g(z1)) -> c1(F(g(z0), z1))
f(z0, a) -> z0
f(z0, g(z1)) -> f(g(z0), z1)

Types:
F :: a:g -> a:g -> c:c1
a :: a:g
c :: c:c1
g :: a:g -> a:g
c1 :: c:c1 -> c:c1
f :: a:g -> a:g -> a:g
hole_c:c11_2 :: c:c1
hole_a:g2_2 :: a:g
gen_c:c13_2 :: Nat -> c:c1
gen_a:g4_2 :: Nat -> a:g


Generator Equations:
gen_c:c13_2(0) <=> c
gen_c:c13_2(+(x, 1)) <=> c1(gen_c:c13_2(x))
gen_a:g4_2(0) <=> a
gen_a:g4_2(+(x, 1)) <=> g(gen_a: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(z0, a) -> c
F(z0, g(z1)) -> c1(F(g(z0), z1))
f(z0, a) -> z0
f(z0, g(z1)) -> f(g(z0), z1)

Types:
F :: a:g -> a:g -> c:c1
a :: a:g
c :: c:c1
g :: a:g -> a:g
c1 :: c:c1 -> c:c1
f :: a:g -> a:g -> a:g
hole_c:c11_2 :: c:c1
hole_a:g2_2 :: a:g
gen_c:c13_2 :: Nat -> c:c1
gen_a:g4_2 :: Nat -> a:g


Lemmas:
F(gen_a:g4_2(a), gen_a:g4_2(n6_2)) -> gen_c:c13_2(n6_2), rt in Omega(1 + n6_2)


Generator Equations:
gen_c:c13_2(0) <=> c
gen_c:c13_2(+(x, 1)) <=> c1(gen_c:c13_2(x))
gen_a:g4_2(0) <=> a
gen_a:g4_2(+(x, 1)) <=> g(gen_a: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_a:g4_2(a), gen_a:g4_2(n281_2)) -> gen_a:g4_2(+(n281_2, a)), rt in Omega(0)

Induction Base:
f(gen_a:g4_2(a), gen_a:g4_2(0)) ->_R^Omega(0)
gen_a:g4_2(a)

Induction Step:
f(gen_a:g4_2(a), gen_a:g4_2(+(n281_2, 1))) ->_R^Omega(0)
f(g(gen_a:g4_2(a)), gen_a:g4_2(n281_2)) ->_IH
gen_a:g4_2(+(+(a, 1), c282_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)