WORST_CASE(Omega(n^1),O(n^1))
proof of input_zTUAgaP67Q.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 43 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), 281 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:

   g(S(x), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 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), y) -> g(x, S(y))
   f(y, S(x)) -> f(S(y), x)
   g(0, x2) -> x2
   f(x1, 0) -> g(x1, 0)

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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
S0(0) -> 0
00() -> 0
g0(0, 0) -> 1
f0(0, 0) -> 2
S1(0) -> 3
g1(0, 3) -> 1
S1(0) -> 4
f1(4, 0) -> 2
01() -> 5
g1(0, 5) -> 2
S1(3) -> 3
S1(5) -> 3
g1(0, 3) -> 2
S1(4) -> 4
g1(4, 5) -> 2
S2(5) -> 6
g2(0, 6) -> 2
g2(4, 6) -> 2
S1(6) -> 3
S2(6) -> 6
0 -> 1
3 -> 1
3 -> 2
5 -> 2
6 -> 2

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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:
   g(S(z0), z1) -> g(z0, S(z1))
   g(0, z0) -> z0
   f(z0, S(z1)) -> f(S(z0), z1)
   f(z0, 0) -> g(z0, 0)
Tuples:
   G(S(z0), z1) -> c(G(z0, S(z1)))
   G(0, z0) -> c1
   F(z0, S(z1)) -> c2(F(S(z0), z1))
   F(z0, 0) -> c3(G(z0, 0))
S tuples:
   G(S(z0), z1) -> c(G(z0, S(z1)))
   G(0, z0) -> c1
   F(z0, S(z1)) -> c2(F(S(z0), z1))
   F(z0, 0) -> c3(G(z0, 0))
K tuples:none
Defined Rule Symbols:   g_2, f_2

Defined Pair Symbols:   G_2, F_2

Compound Symbols:   c_1, c1, c2_1, 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.
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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:

   G(S(z0), z1) -> c(G(z0, S(z1)))
   G(0, z0) -> c1
   F(z0, S(z1)) -> c2(F(S(z0), z1))
   F(z0, 0) -> c3(G(z0, 0))

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

   g(S(z0), z1) -> g(z0, S(z1))
   g(0, z0) -> z0
   f(z0, S(z1)) -> f(S(z0), z1)
   f(z0, 0) -> g(z0, 0)

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:

   G(S(z0), z1) -> c(G(z0, S(z1)))
   G(0', z0) -> c1
   F(z0, S(z1)) -> c2(F(S(z0), z1))
   F(z0, 0') -> c3(G(z0, 0'))

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

   g(S(z0), z1) -> g(z0, S(z1))
   g(0', z0) -> z0
   f(z0, S(z1)) -> f(S(z0), z1)
   f(z0, 0') -> g(z0, 0')

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(12)
Obligation:
Innermost TRS:
Rules:
G(S(z0), z1) -> c(G(z0, S(z1)))
G(0', z0) -> c1
F(z0, S(z1)) -> c2(F(S(z0), z1))
F(z0, 0') -> c3(G(z0, 0'))
g(S(z0), z1) -> g(z0, S(z1))
g(0', z0) -> z0
f(z0, S(z1)) -> f(S(z0), z1)
f(z0, 0') -> g(z0, 0')

Types:
G :: S:0' -> S:0' -> c:c1
S :: S:0' -> S:0'
c :: c:c1 -> c:c1
0' :: S:0'
c1 :: c:c1
F :: S:0' -> S:0' -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c:c1 -> c2:c3
g :: S:0' -> S:0' -> S:0'
f :: S:0' -> S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_S:0'5_4 :: Nat -> S:0'
gen_c2:c36_4 :: Nat -> c2:c3

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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), z1) -> c(G(z0, S(z1)))
G(0', z0) -> c1
F(z0, S(z1)) -> c2(F(S(z0), z1))
F(z0, 0') -> c3(G(z0, 0'))
g(S(z0), z1) -> g(z0, S(z1))
g(0', z0) -> z0
f(z0, S(z1)) -> f(S(z0), z1)
f(z0, 0') -> g(z0, 0')

Types:
G :: S:0' -> S:0' -> c:c1
S :: S:0' -> S:0'
c :: c:c1 -> c:c1
0' :: S:0'
c1 :: c:c1
F :: S:0' -> S:0' -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c:c1 -> c2:c3
g :: S:0' -> S:0' -> S:0'
f :: S:0' -> S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_S:0'5_4 :: Nat -> S:0'
gen_c2:c36_4 :: Nat -> c2:c3


Generator Equations:
gen_c:c14_4(0) <=> c1
gen_c:c14_4(+(x, 1)) <=> c(gen_c:c14_4(x))
gen_S:0'5_4(0) <=> 0'
gen_S:0'5_4(+(x, 1)) <=> S(gen_S:0'5_4(x))
gen_c2:c36_4(0) <=> c3(c1)
gen_c2:c36_4(+(x, 1)) <=> c2(gen_c2:c36_4(x))


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

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:
G(gen_S:0'5_4(n8_4), gen_S:0'5_4(b)) -> gen_c:c14_4(n8_4), rt in Omega(1 + n8_4)

Induction Base:
G(gen_S:0'5_4(0), gen_S:0'5_4(b)) ->_R^Omega(1)
c1

Induction Step:
G(gen_S:0'5_4(+(n8_4, 1)), gen_S:0'5_4(b)) ->_R^Omega(1)
c(G(gen_S:0'5_4(n8_4), S(gen_S:0'5_4(b)))) ->_IH
c(gen_c:c14_4(c9_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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(16)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
G(S(z0), z1) -> c(G(z0, S(z1)))
G(0', z0) -> c1
F(z0, S(z1)) -> c2(F(S(z0), z1))
F(z0, 0') -> c3(G(z0, 0'))
g(S(z0), z1) -> g(z0, S(z1))
g(0', z0) -> z0
f(z0, S(z1)) -> f(S(z0), z1)
f(z0, 0') -> g(z0, 0')

Types:
G :: S:0' -> S:0' -> c:c1
S :: S:0' -> S:0'
c :: c:c1 -> c:c1
0' :: S:0'
c1 :: c:c1
F :: S:0' -> S:0' -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c:c1 -> c2:c3
g :: S:0' -> S:0' -> S:0'
f :: S:0' -> S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_S:0'5_4 :: Nat -> S:0'
gen_c2:c36_4 :: Nat -> c2:c3


Generator Equations:
gen_c:c14_4(0) <=> c1
gen_c:c14_4(+(x, 1)) <=> c(gen_c:c14_4(x))
gen_S:0'5_4(0) <=> 0'
gen_S:0'5_4(+(x, 1)) <=> S(gen_S:0'5_4(x))
gen_c2:c36_4(0) <=> c3(c1)
gen_c2:c36_4(+(x, 1)) <=> c2(gen_c2:c36_4(x))


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

They will be analysed ascendingly in the following order:
G < F
g < 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:
G(S(z0), z1) -> c(G(z0, S(z1)))
G(0', z0) -> c1
F(z0, S(z1)) -> c2(F(S(z0), z1))
F(z0, 0') -> c3(G(z0, 0'))
g(S(z0), z1) -> g(z0, S(z1))
g(0', z0) -> z0
f(z0, S(z1)) -> f(S(z0), z1)
f(z0, 0') -> g(z0, 0')

Types:
G :: S:0' -> S:0' -> c:c1
S :: S:0' -> S:0'
c :: c:c1 -> c:c1
0' :: S:0'
c1 :: c:c1
F :: S:0' -> S:0' -> c2:c3
c2 :: c2:c3 -> c2:c3
c3 :: c:c1 -> c2:c3
g :: S:0' -> S:0' -> S:0'
f :: S:0' -> S:0' -> S:0'
hole_c:c11_4 :: c:c1
hole_S:0'2_4 :: S:0'
hole_c2:c33_4 :: c2:c3
gen_c:c14_4 :: Nat -> c:c1
gen_S:0'5_4 :: Nat -> S:0'
gen_c2:c36_4 :: Nat -> c2:c3


Lemmas:
G(gen_S:0'5_4(n8_4), gen_S:0'5_4(b)) -> gen_c:c14_4(n8_4), rt in Omega(1 + n8_4)


Generator Equations:
gen_c:c14_4(0) <=> c1
gen_c:c14_4(+(x, 1)) <=> c(gen_c:c14_4(x))
gen_S:0'5_4(0) <=> 0'
gen_S:0'5_4(+(x, 1)) <=> S(gen_S:0'5_4(x))
gen_c2:c36_4(0) <=> c3(c1)
gen_c2:c36_4(+(x, 1)) <=> c2(gen_c2:c36_4(x))


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

They will be analysed ascendingly in the following order:
g < f