WORST_CASE(Omega(n^1),O(n^1))
proof of input_u1Dccbbup8.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 6 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 283 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), 41 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(s(x)) -> s(s(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> 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(s(x)) -> s(s(f(p(s(x)))))
   f(0) -> 0
   p(s(x)) -> 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 2. 
The certificate found is represented by the following graph.

"[4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
{(4,5,[f_1|0, p_1|0, 0|1, s_1|1]), (4,6,[s_1|1]), (5,5,[s_1|0, 0|0]), (6,7,[s_1|1]), (7,8,[f_1|1]), (7,10,[s_1|2]), (7,5,[0|2]), (8,9,[p_1|1]), (8,5,[s_1|1, 0|1]), (9,5,[s_1|1]), (10,11,[s_1|2]), (11,12,[f_1|2]), (11,10,[s_1|2]), (11,5,[0|2]), (12,13,[p_1|2]), (12,5,[s_1|1, 0|1]), (13,5,[s_1|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:
   f(s(z0)) -> s(s(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> z0
Tuples:
   F(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
S tuples:
   F(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_2, c1, c2


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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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
   F(0) -> c1
   P(s(z0)) -> c2

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

   f(s(z0)) -> s(s(f(p(s(z0)))))
   f(0) -> 0
   p(s(z0)) -> 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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
   F(0') -> c1
   P(s(z0)) -> c2

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

   f(s(z0)) -> s(s(f(p(s(z0)))))
   f(0') -> 0'
   p(s(z0)) -> 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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(s(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c2 -> c:c1
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
f :: s:0' -> s:0'
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'

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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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(s(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c2 -> c:c1
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
f :: s:0' -> s:0'
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(gen_c:c14_3(x), c2)
gen_s:0'5_3(0) <=> 0'
gen_s:0'5_3(+(x, 1)) <=> s(gen_s:0'5_3(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_s:0'5_3(n7_3)) -> gen_c:c14_3(n7_3), rt in Omega(1 + n7_3)

Induction Base:
F(gen_s:0'5_3(0)) ->_R^Omega(1)
c1

Induction Step:
F(gen_s:0'5_3(+(n7_3, 1))) ->_R^Omega(1)
c(F(p(s(gen_s:0'5_3(n7_3)))), P(s(gen_s:0'5_3(n7_3)))) ->_R^Omega(0)
c(F(gen_s:0'5_3(n7_3)), P(s(gen_s:0'5_3(n7_3)))) ->_IH
c(gen_c:c14_3(c8_3), P(s(gen_s:0'5_3(n7_3)))) ->_R^Omega(1)
c(gen_c:c14_3(n7_3), c2)

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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(s(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c2 -> c:c1
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
f :: s:0' -> s:0'
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(gen_c:c14_3(x), c2)
gen_s:0'5_3(0) <=> 0'
gen_s:0'5_3(+(x, 1)) <=> s(gen_s:0'5_3(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(s(z0)) -> c(F(p(s(z0))), P(s(z0)))
F(0') -> c1
P(s(z0)) -> c2
f(s(z0)) -> s(s(f(p(s(z0)))))
f(0') -> 0'
p(s(z0)) -> z0

Types:
F :: s:0' -> c:c1
s :: s:0' -> s:0'
c :: c:c1 -> c2 -> c:c1
p :: s:0' -> s:0'
P :: s:0' -> c2
0' :: s:0'
c1 :: c:c1
c2 :: c2
f :: s:0' -> s:0'
hole_c:c11_3 :: c:c1
hole_s:0'2_3 :: s:0'
hole_c23_3 :: c2
gen_c:c14_3 :: Nat -> c:c1
gen_s:0'5_3 :: Nat -> s:0'


Lemmas:
F(gen_s:0'5_3(n7_3)) -> gen_c:c14_3(n7_3), rt in Omega(1 + n7_3)


Generator Equations:
gen_c:c14_3(0) <=> c1
gen_c:c14_3(+(x, 1)) <=> c(gen_c:c14_3(x), c2)
gen_s:0'5_3(0) <=> 0'
gen_s:0'5_3(+(x, 1)) <=> s(gen_s:0'5_3(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_s:0'5_3(n183_3)) -> gen_s:0'5_3(*(2, n183_3)), rt in Omega(0)

Induction Base:
f(gen_s:0'5_3(0)) ->_R^Omega(0)
0'

Induction Step:
f(gen_s:0'5_3(+(n183_3, 1))) ->_R^Omega(0)
s(s(f(p(s(gen_s:0'5_3(n183_3)))))) ->_R^Omega(0)
s(s(f(gen_s:0'5_3(n183_3)))) ->_IH
s(s(gen_s:0'5_3(*(2, c184_3))))

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)