WORST_CASE(Omega(n^1),?)
proof of input_bfyiNSxUYa.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, INF).

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 20 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 950 ms]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
(14) BEST
    (15) proven lower bound
        (16) LowerBoundPropagationProof [FINISHED, 0 ms]
        (17) BOUNDS(n^1, INF)
    (18) 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, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1
Tuples:
   F(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4
S tuples:
   F(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4
K tuples:none
Defined Rule Symbols:   f_2, h_3

Defined Pair Symbols:   F_2, H_3

Compound Symbols:   c, c1_1, c2_1, c3_2, c4


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(3) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(4)
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(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4

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

   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1

Rewrite Strategy: INNERMOST
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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
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(a, g(z0)) -> c
   F(g(z0), a) -> c1(F(z0, g(a)))
   F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
   H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
   H(a, z0, z1) -> c4

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

   f(a, g(z0)) -> g(g(z0))
   f(g(z0), a) -> f(z0, g(a))
   f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
   h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
   h(a, z0, z1) -> z1

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
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(8)
Obligation:
Innermost TRS:
Rules:
F(a, g(z0)) -> c
F(g(z0), a) -> c1(F(z0, g(a)))
F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
H(a, z0, z1) -> c4
f(a, g(z0)) -> g(g(z0))
f(g(z0), a) -> f(z0, g(a))
f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
h(a, z0, z1) -> z1

Types:
F :: a:g -> a:g -> c:c1:c2
a :: a:g
g :: a:g -> a:g
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c3:c4 -> c:c1:c2
H :: a:g -> a:g -> a:g -> c3:c4
c3 :: c:c1:c2 -> c3:c4 -> c3:c4
h :: a:g -> a:g -> a:g -> a:g
c4 :: c3:c4
f :: a:g -> a:g -> a:g
hole_c:c1:c21_5 :: c:c1:c2
hole_a:g2_5 :: a:g
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_a:g5_5 :: Nat -> a:g
gen_c3:c46_5 :: Nat -> c3:c4

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, H, h, f

They will be analysed ascendingly in the following order:
F = H
h < H
h = f

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(10)
Obligation:
Innermost TRS:
Rules:
F(a, g(z0)) -> c
F(g(z0), a) -> c1(F(z0, g(a)))
F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
H(a, z0, z1) -> c4
f(a, g(z0)) -> g(g(z0))
f(g(z0), a) -> f(z0, g(a))
f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
h(a, z0, z1) -> z1

Types:
F :: a:g -> a:g -> c:c1:c2
a :: a:g
g :: a:g -> a:g
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c3:c4 -> c:c1:c2
H :: a:g -> a:g -> a:g -> c3:c4
c3 :: c:c1:c2 -> c3:c4 -> c3:c4
h :: a:g -> a:g -> a:g -> a:g
c4 :: c3:c4
f :: a:g -> a:g -> a:g
hole_c:c1:c21_5 :: c:c1:c2
hole_a:g2_5 :: a:g
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_a:g5_5 :: Nat -> a:g
gen_c3:c46_5 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_a:g5_5(0) <=> a
gen_a:g5_5(+(x, 1)) <=> g(gen_a:g5_5(x))
gen_c3:c46_5(0) <=> c4
gen_c3:c46_5(+(x, 1)) <=> c3(c, gen_c3:c46_5(x))


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

They will be analysed ascendingly in the following order:
F = H
h < H
h = f

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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_a:g5_5(n3304_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_a:g5_5(+(1, n3304_5)), rt in Omega(0)

Induction Base:
h(gen_a:g5_5(0), gen_a:g5_5(0), gen_a:g5_5(1)) ->_R^Omega(0)
gen_a:g5_5(1)

Induction Step:
h(gen_a:g5_5(+(n3304_5, 1)), gen_a:g5_5(0), gen_a:g5_5(1)) ->_R^Omega(0)
f(gen_a:g5_5(0), h(gen_a:g5_5(n3304_5), gen_a:g5_5(0), gen_a:g5_5(1))) ->_IH
f(gen_a:g5_5(0), gen_a:g5_5(+(1, c3305_5))) ->_R^Omega(0)
g(g(gen_a:g5_5(n3304_5)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(12)
Obligation:
Innermost TRS:
Rules:
F(a, g(z0)) -> c
F(g(z0), a) -> c1(F(z0, g(a)))
F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
H(a, z0, z1) -> c4
f(a, g(z0)) -> g(g(z0))
f(g(z0), a) -> f(z0, g(a))
f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
h(a, z0, z1) -> z1

Types:
F :: a:g -> a:g -> c:c1:c2
a :: a:g
g :: a:g -> a:g
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c3:c4 -> c:c1:c2
H :: a:g -> a:g -> a:g -> c3:c4
c3 :: c:c1:c2 -> c3:c4 -> c3:c4
h :: a:g -> a:g -> a:g -> a:g
c4 :: c3:c4
f :: a:g -> a:g -> a:g
hole_c:c1:c21_5 :: c:c1:c2
hole_a:g2_5 :: a:g
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_a:g5_5 :: Nat -> a:g
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
h(gen_a:g5_5(n3304_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_a:g5_5(+(1, n3304_5)), rt in Omega(0)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_a:g5_5(0) <=> a
gen_a:g5_5(+(x, 1)) <=> g(gen_a:g5_5(x))
gen_c3:c46_5(0) <=> c4
gen_c3:c46_5(+(x, 1)) <=> c3(c, gen_c3:c46_5(x))


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

They will be analysed ascendingly in the following order:
F = H
h < H
h = f

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
H(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_c3:c46_5(n5218_5), rt in Omega(1 + n5218_5)

Induction Base:
H(gen_a:g5_5(0), gen_a:g5_5(0), gen_a:g5_5(1)) ->_R^Omega(1)
c4

Induction Step:
H(gen_a:g5_5(+(n5218_5, 1)), gen_a:g5_5(0), gen_a:g5_5(1)) ->_R^Omega(1)
c3(F(gen_a:g5_5(0), h(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1))), H(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1))) ->_L^Omega(0)
c3(F(gen_a:g5_5(0), gen_a:g5_5(+(1, n5218_5))), H(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1))) ->_R^Omega(1)
c3(c, H(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1))) ->_IH
c3(c, gen_c3:c46_5(c5219_5))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
F(a, g(z0)) -> c
F(g(z0), a) -> c1(F(z0, g(a)))
F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
H(a, z0, z1) -> c4
f(a, g(z0)) -> g(g(z0))
f(g(z0), a) -> f(z0, g(a))
f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
h(a, z0, z1) -> z1

Types:
F :: a:g -> a:g -> c:c1:c2
a :: a:g
g :: a:g -> a:g
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c3:c4 -> c:c1:c2
H :: a:g -> a:g -> a:g -> c3:c4
c3 :: c:c1:c2 -> c3:c4 -> c3:c4
h :: a:g -> a:g -> a:g -> a:g
c4 :: c3:c4
f :: a:g -> a:g -> a:g
hole_c:c1:c21_5 :: c:c1:c2
hole_a:g2_5 :: a:g
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_a:g5_5 :: Nat -> a:g
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
h(gen_a:g5_5(n3304_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_a:g5_5(+(1, n3304_5)), rt in Omega(0)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_a:g5_5(0) <=> a
gen_a:g5_5(+(x, 1)) <=> g(gen_a:g5_5(x))
gen_c3:c46_5(0) <=> c4
gen_c3:c46_5(+(x, 1)) <=> c3(c, gen_c3:c46_5(x))


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

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

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
Innermost TRS:
Rules:
F(a, g(z0)) -> c
F(g(z0), a) -> c1(F(z0, g(a)))
F(g(z0), g(z1)) -> c2(H(g(z1), z0, g(z1)))
H(g(z0), z1, z2) -> c3(F(z1, h(z0, z1, z2)), H(z0, z1, z2))
H(a, z0, z1) -> c4
f(a, g(z0)) -> g(g(z0))
f(g(z0), a) -> f(z0, g(a))
f(g(z0), g(z1)) -> h(g(z1), z0, g(z1))
h(g(z0), z1, z2) -> f(z1, h(z0, z1, z2))
h(a, z0, z1) -> z1

Types:
F :: a:g -> a:g -> c:c1:c2
a :: a:g
g :: a:g -> a:g
c :: c:c1:c2
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c3:c4 -> c:c1:c2
H :: a:g -> a:g -> a:g -> c3:c4
c3 :: c:c1:c2 -> c3:c4 -> c3:c4
h :: a:g -> a:g -> a:g -> a:g
c4 :: c3:c4
f :: a:g -> a:g -> a:g
hole_c:c1:c21_5 :: c:c1:c2
hole_a:g2_5 :: a:g
hole_c3:c43_5 :: c3:c4
gen_c:c1:c24_5 :: Nat -> c:c1:c2
gen_a:g5_5 :: Nat -> a:g
gen_c3:c46_5 :: Nat -> c3:c4


Lemmas:
h(gen_a:g5_5(n3304_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_a:g5_5(+(1, n3304_5)), rt in Omega(0)
H(gen_a:g5_5(n5218_5), gen_a:g5_5(0), gen_a:g5_5(1)) -> gen_c3:c46_5(n5218_5), rt in Omega(1 + n5218_5)


Generator Equations:
gen_c:c1:c24_5(0) <=> c
gen_c:c1:c24_5(+(x, 1)) <=> c1(gen_c:c1:c24_5(x))
gen_a:g5_5(0) <=> a
gen_a:g5_5(+(x, 1)) <=> g(gen_a:g5_5(x))
gen_c3:c46_5(0) <=> c4
gen_c3:c46_5(+(x, 1)) <=> c3(c, gen_c3:c46_5(x))


The following defined symbols remain to be analysed:
F

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